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Interpreting
Ratios

Master the art of comparing quantities using ratios: learn to read, write, and simplify them for the SSAT.

SSAT Middle Level

What Is a Ratio?

A ratio is a comparison of two quantities that tells you how much of one thing there is relative to another. If a bag has 3 blue marbles and 5 red marbles, the ratio of blue to red is 3 to 5.

Three Ways to Write a Ratio

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

Visualizing Ratios

Blue marbles : Red marbles
Ratio = \(3 : 5\)

Count the groups: 3 blue compared to 5 red. A picture makes the comparison easy to see at a glance.

Part-to-Part vs Part-to-Whole

Part-to-Part
Compares one group to another group.
Blue to Red = \( 3 : 5 \)
Part-to-Whole
Compares one group to the total.
Blue to All = \( 3 : 8 \)

The Core Rule of Ratios

\(3 : 5\)
x 2
\(6 : 10\)
x 3
\(9 : 15\)
All represent the same ratio

Simplifying with the GCF

To simplify a ratio, divide both terms by their Greatest Common Factor (GCF). The result is the ratio in lowest terms.

\(12 : 18\)
÷ GCF(6)
\(2 : 3\)
Check: 12 ÷ 6 = 2 and 18 ÷ 6 = 3 ✓ Lowest terms: no common factor > 1

Ratios vs Fractions

Ratio Fraction
Meaning Compares any two quantities Part of a whole
Example \( 3 : 5 \) \( \dfrac{3}{8} \)
Can exceed 1? Yes (\( 7 : 2 \)) Yes (improper)
Order matters? Yes: \( 3:5 \neq 5:3 \) Yes: \( \frac{3}{5} \neq \frac{5}{3} \)
Simplify how? Divide both terms by GCF Divide numerator & denominator by GCF

How to Interpret & Simplify

1
Identify the two quantities being compared and note their order.
2
Determine the type: part-to-part or part-to-whole?
3
Find the GCF of the two numbers.
4
Divide both terms by the GCF to get lowest terms.
5
Verify: confirm the two simplified numbers share no common factor greater than \(1\).

Example 1: Classroom Ratio

A class has 14 boys and 21 girls. What is the ratio of boys to girls in simplest form?

\(14 : 21\)
÷ GCF(7)
\(2 : 3\)
14 ÷ 7 = 2 and 21 ÷ 7 = 3 Answer: 2 : 3

Example 2: Recipe Ratio

A recipe uses 4 cups of flour for every 6 cups of sugar. Express the ratio of flour to sugar in lowest terms.

1
Identify quantities: flour = 4, sugar = 6.
2
Find GCF: factors of 4 are {1, 2, 4}; factors of 6 are {1, 2, 3, 6}. GCF = 2.
3
Divide: \(4 \div 2 = 2\) and \(6 \div 2 = 3\).
4
Result: \(2 : 3\) (flour to sugar).

Example 3: Part-to-Whole

A bag contains 8 blue, 12 red, and 20 green candies. What is the ratio of red candies to the total?

1
Total: \(8 + 12 + 20 = 40\).
2
Ratio (red to total): \(12 : 40\).
3
GCF of 12 and 40: factors of 12 = {1,2,3,4,6,12}; factors of 40 = {1,2,4,5,8,10,20,40}. GCF = 4.
4
Simplify: \(12 \div 4 = 3\) and \(40 \div 4 = 10\). Answer: \(3 : 10\).

Example 4: Using a Ratio to Find Quantities

The ratio of cats to dogs at a shelter is \(5 : 3\). If there are 40 animals total, how many cats are there?

1
Total parts: \(5 + 3 = 8\) parts.
2
Value per part: \(40 \div 8 = 5\) animals per part.
3
Cats: \(5 \times 5 = 25\) cats.
4
Check: Dogs = \(3 \times 5 = 15\). Total = \(25 + 15 = 40\). ✓

Common Mistakes

✕ Reversing Order
Writing \(5 : 3\) when the question says "dogs to cats" and there are 3 dogs. Order matters.
✕ Wrong Total
Using a part instead of the whole for part-to-whole ratios. Always add all groups first.
✕ Not Simplifying
Leaving \(12 : 18\) instead of \(2 : 3\). The SSAT expects lowest terms.

Summary at a Glance

Concept Key Idea Example
Ratio notation Three equivalent forms \(3 : 5\), "3 to 5", \(\frac{3}{5}\)
Equivalent ratios Multiply/divide both terms \(3:5 = 6:10 = 9:15\)
Simplifying Divide both by GCF \(12:18 \to 2:3\)
Part-to-Part One group vs another Boys : Girls = \(2:3\)
Part-to-Whole One group vs total Boys : All = \(2:5\)
Finding quantities Total ÷ sum of parts \(40 \div 8 = 5\) per part

Question 1 of 5

A jar has 10 red and 15 blue marbles. What is the ratio of red to blue in simplest form?

Question 2 of 5

A class has 12 boys and 18 girls. What is the ratio of boys to the total number of students in simplest form?

Question 3 of 5

Which ratio is equivalent to \(4 : 7\)?

Question 4 of 5

The ratio of fiction to nonfiction books on a shelf is \(5 : 2\). If there are 35 books total, how many are nonfiction?

Question 5 of 5

Simplify the ratio \(36 : 48\).