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Proportional
Relationships

Master ratios, proportions, and cross-multiplication to solve problems efficiently on the ISEE Upper Level Quantitative Reasoning section.

ISEE Upper Level

What Is a Ratio?

A ratio compares two quantities by division. If a class has 12 boys and 8 girls, the ratio of boys to girls is \(\frac{12}{8} = \frac{3}{2}\), often written as 3 : 2.

\(\frac{a}{b}\)
Fraction form: part over part, or part over whole
a : b
Colon form: read as "a to b"
"a to b"
Word form: used in verbal descriptions

What Is a Proportion?

A proportion states that two ratios are equal. When \(\frac{a}{b} = \frac{c}{d}\), the four values form a proportion.

\(2 : 3\)
 2 cups flour to 3 cups water
\(4 : 6\)
 4 cups flour to 6 cups water
\(=\)

Visualizing Proportionality

A proportional relationship always produces a straight line through the origin on a coordinate graph. The slope of the line is the constant of proportionality, \(k\).

\(x\)
\(y\)
\(0\)
\(1\)
\(2\)
\(3\)
\(2\)
\(4\)
\(6\)
\((1, 2)\)
\((2, 4)\)
\((3, 6)\)
\(k = 2\)

The Cross-Multiplication Property

The most powerful tool for solving proportions: if \(\frac{a}{b} = \frac{c}{d}\), then the cross products are equal.

\(a\)
\(b\)
\(=\)
\(c\)
\(d\)
 →
\(\textcolor{#4ade80}{a} \cdot \textcolor{#f87171}{d} = \textcolor{#fbbf24}{b} \cdot \textcolor{#818cf8}{c}\)

Setting Up a Proportion Correctly

The key is to keep consistent units on each side. Make sure "like" quantities are in the same position (both numerators or both denominators).

✓ Correct
\(\frac{\text{miles}}{\text{hours}} = \frac{\text{miles}}{\text{hours}}\)
✕ Incorrect
\(\frac{\text{miles}}{\text{hours}} = \frac{\text{hours}}{\text{miles}}\)

Proportional vs. Non-Proportional

Proportional Non-Proportional
Equation form \(y = kx\) \(y = mx + b\), where \(b \neq 0\)
Passes through origin? Yes, always No
Constant ratio \(\frac{y}{x}\)? Yes No: changes with \(x\)
Example \(y = 5x\) \(y = 5x + 3\)

How to Solve a Proportion

1
Identify the two ratios. Pull out the given quantities and the unknown.
2
Set up the proportion. Ensure matching units are in the same positions on both sides.
3
Cross-multiply. Compute \(a \cdot d = b \cdot c\) to form a single equation.
4
Solve for the unknown. Divide both sides to isolate the variable.
5
Check your answer. Substitute back and verify the two ratios are equal.

Example 1: Basic Proportion

A car travels 150 miles in 3 hours. At the same rate, how far will it travel in 7 hours?

1
Set up: \(\dfrac{150 \text{ mi}}{3 \text{ hr}} = \dfrac{x \text{ mi}}{7 \text{ hr}}\)
2
Cross-multiply: \(150 \times 7 = 3 \times x\) → \(1050 = 3x\)
3
Solve: \(x = \dfrac{1050}{3} = 350\)

Example 2: Scale Drawing

On a map, 2 cm represents 50 km. If two cities are 7.5 cm apart on the map, what is the actual distance?

1
Set up: \(\dfrac{2 \text{ cm}}{50 \text{ km}} = \dfrac{7.5 \text{ cm}}{x \text{ km}}\)
2
Cross-multiply: \(2x = 50 \times 7.5 = 375\)
3
Solve: \(x = \dfrac{375}{2} = 187.5\)

Example 3: Part-to-Whole Ratio

A mixture is made with flour and sugar in the ratio 5 : 3. If you use 40 cups of flour, how many cups of sugar do you need? What is the total amount?

1
Set up: \(\dfrac{5}{3} = \dfrac{40}{x}\)
2
Cross-multiply: \(5x = 3 \times 40 = 120\) → \(x = 24\)
3
Total: \(40 + 24 = 64\) cups of mixture

Example 4: Finding \(k\) from a Table

Determine whether this table represents a proportional relationship and find the constant \(k\).

\(x\) 2 5 8 10
\(y\) 6 15 24 30
\(\frac{y}{x}\) 3 3 3 3

Common Mistakes to Avoid

✕ Flipped Units
Putting miles in the numerator on one side and hours in the numerator on the other. Always keep the same unit in the same position.
✕ Adding Instead of Multiplying
Some students add the cross products instead of setting them equal. Remember: \(ad = bc\), not \(ad + bc\).
✕ Forgetting to Simplify
Leaving an answer like \(\frac{350}{2}\) instead of computing \(175\). ISEE answers are always simplified.

Useful Shortcuts and Strategies

Unit Rate Method
Divide to find the rate per 1 unit, then multiply by the target quantity. Often faster than cross-multiplying.
Scale Factor Method
Find what you multiply one side by to get the other. If \(\frac{3}{5} = \frac{x}{20}\), note that \(5 \times 4 = 20\), so \(x = 3 \times 4 = 12\).
Equivalent Fractions
Simplify both ratios to see if they reduce to the same fraction. Quick way to test proportionality.

Summary of Key Concepts

Concept Key Idea Formula / Test
Ratio Comparison of two quantities \(\frac{a}{b}\) or \(a : b\)
Proportion Two equal ratios \(\frac{a}{b} = \frac{c}{d}\)
Cross-Multiply Solve for unknowns \(ad = bc\)
Constant \(k\) Unit rate / slope \(k = \frac{y}{x}\); \(y = kx\)
Proportional? All \(\frac{y}{x}\) values equal Graph passes through origin

Question 1 of 5

If \(\dfrac{4}{9} = \dfrac{x}{36}\), what is the value of \(x\)?

Question 2 of 5

A recipe calls for 3 cups of flour for every 2 cups of milk. How many cups of milk are needed for 12 cups of flour?

Question 3 of 5

Which table shows a proportional relationship?

Question 4 of 5

On a scale drawing, 3 inches represents 15 feet. A room measures 8 inches on the drawing. What is the actual length of the room?

Question 5 of 5

If \(y\) is proportional to \(x\), and \(y = 18\) when \(x = 6\), what is the value of \(y\) when \(x = 15\)?