Tennessee 6 - 2020 Edition 3.03 Ratio tables
Lesson

A ratio compares the relationship between two values. It compares how much there is of one thing compared to another.

For example, if a pie recipe calls for $2$2 tablespoons of brown sugar per $1$1 apple, we could write this as a ratio

$2:1$2:1

What if we wanted to make more than one pie?

Well, we'd need to keep the ingredients in equivalent ratios. For example, to make two pies, we'd need $4$4 tablespoons of sugar and $2$2 apples.

An easy way to display information about ratios is using a ratio table. Let's display our information from the above example:

 Sugar $2$2 $4$4 $6$6 $8$8 Apples $1$1 $2$2 $3$3 $4$4

Ratio tables display the relationship between the two quantities and help us find the constant multiplicative factor between them. This constant multiplicative factor, which is also called the constant of proportionality, is a constant positive multiple between the two variables. As we've already learned, ratios can also be represented as fractions, and by simplifying the fraction, we can work out the constant of proportionality.

In the table above, $\frac{4}{8}=\frac{3}{6}$48=36$=$=$\frac{2}{4}$24$=$=$\frac{1}{2}$12. In other words, the number of apples will always be half as much as the number of tablespoons of sugar. In other words, the constant multiplicative factor is $\frac{1}{2}$12. That means if I put $20$20 tablespoons of sugar in my recipe, I know I would need $10$10 apples because that is half of $20$20. Similarly, if I used $7$7 apples, I would need double the amount of sugar, which is $14$14 tablespoons.

#### Worked examples

##### Question 1

Consider the ratio table below.

a) Fill in the missing values from the table below

 Pens $10$10 $20$20 $30$30 $40$40 $50$50 $\editable{}$ Cost (dollars) $\editable{}$ $11.60$11.60 $\editable{}$ $\editable{}$ $29.00$29.00 $58.00$58.00

Think: What is the common multiplicative factor between the two quantities?

Do:

$11.60\div20=0.58$11.60÷​20=0.58

$29.00\div50=0.58$29.00÷​50=0.58

So our common multiplicative factor is $0.58$0.58. In other words, one pen costs $58$58c.

To work out the cost, we need to multiply the number of pens by $0.58$0.58.

For example, the cost of $10$10 pens would be $10\times0.58=\$5.80$10×0.58=$5.80.

Conversely, to work out the number of pens, we need to divide the cost by $0.58$0.58.

For example, $\$58$$58 would buy 58\div0.58=10058÷​0.58=100 pens.  Pens 1010 2020 3030 4040 5050 100100 Cost (dollars) 5.805.80 11.6011.60 17.4017.40 23.2023.20 29.0029.00 58.0058.00 b) Calculate the cost of buying 9090 pens. Think: If 1010 pens cost \5.80$$5.80, how much would $90$90 pens cost?

Do:

If we think, $90$90 pens is $9$9 times more than $10$10 pens. This means we need to increase the cost of $10$10 pens by $9$9 as well. $5.80\times9=\$52.20$5.80×9=$52.20.

We can also solve this algebraically. Let's let $x$x be the cost of $90$90 pens.

 $10:5.80$10:5.80 $=$= $90:x$90:x $\frac{10}{5.80}$105.80​ $=$= $\frac{90}{x}$90x​ $\frac{5.80}{10}$5.8010​ $=$= $\frac{x}{90}$x90​ $\frac{5.80\times90}{10}$5.80×9010​ $=$= $x$x $x$x $=$= $\$52.20$$52.20 c) How much would you expect to pay for 55 pens? Think: The cost of 55 pens would be half the price of 1010 pens. Do: 5.80\div2=\2.905.80÷​2=2.90 #### Practice questions ##### Question 2 Ryan and Valerie are preparing for a party. Ryan blows up 1212 balloons in 1515 minutes. Valerie blows up 2424 balloons in 2828 minutes. Who is blowing up balloons faster? 1. Complete the table for the number of balloons Ryan blows up in each time period. Assume that he keeps blowing up balloons at a constant rate.  Time (minutes) Balloons 00 1515 \editable{} 4545 6060 \editable{} 1212 \editable{} \editable{} \editable{} 2. Complete the table for the number of balloons Valerie blows up in each time period. Assume that she keeps blowing up balloons at a constant rate.  Time (minutes) 00 2828 5656 8484 \editable{} Balloons \editable{} \editable{} \editable{} \editable{} 9696 3. Using the table above, who is blowing up balloons the fastest? Valerie A Ryan B Valerie A Ryan B ##### Question 3 Fill in the ratio table below, then use it to answer the following questions. 1.  US dollar () Japanese yen (¥) 11 22 33 44 55 \editable{} \editable{} 320.40320.40 \editable{} \editable{} 2. How many Japanese yen will you be able to buy for$$20$20? 3. Beth wants to buy a pair of jeans that costs$4272$4272 Japanese yen. What is this equivalent to in US dollars? ##### Question 4 Kate and Laura are selling cakes at a bake sale. For every$6$6 cakes that Kate sells, she will make$\$15$$15. For every 2424 cakes that Laura sells, she will make \53$$53. Whose cakes are more expensive? 1. Fill in the missing gaps in the table for Kate.  Cakes sold Earnings (dollars)$6$6$\editable{}18$18$\editable{}30$30$\editable{}30$30$\editable{}60$60$75$75 2. Fill in the missing gaps in the table for Laura.  Cakes sold Earnings (dollars)$\editable{}48$48$72$72$96$96$120$120$53$53$\editable{}159$159$212$212$265\$265
3. Whose cakes are more expensive?

Laura's

A

Kate's

B

Laura's

A

Kate's

B

### Outcomes

#### 6.RP.A.3

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g. By reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

#### 6.RP.A.3.A

Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.