6 Ratio and Proportion

Introduction:

This module will cover proportionality concepts and situations, ratios, equivalent ratios, and proportion equations. You will also learn how to find the missing factor in a proportion. You will be presented with content and examples and then given opportunities to do practice exercises.

Chapter Contents:

• Topic 1: Proportionality Concept and Situations
  • 1.1 Ratio
  • 1.2 Equivalent Ratios
  • 1.3 Rates
• Topic 2: Proportion Equation Scenarios
  • 2.1 Ratios, Rates, and Proportions From a Mathematical Perspective
  • 2.2 Finding the Missing Factor in a Proportion
  • 2.3 Writing the Proportion Equation
• Topic 3: Other Situations of Proportionality
  • 3.1 Percentages
  • 3.2 Unit Conversions
  • 3.3 Similar Triangles
• Test: Outcome 5

Topic 1: Proportionality Concept and Situations

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1.1 Ratio

A ratio is an association between two or more quantities. There are many ways to describe a situation in terms of ratios. For example, look at this collection:

 

Here are some correct ways to describe the collection:

• The ratio of squares to circles is 6:3
• The ratio of circles to squares is 3 to 6.

Notice that the shapes can be arranged in equal groups, which allow us to describe the shapes using other numbers.

 

• There are 2 squares for every 1 circle.
• There is 1 circle for every 2 squares.

A ratio is an association between two or more quantities.

For example, the ratio 3:2 could describe a recipe that uses 3 cups of flour for every 2 eggs, or a boat that moves 3 meters every 2 seconds. One way to represent the ratio 3:2 is with a diagram that has 3 blue squares for every 2 green squares.

 

1.1 Practice Exercises

1.2 Equivalent Ratios

A recipe for fizzy juice says, “Mix 5 cups of cranberry juice with 2 cups of soda water.”

To double this recipe, we would use 10 cups of cranberry juice with 4 cups of soda water. To triple this recipe, we would use 15 cups of cranberry juice with 6 cups of soda water.

This diagram shows a single batch of the recipe, a double batch, and a triple batch:

 

We say that the ratios 5:2, 10:4, and 15:6 are equivalent. Even though the amounts of each ingredient within a single, double, or triple batch are not the same, they would make fizzy juice that tastes the same.

 

When mixing colors, doubling or tripling the amount of each color will create the same shade of the mixed color. In fact, you can always multiply the amount of each color by the same number to create a different amount of the same mixed color.

For example, a batch of dark orange paint uses 4 ml of red paint and 2 ml of yellow paint.

To make two batches of dark orange paint, we can mix 8 ml of red paint with 4 ml of yellow paint.

To make three batches of dark orange paint, we can mix 12 ml of red paint with 6 ml of yellow paint.

Here is a diagram that represents 1, 2, and 3 batches of this recipe.

 

We say that the ratios 4:2, 8:4, and 12:6 are equivalent because they describe the same color mixture in different numbers of batches, and they make the same shade of orange.

 

Summary: Ratio and Equivalent Ratios

All ratios that are equivalent to a:b can be made by multiplying both a and b by the same number.

For example, the ratio 18:12 is equivalent to 9:6 because both 9 and 6 are multiplied by the same number: 2.

3:2 is also equivalent to 9:6, because both 9 and 6 are multiplied by the same number.

Is 18:15 equivalent to 9:6?

No, because 18 is 9 X 2, but 15 is not 6 X 2.

 

 

1.3 Rates

Suppose a train traveled 100 meters in 5 seconds at a constant speed. To find its speed in meters per second, we can find a unit rate. A unit rate is a ratio of distance (in this case) to one second. This can be found by dividing the numbers.

100 meters : 5 seconds

100 divided by 5 is 20.

100 meters : 5 seconds is equivalent to 20 meters:1 second. The unit rate of speed of the train is 20 meters per second, or 20 m/s.

Glossary for Rates:

Definition: Meters per Second

Meters per second is a unit for measuring speed. It tells us how many meters an object goes in one second. For example, a person walking 3 meters per second is going faster than another person walking 2 meters per second.

Definition: Per

The word per means “for each.” For example, if the price is $5 per ticket, that means you will pay $5 for each ticket. Buying 4 tickets would cost $20, because 4 x 5 = 20.

Definition: Unit Price

The unit price is the cost for one item or for one unit of measure. For example, if 10 feet of chain link fencing cost $150, then the unit price is 150 ÷ 10, or $15 per foot.

 

 

Sometimes we want to know whether two situations are described by the same rate. To do that, we can write an equivalent ratio for one or both situations so that one part of their ratios has the same value. Then we can compare the other part of the ratios.

The other option is to divide the two values to find a unit rate, and compare the unit rates.

 

Practice Exercises: Proportionality Concept and Situations

1. A slug travels 3 centimeters in 3 seconds. A snail travels 6 centimeters in 6 seconds. Both travel at constant speeds. “The snail was traveling faster because it went a greater distance.” Do you agree with this statement? Explain or show your reasoning.

2. If you blend 2 scoops of chocolate ice cream with 1 cup of milk, you get a milkshake with a stronger chocolate flavor than if you blended 3 scoops of chocolate ice cream with 2 cups of milk. Explain or show why.

3. There are 2 mixtures of light purple paint. Which mixture is a lighter shade of purple? Explain your reasoning.

Mixture A is made with 5 cups of purple paint and 2 cups of white paint.

Mixture B is made with 15 cups of purple paint and 8 cups of white paint.

 

Solution:

1. No, even though snail went further and it took longer, their speeds are the same: 1 cm/s.

2. Compare unit rate of mixture 1 (2 scoops: 1 cup) and mixture 2 (3 scoops: 2 cups or 1.5 scoops: 1cups). In the first mixture, there is more chocolate per cup of milk. It is the stronger flavored milk shake.

3. Compare the unit rates.

Mixture A

5 purple : 2 white = 2.5 purple : 1 white

Mixture B

15 purple : 8 white = 1.9 purple : 1 white

Mixture B is a lighter shade because it has less purple paint per cup of white.

Topic 2: Proportion Equation and Scenarios

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To solve problems about something happening at the same rate, we often need:

• Two pieces of information that allow us to write a ratio that describes the situation.
• A third piece of information that gives us one number of an equivalent ratio. Solving the problem often involves finding the other number in the equivalent ratio.

Example 1

Suppose we are making a large batch of juice and the recipe says, “Mix 5 cups of cranberry juice with 2 cups of soda water.” We know that the ratio of cranberry juice to soda water is 5:2. As in the example with the distance travelled by the train, we can divide to find how many cups of cranberry juice are required for one cup of water. 5 divided by 2 is 2.5. So, 2 and a half cups of cranberry juice is required for one cup of water. This is the unit rate.

We still need to know something about the size of the large batch. If we use 16 cups of soda water, what number goes with 16 to make a ratio that is equivalent to 5:2?

To make this large batch taste the same as the original recipe, once we have the unit rate, we can multiply by the number of cups of water to fund the required cups of cranberry juice. 2.5 x 16 = 40. we would need to use 40 cups of cranberry juice.

This data can be stored in a table. Tables are useful when you have a rate, but don’t want to do a calculation for every scenario. You can look up the pre calculated values in the table.

cranberry juice (cups)	soda water (cups)
1	2.5
2	5
3	7
4	10
…	…
16	40

For each row of the table, the number on the left is multiplied by 2.5 to obtain the number on the right. We can keep adding rows until we reach the values we are looking for.

Glossary for Proportion Equation and Scenarios 

Definition: Table

A table organizes information into horizontal rows and vertical columns. The first row or column usually tells what the numbers represent. For example, here is a table showing the tail lengths of three different pets. This table has four rows and two columns.

pet	tail length (inches)
dog	22
cat	12
mouse	2

Practice Exercises

1. Create a table: A car has a gas consumption of 9 L to 100 km.

a) What will the two columns be titled?

b) Create 5 rows for values 100 km, 200 km, …, 500 km.

c) Calculate the gas consumption for each row.

Solution:

Kilometers	Gas Consumption
100	9
200	18
300	27
400	36
500	45

2.1 Ratios, Rates, and Proportions From a Mathematical Perspective

Tables can be useful in certain situations, but be cumbersome. Another more direct way to find a number in a proportion is to use algebra to solve a proportion equation.

A proportion is a statement that two ratios or rates are equal. The following two examples show how to read proportions.

 

2.2 Finding the Missing Factor in a Proportion

Many practical problems can be solved by writing the given information as propor­tions. Such proportions will be composed of three specified numbers and one unknown number. It is customary to let a letter, such as x, represent the unknown number. An example of such a proportion is

x/4 = 20/16

This proportion is read as ” x is to 4 as 20 is to 16.”

There is a method of solving these proportions that is based on the equality of fractions. Remember, 2 fractions are equivalent if you can multiply (or divide) the same number to top and bottom to obtain the new fraction. Another way to tell if 2 fractions are equivalent is if their cross products are equal. For example,

 

Notice that in a proportion that contains three specified numbers and a letter representing an unknown quantity, that regardless of where the letter appears, you can find the unknown value.

The missing factor in a proportion can be determined by dividing the product by the known factor, that is, if x represents the missing factor, then

x = (product) ÷ (known factor)

 

A way to remember this might be this diagram:

 

Cross-multiply the two numbers that are diagonally opposite and divide by the third number

What is actually happening, using algebra, is this:

 

But only when you have a proportion equation, you can take the shortcut above.

Example 1

=  (Find the cross-product) 

16x = 3 × 64

16x = 192 (Divide 192 by 16)

x = 12

The means that

16 : 3 = 64 : 12 or, 16 is to 3 as 64 is to 12.

Example 2

=  (Find the cross-product) 

8x = 9 × 40

8x = 360 (Divide 360 by 8)

x = 45

The means that

9 : 8 = 45 : 40 or, 9 is to 8 as 45 is to 40.

2.3 Writing the proportion equation

When setting up a proportion equation, you must have the same units on the top of both sides of the equation, and the same units on the bottom of each side of the equation.

Both cross products produce a statement of the type:

(unit type 1) x (unit type 2) = (unit type 1) x (unit type 2)

However, if we write the same type of units on different sides, such as,

(unit type 1) / (unit type 2) = (unit type 2) / (unit type 1)

We can see that this is an incorrect comparison by observing the following example: It is incorrect to write:

2 hooks/ 3 poles = 6 poles / 4 hooks

for two reasons.

1. The cross product is numerically wrong: (2 x 4 ≠ 3 x 6)
2. The cross product produces the statement “hooks are to hooks as poles are to poles,” which makes no sense.

2.3 Practice Exercises: Proportion Equation and Scenarios

A. For the following 8 problems solve each proportion.

1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 

B. For the following 5 problems, express each sentence as a proportion then solve the proportion

1. 5 hats are to 4 coats as x hats are to 24 coats
2. x cushions are to 2 sofas as 24 cushions are to 16 sofas.
3. 1 spacecraft is to 7 astronauts as 5 spacecraft are to x astronauts.
4. 56 microchips are to x circuit boards as 168 microchips are to 3 circuit boards.
5. 18 calculators are to 90 calculators as x students are to 150 students.
6. x dollars are to $40,000 as 2 sacks are to 1 sack.

C. Indicate whether the proportion is true or false

1. 3/16 = 12/64
2. 2/15 = 10/75
3. 1/9 = 3/30
4. 6 knives/7 forks = 15 forks/12 knives
5. 33 miles/1 gallon = 99 miles/3 gallons
6. 320 feet/5 seconds = 65 feet/1 second

D. Practice

1. A caterer needs to buy 21 pounds of pasta to cater a wedding. At a local store, 8 pounds of pasta cost $12. How much will the caterer pay for the pasta there? Write a ratio for the given information about the cost of pasta and find the answer.
2. Lin is reading a 47-page book. She read the first 20 pages in 35 minutes. If she continues to read at the same rate, will she be able to complete this book in under 1 hour? If so, how much time will she have left? If not, how much more time is needed?
3. Diego can type 140 words in 4 minutes. At this rate, how long will it take him to type 385 words? How many words can he type in 15 minutes?

Solution:

Part A

1. 2
2. 7
3. 8
4. 20
5. 8
6. 35
7. 27
8. 3

Part B

1. 5/4 = x/24, x=30
2. x/2 = 24/16, x=3
3. 1/7 = 5/x, x=35
4. 56/x = 168/3, x=1
5. 18/90 = x/150, x=30
6. x/40000 = 2/1, x=80000

Part C

1. T,
2. T
3. F
4. Proportion improper
5. T
6. F

Part D

1. $12/8 lbs = x/21 lbs, x=31.50. It will cost $31.50.
2. 35 min/20 pages = x/47 pages, x= 82.25. It will take 82 minutes to finish the book. It won’t be done in under 1 hour. She will need 22 more minutes.
3. 4 min/140 words = x min/385 words, x = 11. It will take 11 minutes to write 385 words. 4 min/140 words = 15 min/x words, x=525. He can type 525 words in 15 minutes.

Topic 3: Other Situations of Proportionality

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There are so many situations where the proportion equation in math is useful, although we may not have realized we were using it. Here are some of them.

3.1 Percentages

The word percent actually comes from two words “per” and “cent.” From a previous section, the word “per” means “for every.” The word “cent” means, in French, one hundred. (Remember, when we still had pennies, they were called cents, and one hundred of the made up one dollar.) The word “percent” literally means “for every one hundred.”

In order to find a percentage, we usually just divide. For example, the statement “7 out of 10 dentists recommend using toothpaste” means that 70 percent of dentists recommend using toothpaste.

 

What is really happening, with the proportion equation is:

7/10 = x/100

If 7 out of 10 dentists recommend using toothpaste, then how many out of 100 recommend using toothpaste?

Using our short-cut from before:

3.2 Unit Conversions

A conversion factor can be considered a rate. For example, there are 25.4 millimetres in one inch. This can be expressed as:

25.4mm : 1inch

If we want to convert inches to millimetres, we can use the proportion equation where one side is the conversion rate.

Example 1

Convert 37.627 inches into millimetres.

Set up the proportion equation. Remember to keep the same units on top of each side, and the same units on the bottom of each side. (Note, however, that it doesn’t matter if the inches appear on the top of the equation or the bottom of the equation, so long as both sides are the same.)

25.4 millimetres/1 inch = x millimetres/37.627 inches

Using our short cut from before:

There are roughly 956 millimetres in 37.627 inches.

Here are a couple of things to note:

• Generally, round the answer to the nearest whole number when measuring in millimetres. It is impossible to measure less than a millimeter on a standard ruler.
• When you have a unit rate (25.4 millimetres per one inch) it is cumbersome to use the proportion equation. You would normally just multiply 25.4 X 37.627.

3.3 Similar Triangles

If a triangle has been expanded, the sides are in proportion. (What expanded means in this case is like zooming into a picture on your cell phone. The picture looks the same, only bigger.)

So, when we have two triangles that are similar, we can use the proportion equation to find the unknown side.

Example 2

Find the unknown side.

The unknown side is approximately 29.14. This will come up again in the trigonometry chapter.

3.1 – 3.3 Practice Exercises: Other Situations of Proportionality

1. A student received 37 out of 48 on an exam. What percent did the student achieve?
2. The ratio of pounds to kilograms is 2.2 lbs to every 1 kg. Convert 74.8 lbs to kg.
3. The following two triangles are similar (they have the same angles). Find the unknown side.

Solution:

1. 37/48 = x/100, x=77%
2. 2.2 lbs/1 kg = 74.8 lbs/x, x=34
3. 3/4 = 9/x, x=12

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