Module MU: Measurements and Units
Module MU Learning Objectives:
- Write a comparison of two quantities as a ratio
- Find the unit price
- Solve problems using proportions
- Use dimensional analysis to convert between different units of measurement for length, mass, and capacity
- Use dimensional analysis to convert between units of measurement in the metric system
- Use dimensional analysis to convert between the metric and standard systems of measurement
- Use rates and unit rates to problem solve
Section MU.1 – Proportional Reasoning
Section MU.2 – Length, Weight & Capacity
Section MU.3 – The Metric System & Converting Between Systems
Section MU.4 – Problem Solving Using Rates and Dimensional Analysis
Section MU.1 – Proportional Reasoning
A ratio is an expression that compares two numbers using division.
Example: 7:10 or [latex]\frac{7}{10}[/latex] or “7 to 10” or 7/10
| Example 1 |
| There are 66 freshman and 54 sophomores in a school cafeteria. Express the ratio of the number of freshman to that of sophomores. |
| Given, the number of freshman = 66; and the number of sophomores = 54. The GCF of 66 and 54 is 6. Now, to simplify, divide the two terms by their GCF which is 6. This means, (66 ÷ 6)/(54 ÷ 6) = 11/9. Therefore, the ratio of the number of freshman to that of sophomores = 11 : 9.
[latex]\frac{66}{54}[/latex] ÷ [latex]\frac{6}{6}[/latex] = [latex]\frac{11}{9}[/latex] or 11 : 9 |
| Example 2 |
| Determine the ratio and write it in lowest terms, 17 dollars to 68 dollars. |
| [latex]\frac{17~dollars}{68~dollars}[/latex] ÷ [latex]\frac{17}{17}[/latex] = [latex]\frac{1}{4}[/latex] or 1 : 4 |
| You Try MU.1.A |
| Dale’s Donuts Shop made 88 glazed donuts and 32 crème filled donuts.
a. What is the ratio of glazed to crème filled donuts? b. What is the ratio of crème filled to glazed donuts |
A proportion is a mathematical comparison between two ratios. Two ratios are said to be in proportion when the two ratios are equal.
Example: [latex]\frac{4}{3}[/latex] = [latex]\frac{8}{6}[/latex]
| Solving a Proportion |
| When two ratios are equal to each other, the product obtained by cross multiplication are equal:
[latex]\frac{a}{b}[/latex] = [latex]\frac{c}{d}[/latex] if and only if a • d = b • c |
| Example 3 | ||||||||
| Solve the proportion [latex]\frac{x}{63}[/latex] = [latex]\frac{5}{9}[/latex] for the unknown value of x. | ||||||||
You can check your answer by using your calculator to divide 35 by 63 and 5 by 9, which equals [latex]\frac{35}{63}[/latex] = 0.5555… and [latex]\frac{5}{9}[/latex] = 0.5555… |
| You Try MU.1.B |
| Solve the proportion [latex]\frac{720}{57}[/latex] = [latex]\frac{240}{x}[/latex] for the unknown value of x. |
| Example 4 | ||
| Determine if each proportion is true or false:
a. [latex]\frac{2}{9}[/latex] = [latex]\frac{6}{25}[/latex] b. [latex]\frac{11}{2}[/latex] = [latex]\frac{55}{10}[/latex] |
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| Example 5 |
| Your car can drive 300 miles on a tank of 15 gallons. Express this as a rate and as a unit rate, both in miles per gallon. |
| Expressed as a rate: [latex]\frac{}{}[/latex]
We can divide to find a unit rate: [latex]\frac{20~miles}{1~gallon}[/latex], which can also be written as [latex]20\frac{miles}{gallon}[/latex], or 20 miles per gallon. |
Notice that, had we wanted to find the unit rate in gallons per mile, we would have had to invert the original rate:
[latex]\frac{15~gallons}{300~miles}[/latex] = [latex]\frac{15}{300}[/latex] gallons per mile = [latex]\frac{1}{20}[/latex] gallon per mile
| You Try MU.1.C |
| Find the unit rates in dollars per pound and pounds per dollar for 6 pounds for $5.29. If necessary, round your answers to the nearest hundredth. |
| Example 6 | ||
| You are buying 70 acres of farmland at a cost of $302,000.00.
a. What is the cost per acre? b. What would the cost be for 40 acres |
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Another way to solve this problem is to use the unit rate; as a continuation of the problem from part a. Since the unit rate was rounded, we have to be careful to use the calculator correctly and not round until the final answer of the problem. [latex]\frac{1}{$4,314.285714}[/latex] = [latex]\frac{40~acres}{$~x}[/latex] $302,000/70 acres = $4314.2… in the calculator display, keep the entire result, and multiply by 40. Your calculator will take the EXACT answer for the division 302,000/70 and multiply by 40 to get the correct result of $172,571.43 (rounded to the nearest cent in the very last step). |
| Example 7 | ||||
| Your car can drive 300 miles on a tank of 15 gallons.
a. How far can it drive on 40 gallons? b. How many gallons are needed to drive 50 miles? |
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We earlier found that 300 miles on 15 gallons gives a rate of 20 miles per gallon.
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| Example 8 | |||
| A baseball coach can buy 8 caps for his team at a cost of $66.00.
a. How much would the coach pay to buy caps for 21 players on his team? b. How many caps can the coach buy for $99.00? c. What is the unit price for each cap? |
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| You Try MU.1.D |
| One brand of microwave popcorn has 120 calories per serving. A whole bag of this popcorn has 3.5 servings. How many calories are in a whole bag of this microwave popcorn? |
| Example 9 | |||||||||||||||
A consumer priced potatoes in various quantities at a grocery store. Find the best price for each item based on the price per unit.
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To determine price per unit, divide the price by the size.
The best buy based on the lowest unit cost is 2-lbs at $1.105 dollars per pound. |
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| Population Density |
| Population density refers to the number of people who live in each square mile of an area, and is measured in people per square mile. |
| Example 10 |
| The population of Mesa, Arizona is 509,475 and its area is 133.1 square miles. What is the population density? Round to the nearest tenth. |
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Population Density = [latex]\frac{Population}{Area}[/latex] [latex]\frac{509,475~people}{133.1~mi^2}[/latex] = 3,827.76 people per square mile or [latex]\frac{people}{mi^2}[/latex] |
| Example 11 |
| Compare the electricity consumption per capita in China to the rate in Japan. |
| To address this question, we will first need data. From the CIA[1] website we can find the electricity consumption in 2011 for China was 4,693,000,000,000 KWH (kilowatt-hours), or 4.693 trillion KWH, while the consumption for Japan was 859,700,000,000, or 859.7 billion KWH. To find the rate per capita (per person), we will also need the population of the two countries. From the World Bank[2], we can find the population of China is 1,344,130,000, or 1.344 billion, and the population of Japan is 127,817,277, or 127.8 million.
Computing the consumption per capita for each country: China: [latex]\frac{4,463,000,000,000~KWH}{1,344,130,000~people}[/latex] ≈ 3,491.5 KWH per person Japan: [latex]\frac{859,700,000,000~KWH}{127,817,277~people}[/latex] ≈ 6,726 KWH per person While China uses more than 5 times the electricity of Japan overall, because the population of Japan is so much smaller, it turns out Japan uses almost twice the electricity per person compared to China. |
Section MU.1 Answers to You Try Problems
MU.1.A
a. 11 : 4
b. 4 : 11
MU.1.B
x = 19
MU.1.C
$0.88 per pound, 1.13 pounds per dollar
MU.1.D
420 calories
