problem solving involving conversion of units

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Chapter 1: Algebra Review

1.6 Unit Conversion Word Problems

One application of rational expressions deals with converting units. Units of measure can be converted by multiplying several fractions together in a process known as dimensional analysis.

The trick is to decide what fractions to multiply. If an expression is multiplied by 1, its value does not change. The number 1 can be written as a fraction in many different ways, so long as the numerator and denominator are identical in value. Note that the numerator and denominator need not be identical in appearance, but rather only identical in value. Below are several fractions, each equal to 1, where the numerator and the denominator are identical in value. This is why, when doing dimensional analysis, it is very important to use units in the setup of the problem, so as to ensure that the conversion factor is set up correctly.

Example 1.6.1

 If 1 pound = 16 ounces, how many pounds are in 435 ounces?

[latex]\begin{array}{rrll} 435\text{ oz}&=&435\cancel{\text{oz}}\times \dfrac{1\text{ lb}}{16\cancel{ \text{oz}}} \hspace{0.2in}& \text{This operation cancels the oz and leaves the lbs} \\ \\ &=&\dfrac{435\text{ lb}}{16} \hspace{0.2in}& \text{Which reduces to } \\ \\ &=&27\dfrac{3}{16}\text{ lb} \hspace{0.2in}& \text{Solution} \end{array}[/latex]

The same process can be used to convert problems with several units in them. Consider the following example.

Example 1.6.2

A student averaged 45 miles per hour on a trip. What was the student’s speed in feet per second?

[latex]\begin{array}{rrll} 45 \text{ mi/h}&=&\dfrac{45\cancel{\text{mi}}}{\cancel{\text{hr}}}\times \dfrac{5280 \text{ ft}}{1\cancel{ \text{mi}}}\times \dfrac{1\cancel{\text{hr}}}{3600\text{ s}}\hspace{0.2in}&\text{This will cancel the miles and hours} \\ \\ &=&45\times \dfrac{5280}{1}\times \dfrac{1}{3600} \text{ ft/s}\hspace{0.2in}&\text{This reduces to} \\ \\ &=&66\text{ ft/s}\hspace{0.2in}&\text{Solution} \end{array}[/latex]

Example 1.6.3

Convert 8 ft 3 to yd 3 .

[latex]\begin{array}{rrll} 8\text{ ft}^3&=&8\text{ ft}^3 \times \dfrac{(1\text{ yd})^3}{(3\text{ ft})^3}&\text{Cube the parentheses} \\ \\ &=&8\text{ }\cancel{\text{ft}^3}\times \dfrac{1\text{ yd}^3}{27\text{ }\cancel{\text{ft}^3}}&\text{This will cancel the ft}^3\text{ and replace them with yd}^3 \\ \\ &=&8\times \dfrac{1\text{ yd}^3}{27}&\text{Which reduces to} \\ \\ &=&\dfrac{8}{27}\text{ yd}^3\text{ or }0.296\text{ yd}^3&\text{Solution} \end{array}[/latex]

Example 1.6.4

A room is 10 ft by 12 ft. How many square yards are in the room? The area of the room is 120 ft 2 (area = length × width).

Converting the area yields:

[latex]\begin{array}{rrll} 120\text{ ft}^2&=&120\text{ }\cancel{\text{ft}^2}\times \dfrac{(1\text{ yd})^2}{(3\text{ }\cancel{\text{ft}})^2}&\text{Cancel ft}^2\text{ and replace with yd}^2 \\ \\ &=&\dfrac{120\text{ yd}^2}{9}&\text{This reduces to} \\ \\ &=&13\dfrac{1}{3}\text{ yd}^2&\text{Solution} \\ \\ \end{array}[/latex]

The process of dimensional analysis can be used to convert other types of units as well. Once relationships that represent the same value have been identified, a conversion factor can be determined.

Example 1.6.5

A child is prescribed a dosage of 12 mg of a certain drug per day and is allowed to refill his prescription twice. If there are 60 tablets in a prescription, and each tablet has 4 mg, how many doses are in the 3 prescriptions (original + 2 refills)?

[latex]\begin{array}{rrll} 3\text{ prescriptions}&=&3\cancel{\text{pres.}}\times \dfrac{60\cancel{\text{tablets}}}{1\cancel{\text{pres.}}}\times \dfrac{4\cancel{\text{mg}}}{1\cancel{\text{tablet}}}\times \dfrac{1\text{ dosage}}{12\cancel{\text{mg}}}&\text{This cancels all unwanted units} \\ \\ &=&\dfrac{3\times 60\times 4\times 1}{1\times 1\times 12}\text{ or }\dfrac{720}{12}\text{ dosages}&\text{Which reduces to} \\ \\ &=&60\text{ daily dosages}&\text{Solution} \\ \\ \end{array}[/latex]

Metric and Imperial (U.S.) Conversions

[latex]\begin{array}{rrlrrl} 12\text{ in}&=&1\text{ ft}\hspace{1in}&10\text{ mm}&=&1\text{ cm} \\ 3\text{ ft}&=&1\text{ yd}&100\text{ cm}&=&1\text{ m} \\ 1760\text{ yds}&=&1\text{ mi}&1000\text{ m}&=&1\text{ km} \\ 5280\text{ ft}&=&1\text{ mi}&&& \end{array}[/latex]

Imperial to metric conversions:

[latex]\begin{array}{rrl} 1\text{ inch}&=&2.54\text{ cm} \\ 1\text{ ft}&=&0.3048\text{ m} \\ 1\text{ mile}&=&1.61\text{ km} \end{array}[/latex]

[latex]\begin{array}{rrlrrl} 144\text{ in}^2&=&1\text{ ft}^2\hspace{1in}&10,000\text{ cm}^2&=&1\text{ m}^2 \\ 43,560\text{ ft}^2&=&1\text{ acre}&10,000\text{ m}^2&=&1\text{ hectare} \\ 640\text{ acres}&=&1\text{ mi}^2&100\text{ hectares}&=&1\text{ km}^2 \end{array}[/latex]

[latex]\begin{array}{rrl} 1\text{ in}^2&=&6.45\text{ cm}^2 \\ 1\text{ ft}^2&=&0.092903\text{ m}^2 \\ 1\text{ mi}^2&=&2.59\text{ km}^2 \end{array}[/latex]

[latex]\begin{array}{rrlrrl} 57.75\text{ in}^3&=&1\text{ qt}\hspace{1in}&1\text{ cm}^3&=&1\text{ ml} \\ 4\text{ qt}&=&1\text{ gal}&1000\text{ ml}&=&1\text{ litre} \\ 42\text{ gal (petroleum)}&=&1\text{ barrel}&1000\text{ litres}&=&1\text{ m}^3 \end{array}[/latex]

[latex]\begin{array}{rrl} 16.39\text{ cm}^3&=&1\text{ in}^3 \\ 1\text{ ft}^3&=&0.0283168\text{ m}^3 \\ 3.79\text{ litres}&=&1\text{ gal} \end{array}[/latex]

[latex]\begin{array}{rrlrrl} 437.5\text{ grains}&=&1\text{ oz}\hspace{1in}&1000\text{ mg}&=&1\text{ g} \\ 16\text{ oz}&=&1\text{ lb}&1000\text{ g}&=&1\text{ kg} \\ 2000\text{ lb}&=&1\text{ short ton}&1000\text{ kg}&=&1\text{ metric ton} \end{array}[/latex]

[latex]\begin{array}{rrl} 453\text{ g}&=&1\text{ lb} \\ 2.2\text{ lb}&=&1\text{ kg} \end{array}[/latex]

Temperature

Fahrenheit to Celsius conversions:

[latex]\begin{array}{rrl} ^{\circ}\text{C} &= &\dfrac{5}{9} (^{\circ}\text{F} - 32) \\ \\ ^{\circ}\text{F}& =& \dfrac{9}{5}(^{\circ}\text{C} + 32) \end{array}[/latex]

For questions 1 to 18, use dimensional analysis to perform the indicated conversions.

• 7 miles to yards
• 234 oz to tons
• 11.2 mg to grams
• 1.35 km to centimetres
• 9,800,000 mm to miles
• 4.5 ft 2 to square yards
• 435,000 m 2 to square kilometres
• 8 km 2 to square feet
• 0.0065 km 3 to cubic metres
• 14.62 in. 2 to square centimetres
• 5500 cm 3 to cubic yards
• 3.5 mph (miles per hour) to feet per second
• 185 yd per min. to miles per hour
• 153 ft/s (feet per second) to miles per hour
• 248 mph to metres per second
• 186,000 mph to kilometres per year
• 7.50 tons/yd 2 to pounds per square inch
• 16 ft/s 2 to kilometres per hour squared

For questions 19 to 27, solve each conversion word problem.

• On a recent trip, Jan travelled 260 miles using 8 gallons of gas. What was the car’s miles per gallon for this trip? Kilometres per litre?
• A certain laser printer can print 12 pages per minute. Determine this printer’s output in pages per day.
• An average human heart beats 60 times per minute. If the average person lives to the age of 86, how many times does the average heart beat in a lifetime?
• Blood sugar levels are measured in milligrams of glucose per decilitre of blood volume. If a person’s blood sugar level measured 128 mg/dL, what is this in grams per litre?
• You are buying carpet to cover a room that measures 38 ft by 40 ft. The carpet cost $18 per square yard. How much will the carpet cost?
• A cargo container is 50 ft long, 10 ft wide, and 8 ft tall. Find its volume in cubic yards and cubic metres.
• A local zoning ordinance says that a house’s “footprint” (area of its ground floor) cannot occupy more than ¼ of the lot it is built on. Suppose you own a [latex]\frac{1}{3}[/latex]-acre lot (1 acre = 43,560 ft 2 ). What is the maximum allowed footprint for your house in square feet? In square metres?
• A car travels 23 km in 15 minutes. How fast is it going in kilometres per hour? In metres per second?
• The largest single rough diamond ever found, the Cullinan Diamond, weighed 3106 carats. One carat is equivalent to the mass of 0.20 grams. What is the mass of this diamond in milligrams? Weight in pounds?

Answer Key 1.6

Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Solving Problems Involving Conversion of Units of Measurement

Measuring is one of the human activities that we perform daily. A tailor measures the length of the dress; a butcher measures the weight of the meat; a surveyor measures the area of large land masses, and so on. 

The ability to measure objects is always connected to the history of our civilization. During ancient times, our ancestors used their fingers, hands, and feet to determine the length of an object. In this era, humans had varying ways of measuring things. 

Eventually, after the French Revolution in the late 18th century, a standardized way of measurement was developed (i.e., the metric system). Today, we have a convention of measurement units and advanced technological tools to measure objects. 

In this module, you’ll learn what measurement is, the units of measurement (for length, weight, volume, time, and temperature), and how to convert units in the metric system. 

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What is measurement.

Measurement is the provision of a numerical value to present and describe the magnitude or amount of a particular object.  

We use measurement units to provide a more accurate description of the object’s measurement. Some examples of measurement units are meters, liters, grams, inches, Fahrenheit, and so on. 

For instance, if we want to determine how long a piece of wood is, we measure its length. To do this, we use a particular SI unit of measurement (e.g., meters) and provide a number that describes the length of this wood (e.g., this wood is 3 meters long). 

Since there are a lot of measurement units being used around the world, a standardized set of measurement units have been adopted by several countries. This is called the SI Units of Measurement , more commonly known as the Metric System (e.g., meter, gram, liter). On the other hand, there’s also the Imperial System or US Standard Units , which is also commonly used in the Philippines (e.g., feet, yards, pounds, etc.).

What Are the Different Ways To Measure Objects?

There are different ways to measure objects depending on the particular trait we want to describe or show. 

This refers to the distance from one point to another. In other words, this describes how long or short an object is. Standard measurement units are meters, centimeters, inches, feet, etc.

The SI base unit for length is a meter.

This refers to the amount of space occupied by a two-dimensional figure. In other words, it tells us how much surface a plane figure covers . Commonly used measurement units are square meters, square kilometers, square yards, etc. 

Metric units of the area have an exponent of 2 to indicate that we are measuring the amount of two-dimensional space occupied (e.g., the square meter is written as m 2 ).

The SI base unit for the area is square meters (m 2 ).

3. Volume or Capacity

The volume is the space occupied or enclosed by a three-dimensional figure . Commonly used measurement units are cubic meters, cubic kilometers, cubic yards, etc. 

Metric volume units have an exponent of 3 to indicate that we are measuring the amount of three-dimensional space occupied (e.g., a cubic meter is written as m 3 ).

In Physics, mass and weight mean differently . Mass refers to the amount of matter an object has, while weight refers to the force that gravity exerts on an object. 

The SI base units for mass and weight are different. The kilogram is the SI base unit for mass, while Newton is the SI base unit for weight.

However, outside a Physics classroom, these terms are often used interchangeably. Many people perceive mass and weight as the same thing, which refers to how heavy an object is. To avoid ambiguity and confusion in our discussion, we will strictly use in this article the word “mass” to refer to the heaviness of an object 

This refers to the duration of the sequence of events. For instance, we measure time to determine how long you read this reviewer. 

The SI base unit for time is seconds.

6. Temperature

Temperature tells us how hot or cold an object is. In Physics, the temperature is the average kinetic energy of the particles of an object . The primary way to measure temperature is through a thermometer .

SI base unit for temperature is Kelvin (K). However, Celsius (°C) and Fahrenheit (°F) are the more popular units.

This review will discuss the conversion of measurement units for length, area, volume, mass, time, and temperature. However, remember that these are not just the ways to measure objects. For instance, we can also measure their luminous intensity, electric current, amount of substance, and so on. 

Metric System Units of Length

We learned earlier that meter (m) is the SI base unit for length. This means that other metric units for length are derived from the meter. For instance, a kilometer (km) means 1000 meters. 

In the metric system, we use prefixes to indicate that a particular metric unit is a multiple of the base unit. For example, the prefix “kilo” means 1000 times the base unit. So, 1 km = 1000 meters.

Six prefixes are used in the metric system, and we list them below together with their equivalent value in the base unit. 

An easier way to visualize these prefixes is by using a table:

Whether you want to memorize the conversion table above is up to you. However, it is advisable to remember the equivalent value of each prefix in terms of the base unit. These prefixes also apply to metric units for area and mass. 

Converting Metric Units of Length

The easiest way to convert metric units is by simply moving decimal places. 

For instance, let us convert 375 meters to kilometers by looking at the table of prefixes below. 

Note that the table above has three steps to the left from meter to kilometer. This means we must move three decimal places to the left in 375 meters to get its equivalent in kilometers. 

This means that 375 meters are equal to 0.375 kilometers.

Sample Problem 1: Convert 98.35 decameters to centimeters

Solution: Looking at the table of metric units of length, there are three steps to the right, from decameters to centimeters. 

This implies that we must move three decimal places to the right to convert 98.35 decameters to centimeters.

Hence, 98.35 decameters = 98,350 centimeters

Sample Problem 2: A ribbon was divided into two strips. The first strip measures 176.50 centimeters, while the second measures 89.56 centimeters. What is the length of the original ribbon in meters?

Solution: There are two steps to the left from centimeters to meters. Hence, we move two decimal places to the left to convert centimeters to meters. 

• Moving two decimal places in 176.50 centimeters results in 1.7650 meters.
• Moving two decimal places in 89.56 centimeters results in 0.8956 meters.

Adding the length of the strips converted to meters: 1.7650 m + 0.8956 m = 2.6606 m.

Hence, the length of the original ribbon is 2.6606 or 2.66 meters. 

Metric System Units of Area

The SI base unit for the area is a square meter (m 2 ). Like the metric units of length, the metric units for the area are derived from the base unit (i.e., square meters). 

The prefixes you have learned in the metric units for length also apply to metric units for the area. Again, these prefixes indicate that a particular metric unit is a multiple of the base unit. 

Six prefixes are used in the metric system, and we list them below together with their equivalent value in the base unit. Note that all metric units for the area have a superscript of 2 to indicate that we are dealing with square units. 

Just like for length, it is easier to visualize these prefixes by using a table:

Converting Metric Units of Area

The method of converting metric units of the area is similar to the one we use to convert metric units of length. That is, by moving decimal places.

Let us convert 520 m 2 to km 2 . By looking at the table, there are three steps to the left, from square meters to square kilometers. 

We move three decimal places to the left in 520 m 2 to obtain its equivalent in km 2 . 

Thus, 520 m 2 is equal to 0.52 km 2 .

Since we always use the table of prefixes of metric units, I highly recommend memorizing the horizontal arrangement of these prefixes. They are not that hard to remember since there are only six prefixes. This is much easier than memorizing the conversion units. 

Sample Problem: Every square meter of land in a province costs ₱4,000. Jennie plans to buy a 15-square decameter of land in this province. How much will Jennie have to pay to purchase the land? 

Solution: Since the pricing of the land is expressed as ₱4,000 per square meter (m 2 ), we have to convert 15 square decameters (dam 2 ) to square meters (m 2 ) to calculate the land price accurately. 

From square decameter to square meter, there’s one step to the right.

Hence, we move one decimal to the right in 15 dam 2 to convert it into m 2 :

Thus, 15 dam 2 is equal to 150 m 2 .

Now, since the price of the land is ₱4000 per square meter (m 2 ), then a 150 m 2 of land will cost:

150 x 4000 = 600,000

The answer is ₱600,000.

Metric System Units of Volume

The SI base unit for volume is cubic meters (m 3 ). Like length, the metric units for volume or capacity are derived from cubic meters (m 3 ). 

Six prefixes are used in the metric system, and we list them below together with their equivalent value in the base unit. Note that all metric units for volume have a superscript of 3 to indicate that we are dealing with cubic units. 

Just like for length and area, it is easier to visualize these prefixes by using a table:

The liter is another metric unit used for volume/capacity. The liter is a special name for cubic decimeter (dm 3 ). Thus, 1 liter equals 1 cubic decimeter (1 L = 1 dm 3 ). 

Like any metric unit, prefixes are also used to derive other metric units for volume. For instance, the prefix “milli” in milliliter indicates that this unit is equal to thousandths (0.001) of a liter. 

Although a liter is a metric unit for volume, there’s no need to put a superscript of 3 (which also applies to other metric units based on it). 

Here are the other six prefixes associated with liter:

Again, it’s easier to visualize these prefixes by using a table:

Converting Metric Units of Volume

Converting metric units of volume is similar to converting metric units for length and area. 

Sample Problem 1 : A shoe box has a volume of 750 cm 3 . Determine its volume in m 3 .

Solution : There are two steps to the left from cm 3 to m 3 . Hence, we move two decimal places to the left in 750 cm 3 to transform it into m 3 .

Therefore, 750 cm 3 is equivalent to 7.5 m 3 .

Sample Problem 2 : A large tank can be filled with 250 L of water. A water pipe puts 200 cL of water into the tank per minute. How long can the pipe fill the large tank?

Solution : To determine how long the pipe can fill the large tank, we divide 250L by 200 cL. However, we cannot perform this immediately since the given measurements differ in units.  

First, let us convert 200 cL to L. 

Looking at the table of prefixes, notice two steps to the left from cL to L. Hence, we move two decimal places to the left in 200 cL to convert it into L.

Hence, 200 cL = 2 L

We can now divide 250 L by 2 L. Dividing 250 L by 2 L, we’ll obtain

250 ÷ 2 = 125

This means that the pipe can fill the tank after 125 minutes.

Metric System Units of Mass

The SI base unit for mass is the kilogram (kg). However, note that “gram” is the primary basis for deriving other metric units for mass. “Gram” can be viewed as the “meter” in terms of mass.

Six prefixes are used in the metric system, and we list them here with their equivalent value in the base unit. 

It’s easier to visualize these prefixes by using a table:

Converting Metric Units of Mass

Sample Problem 1 : Myrna bought cough syrup with a mass of 50 grams. Determine its mass in milligrams.

Solution : Referring to the table of prefixes, there are three steps to the right from gram (g) to milligram (mg). Thus, we have to move three decimal places to the right to convert 50 g to mg:

Therefore, the cough syrup’s mass is equal to 50,000 mg.

Sample Problem 2: Rosie bought 1250.50 grams of mangoes. What is the mass of the mangoes that Rosie bought in kilograms?

Solution : Let us convert 1250.50 grams to kilograms. There are three steps to the left, from grams (g) to kilograms (kg). Hence, we must move three decimal places to the left in 1250.50 g to convert it into kilograms (kg). 

Hence, 1250.50 grams is equal to 1.25050 kilograms. 

Converting Units of Time

Let’s now discuss how to convert units of time. Moving decimal places is not applicable for converting time units, unlike the metric units for length, area, volume, or mass. Instead, we have to refer to the conversion values for each unit.

Shown below is the conversion of time units:

To convert one unit of time to another, follow these steps:

• Identify the given and to which unit we will convert it.
• Determine the relationship between the given units.
• Express the relationship between the given units as a conversion factor in a fractional form such that the denominator has a unit that is the same as the original unit.
• Multiply the given measurement by the conversion factor.

Let us apply the steps above to answer some examples.

Sample Problem 1 : How many hours are there in 5 days?

Step 1: Identify the given and to which unit we will convert it. 

The problem is asking us to convert 5 days into hours.

Step 2: Determine the relationship between the given units.

There are 24 hours in one day. In other words, 1 day = 24 hours.

Step 3: Express the relationship between the given units as a conversion factor in a fractional form such that the denominator has a unit that is the same as the original unit.

In the previous step, we’re able to determine that 1 day is equivalent to 24 hours. Express this as a fraction with the unit that matches the original unit as the denominator. Since the original unit is “days,” we must express the conversion factor as 24 hours/1 day.

Step 4: Multiply the given measurement by the conversion factor.

Now, let us multiply 5 days by 24 hours/1 day:

Hence, there are 120 hours in 5 days.

Sample Problem 2 : Rhodora plans to go on a vacation to Lemongate Beach for 8 weeks. How many days will Rhodora be on vacation?

The problem is asking us to convert 8 weeks to days

Step 2 : Determine the relationship between the given units.

There are 7 days in one week or 1 week = 7 days.

In the previous step, we’re able to determine that 1 week is equivalent to 7 days. Express this as a fraction with the unit that matches the original unit as the denominator. Since the original unit is “week,” we must express the conversion factor as 7 days/1 week. 

Now, let us multiply 8 weeks by 7 days/1 week:

Hence, Rhodora will be on vacation for 56 days.

Sample Problem 3 : A worker is paid ₱0.5 per minute for his job. How much will the worker earn if he works for a total amount of time equivalent to 25 days?

Solution: The worker’s wage is expressed as ₱0.5 per minute. Therefore, we must convert 25 days to minutes first before determining the worker’s earnings. 

The problem is asking us to convert 25 days to minutes.

Note that before converting days to minutes, we must first convert days to hours. Afterward, we will convert the result to minutes. This means that we will be dealing with two relationships in this problem. 

• Relationship #1 (days to hours): 24 hours in 1 day or 1 day = 24 hours.
• Relationship #2 (hours to minutes): 60 minutes in 1 hour or 1 hour = 60 minutes.

Express the relationships we derived from Step 2 as conversion factors:

• For relationship #1 (days to hours), the original unit is days, so we have 24 hours/1 day 
• For relationship #2 (hours to minutes), the original unit now is hours, so we have 60 minutes/1 hour 

Multiply the 25 days by the two conversion factors. 

Thus, there are 36,000 minutes in 25 days. 

Remember, the worker’s wage is expressed as ₱0.5 per minute. If the worker renders 36,000 minutes of work, he will earn 36,000 x 0.5 = ₱18,000.

Sample Problem 4 : 504 hours is equivalent to how many weeks?

The problem is asking us to convert 504 hours to weeks.

To convert hours to weeks, we first need to convert hours to days. Afterward, we convert days to weeks. Thus, we will be dealing with two relationships of the unit of time in this problem:

• Relationship #1 (hours to days): 24 hours in 1 day or 1 day = 24 hours
• Relationship #2 (days to weeks): 7 days in 1 week or 1 week = 7 days
• For relationship #1 (hours to days), the original unit is hours, so we have 1 day/24 hours 
• For relationship #2 (days to weeks), the original unit now is days, so we have 1 week/7 days 

Multiply the 504 hours by the two conversion factors. 

Thus, 504 hours is equal to 3 weeks.

Converting Units of Temperature

Kelvin, Celsius, and Fahrenheit are the units of measurement for temperature. This section will focus only on converting Celsius to Fahrenheit and vice versa.

To convert Celsius to Fahrenheit, use a conversion formula. We will discuss these formulas in this section.

How To Convert Celsius to Fahrenheit

Shown below is the formula to convert Celsius to Fahrenheit:

To use this formula, insert the given temperature expressed in Celsius into the formula and perform the calculation. The resulting value is the equivalent temperature in Fahrenheit.

Sample Problem 1: The freezing point of water is 0°C. What is the freezing point of water in Fahrenheit?

Solution : Applying the formula to convert Celsius to Fahrenheit:

Hence, the freezing point of water in Fahrenheit is 32°F.

Sample Problem 2: The average body temperature is 37°C. What is the average body temperature in Fahrenheit?

Hence, the average body temperature in Fahrenheit is 98.6°F.

How To Convert Fahrenheit to Celsius

Shown below is the formula to convert Fahrenheit to Celsius:

The formula above can be derived using the formula for converting Celsius to Fahrenheit. All you have to do is perform some basic algebra (in particular, solving a linear equation ).

To use the formula, insert the given temperature expressed in Fahrenheit into the formula and perform the calculation. The resulting value is the equivalent temperature in Celsius.

Sample Problem 1 : Convert 32 °F to °C

Hence, 32 °F is equal to 0°C.

Sample Problem 2 : The required storage temperature of a particular drug is 59 °F. What is its equivalent in °C?

Solution : 

Therefore, 59 °F is equal to 15 °C.

Next topic:  Perimeter and Area of Plane Figures

Previous topic:  Triangles: Classification and Theorems

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LESSON | Converting Units & Solving Problems Involving Conversion

The complete guide to converting units & solving problems involving conversion.

In this post, you'll learn how to solve problems that involve converting units and how to convert measurements from one unit to another in both the English and Metric systems. Since these systems are used by a lot of people, having a good understanding of them will help you solve these problems more accurately. in everyday life.

(toc) Table of Contents

CONVERSION OF MEASUREMENTS FROM ONE UNIT TO ANOTHER

6 Steps on How to Convert A Unit of Measurement to Another Unit

1. Compare the two units.

2. Find the conversion factors that gives the appropriate ratio to the given unit.

3. Write the conversion as a fraction, where the denominator is in the same unit as the given unit.

4. Write a multiplication problem with the original number and the fraction.

5. Cancel out similar units that appears on the numerator and denominator.

Convert Lengths or Distance

Convert mass, convert area, convert volume, convert time, try this unit conversion calculator.

This unit converter is a free and easy to use tool for converting units of measurements.

It converts length, area, volume, weight, speed, density and temperature. It also converts between different units of measurement.

The converter is available in metric or imperial units. You can type in the unit you want to convert into the search bar or click on the links below to find your desired unit.

Conclusion: Use these Tips to Succeed at Unit Conversions with Ease

When changing from one unit of measurement to another, it is very important to know the table of conversion because this will be your guide.

There are some measurements in the table that can't be changed directly, so we should know which conversion factor is the easiest to use.

 You need to know and be good at converting units before you can use the different problem-solving strategies to solve problems that involve converting units.

Unit conversions are a necessary skill in the workplace. Whether you are a student, engineer, or an accountant, you will need to know how to convert units of measurement.

Use these tips to succeed at unit conversions with ease:

• Use a calculator or an online conversion tool.
• Memorize the metric system conversion chart!
• Pay attention to units of measurements in context.
• Think about what units you want and what units you have before starting the conversion process.
• Never forget that there are always two values when converting from one unit to another - one from the original and one from the destination unit!

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Course: 5th grade   >   Unit 14

• Measurement word problem: tea party
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• Convert units word problems (metrics)

Convert units multi-step word problems (metric)

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• an integer, like 6 ‍  
• an exact decimal, like 0.75 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  

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Unit Conversion Example Problems

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Unit conversion problems constitute the first week of nearly every science course. Measurement is a basic skill in science, but not all measurement methods are the same. People measure the same thing in different ways and use different units. To communicate between scientists, there has to be a method of converting between the different measurement units.

This is a collection of unit conversion example problems to help you learn the general method and mindset of converting units. Most of these include an example going the opposite direction (for example: cm to m and m to cm).

How to Do Unit Conversions

• How to Convert Between Units – Ladder Method

Use the ladder method or dimensional analysis to tackle conversions between units. See how to use a conversion factor to step-by-step reach the desired end units. This method is especially useful for measurements with more than one conversion like kilometers/hour to meters/sec. The L/min to m3/hr Conversion Example is a conversion example which uses the Ladder Method.

Length Unit Conversion Example Problems

• Feet to Inches (ft to in) Conversion
• Inches to Centimeters (in to cm) Conversion
• Miles to Kilometers (mi to km) Conversion
• Meters to Yards (and back again)
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Area Unit Conversion Example Problems

• Square Centimeter to Square Meter (cm 2 to m 2 ) Conversion

Volume Unit Conversion Example Problems

• General Volume Example – Step by step conversion between 3 linear measurements to find the volume.
• Cubic Centimeters to Cubic Meters (cm 3 to m 3 ) Conversion
• Cubic Centimeters to Liters (cm 3 to L) Conversion
• Cubic Feet to Liters (ft 3  to L) Conversion
• Cubic Feet to Cubic Meters (ft 3 to m 3 )
• Cubic Inches to Cubic Feet (in 3 to ft 3 ) Conversion
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• Pounds to Kilograms (lbs to Kg) Conversion
• Pounds to Grams (lb to g)
• Grams to Ounces (g to oz)
• Ounces to Grams (oz to g)

Temperature Unit Conversion Example Problems

• Celsius to Fahrenheit (°C to °F) Conversion
• Celsius to Kelvin (°C to K) Conversion
• How to Convert Fahrenheit to Celsius
• Fahrenheit to Celsius (°F to °C) Example
• Fahrenheit to Kelvin (°F to K) Conversion
• Kelvin to Celsius (K to °C) Conversion
• When does Celsius Equal Fahrenheit?
• When does Fahrenheit Equal Kelvin?

Pressure Unit Conversion Example Problems

• Atmospheres to Kilopascals (atm to kPa) Conversion
• Atmospheres to Pounds per Square Inch (atm to PSI) Conversion
• PSI to atm (pounds per square inch to atmospheres)
• Millimeters of mercury to millibars (mmHg to mbar) Conversion
• Millimeters of mercury to kilopascals (mmHg t kPa) Conversion

Mole and Concentration Conversion Example Problems

• Grams to Moles (g to mol) Conversion

Time Conversion Examples

• Days to Years Conversion

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Word Problems on Conversion of Units: Definitions, Types

Word Problems on Conversion of Units : Conversion of units is a multi-step process that involves multiplication or division by a numerical factor. In Mathematics, while solving numerical problems, it is required to convert the units. Thus the conversion of units should be needed to solve the required calculations wherever it is necessary.

For example, if we need to calculate the area of the rectangle, in which length is given in centimetres \(\left( {{\rm{cm}}} \right)\), and the width is given in metres \({\rm{m}}\), then it is necessary to convert any one unit either length or width, so that both the units become the same.

Therefore, to solve the word problems in mathematics, learning the conversion of units is necessary. This article will discuss the conversion of units in more detail.

Definition of Word Problems on Conversion of Units

Conversion of units is a multi-step process that involves multiplication or division by a numerical factor. Word problems on the conversion of units consist of a few sentences describing a real-life scenario where mathematical definitions and concepts of converting units from one unit to another unit are used to solve a problem.

The conversion of units may also require selecting the correct number of significant digits and rounding off. In mathematics, we convert the units from one unit to the other unit for better understanding. 

For example, the length of a garden is measured in yards, whereas the length of a table is measured in inches. But we cannot measure the length of a finger in miles. To measure different quantities, different units of measurements are needed.

The conversion of units should be needed to solve the required calculations wherever it is necessary. For example, if we need to calculate the area of the rectangle, in which length is given in centimetres \(\left( {{\rm{cm}}} \right)\), and the width is given in metres \(\left( {{\rm{m}}} \right)\), then it is necessary to convert any one unit, either length or width, to make them uniform.

Converting Metric Units Word Problems

In mathematics, we have metric systems such as measuring units of length and distance, weight, and capacity. As we discussed, to solve the word problems correctly, we need to learn the conversion of units, and it is necessary too.

The metric system was introduced in France in the year \(1790\). The metric system of measuring units is based on the decimal system. The base units for length is metre, for weight kilogram and seconds for the time.

Unit conversion is a multi-step process that involves multiplication or division by a numerical factor. To convert any bigger unit to a smaller unit, we should multiply with the conversion factor. Similarly, to convert any smaller unit to a bigger unit, we should divide it with the conversion factor.

Word Problems on Conversion of Units of Length

Length is a one-dimensional scalar quantity, which measures the line segment. The basic unit used for measuring the length is a metre \({\rm{(m)}}\). Depending on the specimen or object used for measurement, we have different types of units like \({\rm{cm,}}\,{\rm{km,}}\,{\rm{inches,}}\,{\rm{ft}}\), etc.

For example, the length of a garden is measured in yards, whereas the length of a table is measured in inches. But we cannot measure the length of a finger in miles. To measure different quantities, different units of measurements are needed. 

The below chart gives the conversion of length:

To convert a unit from a metre to a centimetre, we should multiply by \(100\) such that \(1\,{\rm{m}}\, = \,100\,{\rm{cm}}\)

Asit and Keerthi each ran on a treadmill exactly for \(90\) minutes. Asit’s treadmill showed he had run \(18500\) meters. Keerthi’s treadmill showed she had run \(20\) kilometres. Who ran farther, and how much?

The time took by the Keerthi and Asit are the same, that is \(90\) minutes.

Here, the distance covered by Asit and Keerthi has different measuring units. For uniformity in the calculation, we have to convert the units from \({\rm{km}}\) to \({\rm{m}}\) or \({\rm{m}}\) to \({\rm{km}}\).

Let us convert \({\rm{km}}\,\) to \({\rm{m}}\,\).

So, the distance covered by the Asit on the treadmill \( = 18,500\,{\rm{m}}\)

And, the distance covered by the Keerthi on treadmill \( = 20\,\,{\rm{km}}{\rm{ = }}{\rm{20}} \times {\rm{1000}}{\rm{m  = }}{\rm{20,}}\,{\rm{000}}\,{\rm{m}}\)

The difference in their distances is \({\rm{20,}}\,{\rm{000}}\,{\rm{m}}\,{\rm{ – }}\,{\rm{18500}}\,{\rm{m}}\,{\rm{ = 1500}}\,{\rm{m}}\)

So, Keerthi ran farther as compared with Asit by \({\rm{20,}}\,{\rm{000}}\,{\rm{m}}\,{\rm{ – }}\,{\rm{18500}}\,{\rm{m}}\,{\rm{ = 1500}}\,{\rm{m}}\)

So, Keerthi ran farther as compared with Asit by \({\rm{1500}}\,{\rm{m}}\) or \({\rm{1.5}}\,{\rm{km}}\).

Word Problems on Conversion of Units of Weight

Weight is the one-dimensional vector quantity, which is used for the measurement of objects. Generally, the weight of the object is measured in the base unit kilogram \(\left( {{\rm{kg}}} \right).\)  We know that weight of the person is measured in \({\rm{(kg)}}\) and the weight of the small pieces of gold is measured in grams. So, it is important to convert the units of the weights while solving word problems for uniformity in the calculation.

The below figure shows the conversion of weights from one unit to another unit:

Nag is carrying \(2.5\,{\rm{kg}}\) of apples and \(5\,{\rm{g}}\) of carrying bag. Find the total weight she is holding in her hand.

The total weight \( = 2.5 \times 1000\,{\rm{g}}\,{\rm{ +  5g}}\,{\rm{ = }}\,{\rm{2505}}\,{\rm{g}}\)

Word Problems on Conversion of Units of Time

We know that seconds are the basic unit for measuring time. We have to convert the units of time from one unit to another unit for solving the problems. The below chart gives the conversion of time:

Example: The time taken to reach the shop is \(30\) minutes and from the due to heavy traffic, the time taken to reach the house is \(1\) hour. Find the total time taken?

The total time taken \( = 30\min {\rm{utes}} + 1 \times 60\min {\rm{utes}} = 90\min {\rm{utes}}\,\)

Word Problems on Conversion of Units of Area

The area is the two-dimensional property. We have different units for measuring the area. The below figure shows the conversion of area units:

The area of the parking lot is \(12{{\rm{m}}^2}\) and \(50\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\) Find the total area?

The total area \( = \,12\, \times \,{100^2}\,{\rm{c}}{{\rm{m}}^2}\, + \,500\,{\rm{c}}{{\rm{m}}^2}\, = \,1440000\, + 500\, = \,1440500\,{\rm{c}}{{\rm{m}}^2}\)

Word Problems on Conversion of Units of Capacity

Capacity describes the volume of the object. The different types of units used for measuring the capacity are given below:

The capacity of the water bottle is \(1\,{\rm{l}}\) and the cap is \(1\,{\rm{ml}}\) Find the total capacity of the bottle.

The total capacity \( = 1 \times 1000\,{\rm{ml}} + 1\,{\rm{ml}} = 1001\,{\rm{ml}}\)

Solved Examples – Word Problems on Conversion of Units

Q.1. The distance between the two places in a city is \(31,\,680\, feet\) Express the distance in miles. Ans: Given the distance between the places \({\rm{ = 31,680}}\,{\rm{feet}}\) We know that \({\rm{1feet}} = \frac{1}{{5280}}{\rm{miles}}\) So, total distance in miles \( = \frac{{31,680}}{{5280}} = 6\,{\rm{miles}}\)

Q.2. Asit’s mom took \(30\) minutes to cut the vegetables, and she took \(1\) hour in cooking. Find how many minutes she took to complete the whole cooking? Ans: The time taken for cutting the vegetables \( = 30\,{\rm{minutes}}\) The time taken for cooking \( = 1\,{\rm{hour}} = 60\,{\rm{minutes}}\) Total time taken for whole cooking \( = \,30\, + \,60\, = \,90\,{\rm{minutes}}\)

Q.3. Vicky has \(14,500\,{\rm{g}}\) of sand in his sandbox, and he bought \(7500\,{\rm{g}}\) of sand from the beach. Total how many kilograms of sand Vicky has in his sandbox. Ans: Initial sand in the box is \(14,500\,{\rm{g}}\,{\rm{ = }}\,\frac{{14,500}}{{1000}}\,{\rm{kg}}\, = \,14.\,5\,{\rm{kg}}\) The sand bough from the beach \( = 7500\,{\rm{g}}\,{\rm{ = }}\,\frac{{7500}}{{1000}}\,{\rm{kg}}\, = \,7.\,5\,{\rm{kg}}\) Total sand in the box \( = 14.5\,{\rm{kg}}\,{\rm{ + }}\,7.5\,{\rm{kg}}\, = \,22\,{\rm{kg}}\)

Q.4. Jessica measures two line segments. The first line segment is \(30\,{\rm{cm}}\) long. The second line segment is \(500\,{\rm{mm}}\) long. How long are the two line segments together? (answer in cm ) Ans: The length of the first line segment \( = \,\,30\,{\rm{cm}}\) The length of the second line segment \({\rm{ = }}\,{\rm{500}}\,{\rm{mm}}\,{\rm{ = }}\,\frac{{500}}{{10}}\,{\rm{cm}}\,{\rm{ = }}\,{\rm{50}}\,{\rm{cm}}\) The total length of two-line segments \( = \,\,30\,{\rm{m}}\,{\rm{ + }}\,{\rm{50}}\,{\rm{m}}\, = \,80\,{\rm{m}}\)

Q.5. The length of the box is \(2\,{\rm{m}}\) and the width is \(40\,{\rm{cm}}\) Find the area of the box in \({\rm{c}}{{\rm{m}}^2}\) Ans: Given the length of the box \(2\,{\rm{m}}\) Width of the box \({\rm{ = }}\,{\rm{40}}\,{\rm{cm}}\) Area of the box \({\rm{ = }}\,{\rm{length}}\, \times \,{\rm{width}}\,{\rm{ = }}\,{\rm{200}}\,\, \times \,40\, = \,8000\,{\rm{c}}{{\rm{m}}^2}\)

In this article, we have discussed various methods of the conversion of units from one unit to another unit. This article also discusses the conversion of metric units and word problems on the conversion of metric units. In this article, we have studied the word problems on the conversion of units of length, weight, area, time and capacity, along with the solved examples that help us solve the numerical problems easily.

Frequently Asked Questions – Word Problems on Conversion of Units

Q.1. How do you solve unit conversion problems? Ans: The following steps are to be followed to do unit conversion problems. 1. Read the data and the given units. 2. Now, multiply or divide as required conversion with the conversion factor. 3. Then, solve the problems using the proper formulas and operations.

Q.2. How do you solve metric word problems? Ans: The metric word problems are to be solved by using the unit conversion. Convert any bigger unit to the smaller unit, and we should multiply with the conversion factor. Similarly, to convert any smaller unit to a bigger unit, we should divide it with the conversion factor.

Q.3. Why is unit conversion important? Ans: To solve many real-life problems, unit conversion is very important because we cannot perform basic operations like addition, subtraction, multiplication, division etc., unless the two quantities are in the same units.

Q.4. What are the three basic metric units? Ans: The three basic metric units are metre for length, gram for weight and litre for capacity.

Q.5. What is unit conversion? Ans: Conversion of units is a multi-step process that involves multiplication or division by a numerical factor.

Learn about Measurement here

We hope you find this article on ‘Word Problems on Conversion of Units ‘ helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them. 

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6.3.3 Converting & Estimating Measurements

Standard 6.3.3.

Solve problems in various contexts involving conversion of weights, capacities, geometric measurements and times within measurement systems using appropriate units.

Estimate weights, capacities and geometric measurements using benchmarks in measurement systems with appropriate units.

For example : Estimate the height of a house by comparing to a 6-foot man standing nearby.

Big Ideas and Essential Understandings

Standard 6.3.3 Essential Understandings

Students have years of diverse experiences with measurement from prior classroom instruction and from using measurement in their everyday lives. These formal and informal experiences become the building blocks for students to estimate measurements and choose appropriate size and units of measurement for various situations.

Understanding the relationships among units within a measurement system is essential for students to use ratios to convert from one unit to another in solving problems. Students at this level use ratio and reasoning about multiplication and division to convert units within measurement systems and solve real-world and mathematical problems. This standard starts the learning progression of algebra concepts that moves students from ratios to proportions to functions, as well as the use of algebra in geometry.

6.3.3 Converting and Estimating Measurements

6.3.3.1  Solve problems in various contexts involving conversion of weights, capacities, geometric measurements and times within measurement systems using appropriate units.

6.3.3.2  Estimate weights, capacities and geometric measurements using benchmarks in measurement systems with appropriate units.

What students should know and be able to do [at a mastery level] related to these benchmarks:

• Know what is meant by length, capacity and weight;
• Select appropriate size and type of unit for a given measurement situation;
• Understand the size of each benchmark unit and the relationship among benchmark units within a measurement system;
• Identify equivalent ratios within a measurement system;
• Construct and analyze tables to show quantities that are in equivalent ratios;
• Use ratios and reasoning about multiplication and division to solve problems that require conversion of measurement units within a measurement system;
• Use common benchmarks to make reasonable estimates of measurements;
• Recognize new equivalences, such as 1 square yard equals 9 square feet and 1 cubic meter equals 1,000,000 cubic centimeters;
• Know useful approximate equivalents between the customary and metric systems, such as a quart is a little less than a liter, and a yard is a little less than a meter.

Work from previous grades that supports this new learning includes:

• Recognize and generate equivalent decimals, fractions, mixed numbers and improper fractions in various contexts;
• Solve real-world and mathematical problems requiring addition, subtraction, multiplication and division of multi-digit whole numbers. Use various strategies, including the inverse relationships between operations, the use of technology, and the context of the problem to assess the reasonableness of results;
• Create and use rules, tables, spreadsheets and graphs to describe patterns of change and solve problems;
• Represent real-world situations using equations and inequalities involving variables.

NCTM Standards

• Understand measurable attributes of objects and the units, systems, and processes of measurement:
• Understand both metric and customary systems of measurement;
• Understand relationships among units and convert from one unit to another within the same system;
• Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.
• Apply appropriate techniques, tools, and formulas to determine measurements:
• Use common benchmarks to select appropriate methods for estimating measurements;
• Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision;
• Solve problems involving scale factors, using ratio and proportion.

Common Core State Standards (CCSS)

• 5MD (Measurement & Data) Convert measurement units within a given measurement system.
• 5MD.1.   Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.
• 6RP (Ratio & Proportional Relationships) Understand ratio concepts and use ratio reasoning to solve problems.
• 6RP.3.d. Use rate reasoning to convert measurement units; manipulate and transform units appropriately when multiplying and dividing quantities.

Misconceptions

• Students may multiply when converting to larger units and divide when converting to smaller units.
• Students may not know the prefixes used in the metric system or their meanings.
• Students may not be able to form mental pictures of measurement units, especially metric units, making it difficult for them to choose appropriate units or make reasonable estimates.

Ms. Rykken's students were excited about the recent CO 2 car competition in their industrial technology class, where they raced the cars they had designed and built. She decides to use this opportunity for them to use their own data and to review working with rates and unit conversion.

She began the class by asking for the velocity of the winning car. Students could only respond with the winning time (0.971 seconds).

Teacher: How is the velocity of real cars expressed?

Student: Speedometers tell us how many miles per hour we're going.

Teacher: That sounds like a relationship between distance and time. What is the relationship between distance and time for the winning CO 2 car?

Student: Well, the track was 66 feet long and it took the car 0.971 seconds to travel that distance.

Teacher: So the winning car traveled 66 feet in 0.971 seconds. We can write that relationship as a ratio, or fraction $\frac{66\ feet}{0.971\ seconds}$ . But you said that real cars express velocity as miles per hour. How can we convert feet per second to miles per hour?

Student : The feet need to be changed to miles, and the seconds need to be changed to hours.

Teacher: Yes, but how do we do that?

Student: I know how to change the time. There are 60 seconds in every minute, and 60 minutes in every hour. We can just multiply.

Teacher: Tell me more.

Student : If I multiply $\frac{66\ feet}{0.971\ seconds}\times \frac{60\ seconds}{1\ minute}\times \frac{60\ minutes}{1\ hour}$, I'll know how many feet the car went in 1 hour.

Teacher: How do we know that multiplying by those fractions didn't change the original velocity?

Student: Because each of the fractions is equivalent to 1, and multiplying by 1 gives you the same value.

Teacher: Since the fractions are equivalent to 1, could I have inverted them and gotten the same result from multiplying?

Student: No, that would mess up the units. The fractions need to be set up so that the units cancel, like seconds with seconds and minutes with minutes. After multiplying, you're left with $\frac{237,600\ feet}{0.971\ hour}$.

Teacher: That makes sense. Now we've converted the time, but we're not finished. For real cars, we express velocity in miles per hour. How do we convert the feet to miles?

Student: The same way. We multiply using the relationship that every mile has 5,280 feet.

Teacher: Do I multiply by $\frac{5,280\ feet}{1\ mile}$ or $\frac{1\ mile}{5,280\ feet}$?

Student: If you want the feet to cancel, you'll need to use $\frac{1\ mile}{5,280\ feet}$ .

Teacher: Like this? $\frac{237,600\ feet}{0.971\ hour}\times \frac{1\ mile}{5,280\ feet}$?

Student: Yes.

Teacher: What is the result?

Student: About 46.3 miles per hour.

Teacher: WOW! That's amazing! It's a good think you didn't race your cars outside the school, because the speed limit there is 30 miles per hour.

Student: Could we have changed the feet to miles first and gotten the same answer?

Teacher: Good question. Does it change the result if we multiply in a different order?

Student: No, multiplication is commutative, which means that the order doesn't matter.

Teacher: Yes, that's right.

Student: I did all those steps at the same time and got the same result.

Teacher: Like this:  $\frac{66\ feet}{0.971\ seconds}\times \frac{60\ seconds}{1\ minute}\times \frac{60\ minutes}{1\ hour}\times\frac{1\ mile}{5,280\ feet} $? Excellent! That's very efficient. Now I'd like each of you to find the velocity of your CO 2 car in miles per hour.

Ms. Rykken circulates the room to assist students and hear tales of their race experiences.

Teacher Notes

• Students incorrectly multiply when converting to larger units and divide when converting to smaller units when they do not recognize that the size of the unit and the number of units needed to measure an object are inversely related; as one increases, the other decreases. Understanding this relationship is essential for students to be able to make reasonable estimates. Students can develop this understanding by using familiar contexts. For example, students will understand that exchanging five dollars for pennies will result in more than five pennies, since you're moving to a smaller unit. Similarly, converting 50 inches into feet will result in fewer feet since you're moving to a larger unit.
• Students may find the acronym " K ing (kilo) H enry(hecto) D ied (deka) D rinking (deci) C hocolate (centi) M ilk (milli)" helpful to remember the order of commonly used metric prefixes. Making explicit connections to place value and the fact that next-smaller and next-larger units differ by a power of 10 will support students in understanding the relationship between units.
• Students need multiple hands-on experiences in measuring in order to understand the size of various units, form mental pictures, and make reasonable estimates.
• Students will benefit from teacher use of multiple approaches to converting units within a measurement system. Creating tables is especially helpful for students to see equivalent ratios and use reasoning about multiplication and division when making unit conversions. By viewing equivalent ratios and rates as deriving from and extending pairs of rows (or columns) in multiplication tables, students connect their understanding of multiplication and division with ratios and rates.
• Students need to understand that they're actually multiplying by one when using unit analysis to make conversions. Reminding them that the goal is to "cancel" the old units, similar to "canceling" common factors when simplifying fractions, will help them decide how to write the ratio used for conversion ($\frac{1\ yard}{3\ feet}$ or $\frac{3\ feet}{1\ yard}$).
• Although students are more familiar with the customary measurement system, this system can be especially challenging as the relationships between units are cumbersome and require an understanding of fractions. Measurement in the customary system provides an opportunity for students use fractions in real-world situations.

Weighing Your Car

This lesson asks students to measure the area of a car tire's footprint and its air pressure in order to estimate the car's weight.

Have You Ever Seen a Tree Big Enough to Drive a Car Through?

This challenge allows students explore how foresters use measurement.

Information |Description= Henry Cowell Redwoods State Park, Redwood Grove. Man standing beside a particularly large Coastal Redwood tree. |Source= Created this image on Honeymoon in Santa Cruz, CA |Date= October, 2006 |Author= Larry McElhiney |Permiss)

Does Polygon Need a Jacket?

This challenge asks students to estimate the temperature in Fahrenheit given Celsius.

Additional Instructional Resources

Learning Measurement Through Practice

This article describes a team-taught activity-based format to teach students about measurement.

New Vocabulary

capacity: measure of how much a container can hold. Units of capacity are special types of volume units used for containers whose length, width, and height cannot be measured. Capacity is usually used for things such as liquids or pourable substances, such as grains of sugar.

Example: The capacity of a soda pop can is 12 fluid ounces or 355 milliliters.

customary system: measurement system used most often in the United States.

length: inch, foot, yard, and mile

weight : ounce, pound, and ton

capacity : cup, pint, quart, and gallon

temperature : degrees Fahrenheit

metric system: measurement system used in most countries around the world based on the base-ten numeration system.

length: millimeter, centimeter, meter, and kilometer

weight: gram and kilogram

capacity: milliliter and liter

temperature: degrees Celsius

metric system prefixes: kilo means 1000, hecto means 100, deka means 10, deci means $\frac{1}{10}$, centi means $\frac{1}{100}$, and milli means $\frac{1}{1000}$.

weight: measure of how heavy something is.            

Example : The weight of a bag of sugar may be 4 pounds or 1.8 kilograms.

Reflection - Critical Questions regarding the teaching and learning of these benchmarks

• How were multiple approaches and strategies modeled for solving problems that required converting measurement units?
• How was measurement connected to other areas of mathematics?
• How was measurement connected to other content areas?
• How were measurement concepts and skills used to solve real-world problems?
• What strategies were used to develop a sense of reasonableness when measuring?
• Did students have frequent opportunities for hands-on measurement?
• How were important social and scientific contexts used to make measurement relevant to students' lives?
• NCTM: Principles and Standards for School Mathematics
• Grades 6-8 Measurement Standard for Grades 6-8 (pp. 240-247).
• NCTM: A Research Companion to Principles and Standards for School Mathematics
• Chapter 12, Developing Understanding of Measurement (p. 181).
• Integrating Measurement Across the Curriculum
• Measurement: What's the Big Idea ?
• Van Hiele Levels of Geometric Reasoning
• Keeley, P., & Rose, C. (2006). Mathematics curriculum topic study . Thousand Oaks, CA: Corwin Press.
• Kilpatrick, J., Martin, W., & Schifter, D. (Eds.). (2003). A research companion to principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
• Mathematics Curriculum Framework. (2000). Malden, MA: Massachusetts Department of Education.
• Mathematics Framework for the 2009 National Assessment of Educational Progress. (2009). Washington, D.C.: National Assessment Governing Board, U.S. Department of Education.
• Minnesota's K-12 Mathematics Frameworks . (1998). St. Paul, MN: SciMathMN.
• Developing essential understanding of ratios, proportions & proportional reasoning: Grades 6-8. (2010). Reston: VA: National Council of Teachers of Mathematics, Inc.
• Principles and standards for school mathematics . (2000). Reston, VA: National Council of Teachers of Mathematics, Inc.

(DOK: Level 1)

1. Last week, Harry walked his dog 4 days at a nearby park. Each day they walked on a trail 2,400 meters long. How many kilometers did they walk at the park last week?

            a.  9,600 kilometers     b.  960 kilometers

            c.  96 kilometers          d.  9.6 kilometers

(DOK: Level 2)

2. The pilot of a passenger jet announced that their cruising altitude was approximately 31,750 feet. About how many miles above the ground is the plane? Explain your reasoning.

Answer: The plane is about 6 miles above the ground. There are 5,280 feet in one mile, so I divided 31,750 by 5,280 to find the number of miles in 31,750 feet. The answer is a little more than 6.

3. A delivery truck brought several bags of concrete mix to a home improvement store. In all, 10.8 tons of concrete mix were delivered. If each bag of concrete mix weighed 60 pounds, about how many bags did the truck deliver?

            a.  556 bags                 b.  360 bags

            c.  270 bags                 d.  180 bags

(DOK: Level 3)

Answer: 200 m or 0.2 km

5. Figure out which number goes in which place, using reasonable estimates.

A man weighs _____ kg. This is _____ times the amount he weighed at birth.

He drives his car _____ miles in one hour. This is _____ times the distance he walks in one hour.                                      15        22        60        88

Possible answer: 88, 22, 60, 15

(DOK Level 4)

6. Mike uses a pedometer to keep track of how many steps he takes each day. He plans to hike 8 miles next Saturday. Predict how many steps Mike's pedometer will show from hiking 8 miles. Explain your reasoning.                                                                                           

Differentiation

Struggling Learners

• Provide a graphic organizer showing units used for measuring length, capacity, and weight in measurement systems.

            Example:

• Provide key benchmark relationships for measurement systems.

            Examples:

• Post "rules of thumb" to help students develop a sense of reasonableness when converting between measurement systems.

• To explore the concept of capacity, have students compare the capacity of five different-size containers. Ask students to predict which containers have the largest and smallest capacities. Then have students use fill material (macaroni, centimeter cubes) to compare the capacities of the five containers and to determine if their predictions were correct. Encourage students to continue to develop their personal references as they make and check their predictions.
• Students new to the United States will most likely be unfamiliar with the customary system. It is critical for them to have hands-on experiences measuring length, weight and capacity using both customary and metric units.
• Measure distances using rulers and tape measures that show both customary and metric units. Post pictures and maps showing distances labeled with appropriate customary and metric units.
• Label containers appropriately with cup, pint, quart, gallon, and liter, and use graduated cylinders and measuring cups to measure capacity. Post pictures of containers and their capacities. Ask students to find real-world examples that show capacity in both customary and metric units; e.g., a soda pop can contains 12 fluid ounces or 355 milliliters.
• Use scales and balances to measure weight. Label objects that weigh 1 ounce, 1 pound, 1 gram, and 1 kilogram. Post pictures of objects and their weight in both measurement system. Ask students to find real-world examples that show weight in both customary and metric units; e.g., a bag of sugar weighs 4 pounds or 1.8 kilograms.
• Post charts showing the relationships among benchmark units within a measurement system.
• Encourage students to look for the prefixes in number names when using the metric system. Consider having students illustrate some of the words and prefixes they discuss.
• Use graphic organizers, such as the Frayer model shown below, for vocabulary development.

• Can You Run as Fast as a Car?

This challenge requires students compare rates between customary and metric measurement systems.

• Converting Units of Length

The video on this site will help students convert between measurement systems.

Parents/Admin

Administrative/Peer Classroom Observation

Parent Resources

• Metric Units & Converting Between Them
• Conversion Between Metric Units
• Converting within the Metric System
• Cancelling/Converting Units
• Simplifying Rates and Ratios

Related Frameworks

• 6.3.3.1 Conversions Within Measurement Systems
• 6.3.3.2 Estimate Measurements

Using Metric Conversions to Solve Problems

Learning Objective(s)

·          Solve application problems involving metric units of length, mass, and volume.

Introduction

Learning how to solve real-world problems using metric conversions is as important as learning how to do the conversions themselves. Mathematicians, scientists, nurses, and even athletes are often confronted with situations where they are presented with information using metric measurements, and must then make informed decisions based on that data.

To solve these problems effectively, you need to understand the context of a problem, perform conversions, and then check the reasonableness of your answer. Do all three of these steps and you will succeed in whatever measurement system you find yourself using.

Understanding Context and Performing Conversions

The first step in solving any real-world problem is to understand its context. This will help you figure out what kinds of solutions are reasonable (and the problem itself may give you clues about what types of conversions are necessary). Here is an example.

This may not be likely to happen (a runner would have to be quite an athlete to compete in all of these races) but it is an interesting question to consider. The problem required you to find the total distance that the runner would run (in kilometers). The example showed how to add the distances, in meters, and then convert that number to kilometers.

An example with a different context, but still requiring conversions, is shown below.

This problem asked for the difference between two quantities. The easiest way to find this is to convert one quantity so that both quantities are measured in the same unit, and then subtract one from the other.

Checking your Conversions

Sometimes it is a good idea to check your conversions using a second method. This usually helps you catch any errors that you may make, such as using the wrong unit fractions or moving the decimal point the wrong way.

Having come up with the answer, you could also check your conversions using the quicker “move the decimal” method, shown below.

The initial answer checks out—0.72 liter of liquid is needed to fill the bottle. Checking one conversion with another method is a good practice for catching any errors in scale.

Understanding the context of real-life application problems is important. Look for words within the problem that help you identify what operations are needed, and then apply the correct unit conversions. Checking your final answer by using another conversion method (such as the “move the decimal” method, if you have used the factor label method to solve the problem) can cut down on errors in your calculations.

Solving problems involving converting between units of time

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Lesson details

Key learning points.

• In this lesson, we will solve word problems which compare different units of time.

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Starter quiz

5 questions, lesson appears in, unit maths / converting units of measure.

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1.6 Unit Conversion Word Problems

One application of rational expressions deals with converting units. Units of measure can be converted by multiplying several fractions together in a process known as dimensional analysis.

The trick is to decide what fractions to multiply. If an expression is multiplied by 1, its value does not change. The number 1 can be written as a fraction in many different ways, so long as the numerator and denominator are identical in value. Note that the numerator and denominator need not be identical in appearance, but rather only identical in value. Below are several fractions, each equal to 1, where the numerator and the denominator are identical in value. This is why, when doing dimensional analysis, it is very important to use units in the setup of the problem, so as to ensure that the conversion factor is set up correctly.

Example 1.6.1

 If 1 pound = 16 ounces, how many pounds are in 435 ounces?

[latex]\begin{array}{rrll} 435\text{ oz}&=&435\text{ \cancel{oz}}\times \dfrac{1\text{ lb}}{16\text{ \cancel{oz}}} \hspace{0.2in}& \text{This operation cancels the oz and leaves the lbs} \\ \\ &=&\dfrac{435\text{ lb}}{16} \hspace{0.2in}& \text{Which reduces to } \\ \\ &=&27\dfrac{3}{16}\text{ lb} \hspace{0.2in}& \text{Solution} \end{array}[/latex]

The same process can be used to convert problems with several units in them. Consider the following example.

Example 1.6.2

A student averaged 45 miles per hour on a trip. What was the student’s speed in feet per second?

[latex]\begin{array}{rrll} 45 \text{ mi/h}&=&\dfrac{\text{45 \cancel{mi}}}{\text{\cancel{hr}}}\times \dfrac{5280 \text{ ft}}{1\text{ \cancel{mi}}}\times \dfrac{1\text{ \cancel{hr}}}{3600\text{ s}}\hspace{0.2in}&\text{This will cancel the miles and hours} \\ \\ &=&45\times \dfrac{5280}{1}\times \dfrac{1}{3600} \text{ ft/s}\hspace{0.2in}&\text{This reduces to} \\ \\ &=&66\text{ ft/s}\hspace{0.2in}&\text{Solution} \end{array}[/latex]

Example 1.6.3

Convert 8 ft 3 to yd 3 .

[latex]\begin{array}{rrll} 8\text{ ft}^3&=&8\text{ ft}^3 \times \dfrac{(1\text{ yd})^3}{(3\text{ ft})^3}&\text{Cube the parentheses} \\ \\ &=&8\text{ }\cancel{\text{ft}^3}\times \dfrac{1\text{ yd}^3}{27\text{ }\cancel{\text{ft}^3}}&\text{This will cancel the ft}^3\text{ and replace them with yd}^3 \\ \\ &=&8\times \dfrac{1\text{ yd}^3}{27}&\text{Which reduces to} \\ \\ &=&\dfrac{8}{27}\text{ yd}^3\text{ or }0.296\text{ yd}^3&\text{Solution} \end{array}[/latex]

Example 1.6.4

A room is 10 ft by 12 ft. How many square yards are in the room? The area of the room is 120 ft 2 (area = length × width).

Converting the area yields:

[latex]\begin{array}{rrll} 120\text{ ft}^2&=&120\text{ }\cancel{\text{ft}^2}\times \dfrac{(1\text{ yd})^2}{(3\text{ }\cancel{\text{ft}})^2}&\text{Cancel ft}^2\text{ and replace with yd}^2 \\ \\ &=&\dfrac{120\text{ yd}^2}{9}&\text{This reduces to} \\ \\ &=&13\dfrac{1}{3}\text{ yd}^2&\text{Solution} \\ \\ \end{array}[/latex]

The process of dimensional analysis can be used to convert other types of units as well. Once relationships that represent the same value have been identified, a conversion factor can be determined.

Example 1.6.5

A child is prescribed a dosage of 12 mg of a certain drug per day and is allowed to refill his prescription twice. If there are 60 tablets in a prescription, and each tablet has 4 mg, how many doses are in the 3 prescriptions (original + 2 refills)?

[latex]\begin{array}{rrll} 3\text{ prescriptions}&=&3\text{ \cancel{pres.}}\times \dfrac{60\text{ \cancel{tablets}}}{1\text{ \cancel{pres.}}}\times \dfrac{4\text{ \cancel{mg}}}{1\text{ \cancel{tablet}}}\times \dfrac{1\text{ dosage}}{12\text{ \cancel{mg}}}&\text{This cancels all unwanted units} \\ \\ &=&\dfrac{3\times 60\times 4\times 1}{1\times 1\times 12}\text{ or }\dfrac{720}{12}\text{ dosages}&\text{Which reduces to} \\ \\ &=&60\text{ daily dosages}&\text{Solution} \\ \\ \end{array}[/latex]

Metric and Imperial (U.S.) Conversions

\[\begin{array}{rrlrrl} 12\text{ in}&=&1\text{ ft}\hspace{1in}&10\text{ mm}&=&1\text{ cm} \\ 3\text{ ft}&=&1\text{ yd}&100\text{ cm}&=&1\text{ m} \\ 1760\text{ yds}&=&1\text{ mi}&1000\text{ m}&=&1\text{ km} \\ 5280\text{ ft}&=&1\text{ mi}&&& \end{array}\]

Imperial to metric conversions:

\[\begin{array}{rrl} 1\text{ inch}&=&2.54\text{ cm} \\ 1\text{ ft}&=&0.3048\text{ m} \\ 1\text{ mile}&=&1.61\text{ km} \end{array}\]

\[\begin{array}{rrlrrl} 144\text{ in}^2&=&1\text{ ft}^2\hspace{1in}&10,000\text{ cm}^2&=&1\text{ m}^2 \\ 43,560\text{ ft}^2&=&1\text{ acre}&10,000\text{ m}^2&=&1\text{ hectare} \\ 640\text{ acres}&=&1\text{ mi}^2&100\text{ hectares}&=&1\text{ km}^2 \end{array}\]

\[\begin{array}{rrl} 1\text{ in}^2&=&6.45\text{ cm}^2 \\ 1\text{ ft}^2&=&0.092903\text{ m}^2 \\ 1\text{ mi}^2&=&2.59\text{ km}^2 \end{array}\]

\[\begin{array}{rrlrrl} 57.75\text{ in}^3&=&1\text{ qt}\hspace{1in}&1\text{ cm}^3&=&1\text{ ml} \\ 4\text{ qt}&=&1\text{ gal}&1000\text{ ml}&=&1\text{ litre} \\ 42\text{ gal (petroleum)}&=&1\text{ barrel}&1000\text{ litres}&=&1\text{ m}^3 \end{array}\]

\[\begin{array}{rrl} 16.39\text{ cm}^3&=&1\text{ in}^3 \\ 1\text{ ft}^3&=&0.0283168\text{ m}^3 \\ 3.79\text{ litres}&=&1\text{ gal} \end{array}\]

\[\begin{array}{rrlrrl} 437.5\text{ grains}&=&1\text{ oz}\hspace{1in}&1000\text{ mg}&=&1\text{ g} \\ 16\text{ oz}&=&1\text{ lb}&1000\text{ g}&=&1\text{ kg} \\ 2000\text{ lb}&=&1\text{ short ton}&1000\text{ kg}&=&1\text{ metric ton} \end{array}\]

\[\begin{array}{rrl} 453\text{ g}&=&1\text{ lb} \\ 2.2\text{ lb}&=&1\text{ kg} \end{array}\]

Temperature

Fahrenheit to Celsius conversions:

\[\begin{array}{rrl} ^{\circ}\text{C} &= &\dfrac{5}{9} (^{\circ}\text{F} – 32) \\ \\ ^{\circ}\text{F}& =& \dfrac{9}{5}(^{\circ}\text{C} + 32) \end{array}\]

For questions 1 to 18, use dimensional analysis to perform the indicated conversions.

• 7 miles to yards
• 234 oz to tons
• 11.2 mg to grams
• 1.35 km to centimetres
• 9,800,000 mm to miles
• 4.5 ft 2 to square yards
• 435,000 m 2 to square kilometres
• 8 km 2 to square feet
• 0.0065 km 3 to cubic metres
• 14.62 in 3 to square centimetres
• 5500 cm 3 to cubic yards
• 3.5 mph (miles per hour) to feet per second
• 185 yd per min. to miles per hour
• 153 ft/s (feet per second) to miles per hour
• 248 mph to metres per second
• 186,000 mph to kilometres per year
• 7.50 tons/yd 2 to pounds per square inch
• 16 ft/s 2 to kilometres per hour squared

For questions 19 to 27, solve each conversion word problem.

• On a recent trip, Jan travelled 260 miles using 8 gallons of gas. What was the car’s miles per gallon for this trip? Kilometres per litre?
• A certain laser printer can print 12 pages per minute. Determine this printer’s output in pages per day.
• An average human heart beats 60 times per minute. If the average person lives to the age of 86, how many times does the average heart beat in a lifetime?
• Blood sugar levels are measured in milligrams of glucose per decilitre of blood volume. If a person’s blood sugar level measured 128 mg/dL, what is this in grams per litre?
• You are buying carpet to cover a room that measures 38 ft by 40 ft. The carpet cost $18 per square yard. How much will the carpet cost?
• A cargo container is 50 ft long, 10 ft wide, and 8 ft tall. Find its volume in cubic yards and cubic metres.
• A local zoning ordinance says that a house’s “footprint” (area of its ground floor) cannot occupy more than ¼ of the lot it is built on. Suppose you own a [latex]\frac{1}{3}[/latex]-acre lot (1 acre = 43,560 ft 2 ). What is the maximum allowed footprint for your house in square feet? In square metres?
• A car travels 23 km in 15 minutes. How fast is it going in kilometres per hour? In metres per second?
• The largest single rough diamond ever found, the Cullinan Diamond, weighed 3106 carats. One carat is equivalent to the mass of 0.20 grams. What is the mass of this diamond in milligrams? Weight in pounds?

Answer Key 1.6

Long Descriptions

Celsius to Fahrenheit conversion scale long description: Scale showing conversions between Celsius and Fahrenheit. The following table summarizes the data:

[Return to Celsius to Fahrenheit conversion scale]

Intermediate Algebra (Convert to MathJax) Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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