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Ratios &
Comparisons

Master the art of using ratios to compare quantities, simplify expressions, and solve real-world problems on the ISEE Upper Level.

ISEE Upper Level

What Is a Ratio?

A ratio is a comparison of two quantities by division. It tells you how much of one thing there is relative to another. If a basket has 3 apples and 5 oranges, the ratio of apples to oranges is \(\frac{3}{5}\), written as \(3 : 5\).

Visualizing Ratios with Bar Models

Ratio 3 : 5 Apples 1 1 1 Oranges 1 1 1 1 1 Each block = 1 equal part

Bar models make ratios concrete. Each block is one equal part. To find actual quantities, multiply each part by the same value.

Why Ratios Matter

Recipes
Scaling a recipe from 4 to 12 servings requires multiplying every ingredient by the same ratio.
Maps
A scale of \(1 : 50{,}000\) means 1 cm on the map equals 50,000 cm in real life.
Finance
Price-to-earnings ratios help investors compare the value of different companies.

Three Ways to Write a Ratio

Colon Notation
\(a : b\)
Example: \(3 : 5\)
Fraction Form
\(\dfrac{a}{b}\)
Example: \(\dfrac{3}{5}\)
Word Form
"a to b"
Example: "3 to 5"

Simplifying with the GCF

To simplify a ratio, divide both terms by their greatest common factor (GCF). A simplified ratio has no common factors other than 1.

12 : 18 GCF = 6 ÷ 6 each 2 : 3 \(\frac{12}{6} : \frac{18}{6} = 2 : 3\)

Part-to-Part vs Part-to-Whole

Part-to-Part 3 : 5 Apples to Oranges 3 : 5 Part-to-Whole 3 : 8 Apples to Total Fruit 3 : 8

Solving Ratio Problems: Step by Step

1
Identify the ratio. Determine which quantities are being compared and in what order.
2
Write it in fraction form. Express \(a : b\) as \(\frac{a}{b}\) for easier manipulation.
3
Simplify. Divide both terms by their GCF to reach lowest terms.
4
Scale if needed. Multiply both terms by the same factor to find actual quantities.
5
Verify. Check that your answer matches the original relationship and units.

Example 1: Finding Quantities

The ratio of red marbles to blue marbles in a bag is \(2 : 7\). If there are 36 marbles total, how many are red?

1
Total parts = \(2 + 7 = \) 9 parts
2
Value of one part: \(\frac{36}{9} = \) 4 marbles
3
Red marbles = \(2 \times 4 = \) 8 red marbles

Example 2: Equivalent Ratios

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

5 : 2 × 3 each (15 ÷ 5 = 3) 15 : 6 You need 6 cups of sugar.

Example 3: Quantitative Comparison

In a class, the ratio of boys to girls is \(3 : 4\). There are 28 students total. Which is greater: the number of girls, or 15?

1
Total parts: \(3 + 4 = 7\)
2
One part: \(\frac{28}{7} = 4\) students
3
Girls: \(4 \times 4 = \) 16
4
Compare: \(\textcolor{#4ade80}{16} > \textcolor{#fbbf24}{15}\), so Column A is greater.

Solving Proportions with Cross-Multiplication

When two ratios are equal, they form a proportion. Cross-multiplication lets you solve for an unknown quickly.

\(\text{If } \frac{a}{b} = \frac{c}{d} \text{ then } a \times d = b \times c\)
\(\frac{3}{x} = \frac{9}{15}\)
\(3 \times 15 = 9 \times x\)
\(45 = 9x\)
\(x = 5\)

Extended Ratios: Three or More Terms

Some problems give ratios with three quantities, like \(a : b : c\). The same strategy applies: add all parts, find one part's value, then scale.

1
Problem: Red, blue, and green beads are in the ratio \(2 : 3 : 5\). There are 60 beads total.
2
Total parts: \(2 + 3 + 5 = 10\). One part: \(\frac{60}{10} = 6\).
3
Red: \(2 \times 6 =\) 12, Blue: \(3 \times 6 =\) 18, Green: \(5 \times 6 =\) 30.

Combining Two Ratios

Sometimes you know \(A : B\) and \(B : C\) separately and need \(A : B : C\). The trick: make the shared term match.

\(A : B = 2 : 3\)
\(B : C = 4 : 5\)
Make B the same: \(\text{LCM}(3, 4) = 12\)
\(A : B = 8 : 12 \; (\times 4)\)
\(B : C = 12 : 15 \; (\times 3)\)
\(A : B : C = 8 : 12 : 15\)

Common Mistakes to Avoid

✕ Reversing Order
Writing "boys to girls" as \(4 : 3\) when it should be \(3 : 4\). Order matters in a ratio.
✕ Part vs Whole
Confusing part-to-part with part-to-whole. If the ratio is \(3 : 5\), the total is 8 parts, not 5.
✕ Forgetting to Simplify
Leaving a ratio as \(12 : 18\) instead of simplifying to \(2 : 3\). Always reduce to lowest terms.

Quick Reference Summary

Concept Key Idea Example
Ratio Comparison by division \(3 : 5\)
Simplify Divide by GCF \(12 : 18 \to 2 : 3\)
Proportion Two equal ratios \(\frac{3}{5} = \frac{9}{15}\)
Cross-Multiply \(ad = bc\) \(3 \times 15 = 5 \times 9\)
Find quantity \(\text{Total} \div \text{sum of parts}\) \(36 \div 9 = 4\) per part
Extended ratio Match shared term \(2:3, 4:5 \to 8:12:15\)

Question 1 of 5

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

Question 2 of 5

Simplify the ratio \(45 : 60\) to its lowest terms.

Question 3 of 5

If \(\dfrac{x}{12} = \dfrac{5}{4}\), what is the value of \(x\)?

Question 4 of 5

Red, blue, and green candies are in the ratio \(1 : 2 : 3\). If there are 48 candies total, how many blue candies are there?

Question 5 of 5

In a class, the ratio of students who passed to those who failed is \(7 : 3\). What fraction of the class failed?