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Ratios & Rates
in Context

Learn to interpret and apply ratios and rates in real-world situations: from recipes to speed to unit pricing.

ISEE Middle Level

What Is a Ratio?

A ratio is a comparison of two quantities. It tells you how much of one thing there is relative to another. You can write a ratio three ways:

Word Form
"3 to 5"
Colon Form
\( 3 : 5 \)
Fraction Form
\( \dfrac{3}{5} \)

Seeing Ratios Visually

3 blue 5 green :

In this picture, the ratio of blue circles to green circles is \( 3 : 5 \). The order matters: reversing it to \( 5 : 3 \) changes the meaning entirely.

What Is a Rate?

A rate is a special ratio that compares two quantities with different units. Rates tell you how one measurement changes relative to another.

Speed
\( 60 \) miles per \( 1 \) hour
Price
\( \$2.50 \) per \( 1 \) pound
Heart Rate
\( 72 \) beats per \( 1 \) minute

The Unit Rate

A unit rate is a rate where the second quantity is exactly 1. To find a unit rate, divide both quantities so the denominator becomes \( 1 \).

Rate
\( \frac{\$7.50}{3 \text{ lb}} \)
 Divide
\( \div 3 \)
 Unit Rate
\( \frac{\$2.50}{1 \text{ lb}} \)

Ratio vs. Rate

Ratio Rate
Definition Compares two quantities Compares quantities with different units
Units Same or no units Different units (miles/hour, $/lb)
Example \( 2 : 3 \) boys to girls \( 60 \) miles per hour
Key Word "to" "per"

How to Interpret in Context

1
Identify the quantities. What two things are being compared? Read for labels like "for every," "per," or "to."
2
Note the order. The first number named goes on the left (or top). Ratios are order-dependent.
3
Check for different units. If the units differ, it is a rate. If same or no units, it is a ratio.
4
Simplify if needed. Divide both parts by their GCF to write the ratio in simplest form.
5
Find the unit rate if the problem asks "per one." Divide the first quantity by the second.

Example 1: Recipe Ratio

A lemonade recipe calls for 4 cups of water and 1 cup of lemon juice. What is the ratio of water to lemon juice?

4 cups water : 1 cup juice

Example 2: Speed Rate

A car travels 180 miles in 3 hours. What is the car's unit rate (speed)?

0 hr 0 mi 1 hr 60 mi 2 hr 120 mi 3 hr 180 mi
\( \frac{180 \text{ miles}}{3 \text{ hours}} = 60 \text{ mph} \)

Example 3: Best Buy Problem

Which is a better deal: 6 apples for $3.00 or 10 apples for $4.50?

Deal A
\( \dfrac{\$3.00}{6} = \$0.50 \) per apple
Deal B
\( \dfrac{\$4.50}{10} = \$0.45 \) per apple

Equivalent Ratios

Two ratios are equivalent if you can multiply or divide both parts by the same number to get from one to the other.

2 : 3 original × 2 4 : 6 × 2 × 5 2 : 3 original 10 : 15 × 5 = = All equivalent!

Part-to-Part vs. Part-to-Whole

A class has 12 boys and 8 girls. There are different ratios you can form:

Part-to-Part
Boys to girls: \( 12 : 8 = 3 : 2 \). Compares one group to another group.
Part-to-Whole
Boys to total: \( 12 : 20 = 3 : 5 \). Compares one group to the entire set.

Scaling Ratios in Real Life

A map uses a scale of \( 1 \) inch \( = 50 \) miles. If two cities are \( 3.5 \) inches apart on the map, how far apart are they in reality?

0 in 1 in 2 in 3 in 3.5
\( 3.5 \times 50 = 175 \text{ miles} \)

Setting Up a Proportion

A proportion is an equation saying two ratios are equal. Use proportions to find unknown values.

Known Ratio
\( \frac{3 \text{ cups}}{2 \text{ batches}} \)
 sugar to batches = Unknown
\( \frac{x \text{ cups}}{5 \text{ batches}} \)
 same rate!

Common Mistakes

✕ Flipping the Order
Writing \( 5 : 3 \) when the problem says "3 to 5." The order of a ratio must match the order stated in the problem.
✕ Confusing Ratio and Rate
Saying "60 to 3" instead of "60 miles per 3 hours." Rates need both units labeled to be meaningful.
✕ Not Simplifying
Leaving a ratio as \( 6 : 10 \) when it should be \( 3 : 5 \). Always check if both parts share a common factor.

Summary Reference

Concept Key Idea Example
Ratio Compare same-unit quantities \( 3 : 5 \)
Rate Compare different-unit quantities \( 60 \text{ mph} \)
Unit Rate Denominator equals 1 \( \$2.50/\text{lb} \)
Equivalent Ratios Same value, different numbers \( 2:3 = 4:6 \)
Proportion Two equal ratios: cross-multiply \( \frac{3}{2} = \frac{x}{5} \)

Question 1 of 5

A fruit basket has 8 apples and 12 oranges. What is the ratio of apples to oranges in simplest form?

Question 2 of 5

A cyclist rides 45 miles in 3 hours. What is the cyclist's unit rate?

Question 3 of 5

Store A sells 5 notebooks for $8.00. Store B sells 3 notebooks for $5.25. Which store has the better price per notebook?

Question 4 of 5

A recipe uses flour and sugar in the ratio \( 5 : 2 \). If you use 15 cups of flour, how many cups of sugar do you need?

Question 5 of 5

A class has 18 students. The ratio of students who play sports to those who do not is \( 2 : 1 \). How many students play sports?