# 7-4 Comparing and Scaling - Concepts and Explanation

## Ratio

A comparison of two quantities

### Example

Ratios can be written in several forms. You can write the ratio of 3 cups of water
to 2 cups of lemonade concentrate as 2 to 3, 2 : 3, or ^{2}/_{3}

## Proportions

A proportion is a statement of equality between two ratios.

### Example

Kendra takes 70 steps on the treadmill to run 0.1 mile. When her workout is done, she has run 3 miles. How many steps has she taken?

Proportion: ^{70 steps}/_{0.1 mile} = ^{x steps}/_{3 miles}

^{70 steps x 30}/_{0.1 miles x 30} = ^{21.00 steps}/_{3 miles} Solution of the proportion

## Two Types of Ratios

Ratios can be *part-to-part* or *part-to-whole* comparisons. Part-to-whole comparisons can be written as fractions or percents. Part-to-part
comparisons can be written in fraction form, but do not represent a fraction.

### Example

The ratio of concentrate to water in a mix for lemonade is 3 cups concentrate to 16 cups water. What fraction of the mix is concentrate?

^{3/}_{16} is the part-to-part comparison. This does not mean that the fraction of mix that
is concentrate is ^{3/}_{16}. Find the total, 19 cups, to write the fraction of the mix that is concentrate. Write
a part-to-whole comparison using a fraction, ^{3}/_{19}, or a percent, 3 ÷ 19 = 0.15789 ≈ 15.8%, to describe the part that is concentrate.

## Rate

A comparison of measures with two different units

### Example

Examples of rates: miles to gallons, sandwiches to people, dollars to hours, calories to ounces, kilometers to hours

## Unit Rate

A rate in which the second quantity is 1 unit

### Example

Students sometimes find unit rates difficult because they have two options when dividing the two numbers of a rate. Tracking the units helps students think through such situations. The goal is to build flexibility in using either set of unit rates to compare the quantities.

Sascha rides 6 miles in 20 minutes during the first leg of his bike ride. During the second leg, he rides 8 miles in 24 minutes. During which leg is Sascha faster?

^{6 miles}/_{20 minutes} = 0.3 miles per minute^{8 miles}/_{24 minutes} = 0.333 miles per minute

The times, 1 minute, are the same, so 8 miles in 24 minutes is faster.

You can divide the other way as well:

^{20 minutes}/_{6 miles} = 3.333 minutes per mile^{24 minutes}/_{8 miles} = 3 minutes per mile

The distances, 1 mile, are the same, and 3 minutes per mile is faster.

## Scaling Ratios (and Rates)

Finding a common denominator or common numerator to make comparisons easier

### Example

Which is cheaper, 3 roses for $5 or 7 roses for $9 ?

Scale the costs to be the same by finding a common denominator. Use a common multiple of 5 and 9:

^{3 roses/}_{$5} = ^{3 roses x 9}/ _{$5 x 9 }= ^{27 roses}/_{$45}; ^{7 roses}/_{$9} = ^{7 roses x 5}/_{$9 x 5 }= ^{35 roses}/_{$45}

7 roses for $9 gives more roses for the same amount of money. Or, scale the numerators to be the same:

^{3 roses}/_{$5} = ^{3 roses x 7}/_{$5 x 7} = ^{21 roses}/_{$35}; ^{7 roses}/_{$9} = 7 roses x 3/$9 x 3 = 21 roses/$27

21 roses for $27 is cheaper than 21 roses for $35.

## Proportional Relationship

A relationship in which you multiply one variable by a constant number to find the value of another variable

### Example

The price of one pizza is $13.

To find the cost C of any number of pizzas n, multiply the number of pizzas by 13.
The unit rate 13 is also called the constant of proportionality, *k*. The relationship appears as a straight line on a graph. The equation can be written
as *y* = *kx*. In this case,* C *= 13*n*.