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Proportional
Relationships

Master the art of ratios, rates, and proportions to solve real-world problems with confidence.

ISEE Middle 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 \(\frac{12}{8}\), which simplifies to \(\frac{3}{2}\). We can write this as 3 : 2 or 3 to 2.

3 parts : 2 parts Boys to Girls = 3 : 2

From Ratios to Proportions

A proportion is an equation stating that two ratios are equal. If 3 apples cost $2, then 9 apples cost $6, because the ratios are the same:

In general, a proportion takes the form:

Visualizing Proportional Graphs

When two quantities are proportional, their graph is a straight line through the origin \((0,\,0)\). The constant of proportionality \(k\) is the slope.

1 2 3 4 2 4 6 8 x y (0, 0) (1, 2) (2, 4) (3, 6) (4, 8) k = 2

The Cross-Multiplication Rule

To solve a proportion with an unknown, cross-multiply. If \(\frac{a}{b} = \frac{c}{d}\), then \(a \cdot d = b \cdot c\).

a b = c d → a · d = b · c

Unit Rates: The Special Ratio

A unit rate is a ratio where the second quantity is 1. It answers "how much per one?"

300 miles in 5 hours 300 ÷ 5 = ?
\(60 \text{ mi/hr}\)
→ Unit Rate 60 miles per 1 hour

Proportional vs. Non-Proportional

Proportional Non-Proportional
Equation \(y = kx\) \(y = kx + b\), \(b \neq 0\)
Graph Straight line through (0, 0) Straight line, misses origin
Ratio \(\frac{y}{x}\) Constant for all points Changes from point to point
Table check All \(\frac{y}{x}\) values equal At least one \(\frac{y}{x}\) differs
Example \(y = 3x\) \(y = 3x + 5\)

How to Solve a Proportion

1
Set up the proportion. Write two equivalent ratios with the unknown as a variable, e.g. \(\frac{3}{5} = \frac{x}{20}\).
2
Cross-multiply. Multiply diagonally: \(3 \times 20 = 5 \times x\), giving \(60 = 5x\).
3
Solve for the variable. Divide both sides: \(x = \frac{60}{5} = 12\).
4
Check your answer. Verify: \(\frac{3}{5} = \frac{12}{20} = 0.6\). ✓ Both ratios match.

Example 1: Finding a Missing Value

A recipe uses 4 cups of flour for every 6 cups of sugar. How many cups of flour are needed for 15 cups of sugar?

1
Set up: \(\frac{4}{6} = \frac{x}{15}\)
2
Cross-multiply: \(4 \times 15 = 6 \times x\) → \(60 = 6x\)
3
Solve: \(x = \frac{60}{6} = 10\)
4
Check: \(\frac{4}{6} = \frac{2}{3}\) and \(\frac{10}{15} = \frac{2}{3}\). ✓

Example 2: Scale Drawing

On a map, 2 cm represents 50 km. If two cities are 7 cm apart on the map, what is the actual distance?

City A City B 7 cm on map Scale: 2 cm = 50 km
1
Set up: \(\frac{2\text{ cm}}{50\text{ km}} = \frac{7\text{ cm}}{x\text{ km}}\).
2
Cross-multiply: \(2x = 50 \times 7 = 350\)
3
Solve: \(x = \frac{350}{2} = 175\text{ km}\)

Example 3: Unit Rate Problem

A store sells 5 notebooks for $8.75. What is the cost of 12 notebooks?

1
Find the unit rate: \(\frac{\$8.75}{5} = \$1.75\) per notebook
2
Multiply by 12: \(\$1.75 \times 12 = \$21.00\)

Example 4: Similar Figures

Two rectangles are similar. The small one is 6 cm by 4 cm. The large one has a length of 15 cm. Find its width.

6 cm 4 cm 15 cm w = ? ~
1
Set up: \(\frac{6}{15} = \frac{4}{w}\)
2
Cross-multiply: \(6w = 15 \times 4 = 60\)
3
Solve: \(w = \frac{60}{6} = 10\text{ cm}\)

Common Mistakes to Avoid

✕ Mismatched Units
Writing \(\frac{\text{miles}}{\text{hours}} = \frac{\text{hours}}{\text{miles}}\). Always keep the same unit on top in both ratios.
✕ Flipping the Ratio
Setting up \(\frac{3}{5} = \frac{20}{x}\) when it should be \(\frac{3}{5} = \frac{x}{20}\). Check which quantity corresponds to which.
✕ Adding Instead of Multiplying
Thinking "5 became 20 by adding 15, so add 15 to 3 to get 18." Proportions use multiplication, not addition.

Summary: Key Concepts

Concept Definition Example
Ratio Comparison of two quantities \(3 : 5\)
Proportion Two equal ratios \(\frac{3}{5} = \frac{6}{10}\)
Unit Rate Ratio with denominator 1 \(60\text{ mph}\)
Cross-Multiply \(a \cdot d = b \cdot c\) \(3 \times 10 = 5 \times 6\)
Constant \(k\) \(y = kx\), slope of line \(k = \frac{y}{x}\)

ISEE Strategy Recap

Label Units
Write units above and below the fraction bar to avoid mix-ups.
Simplify First
Reduce ratios before cross-multiplying to keep numbers small.
Check with Estimation
If \(\frac{3}{5} = \frac{x}{20}\), then \(x\) should be a bit more than half of 20.

Question 1 of 5

If \(\frac{4}{7} = \frac{x}{28}\), what is the value of \(x\)?

Question 2 of 5

A car travels 180 miles in 3 hours. At the same rate, how far will it travel in 7 hours?

Question 3 of 5

Which table shows a proportional relationship?

Table A
\(x\): 1, 2, 3
\(y\): 3, 6, 9
Table B
\(x\): 1, 2, 3
\(y\): 4, 7, 10

Question 4 of 5

A map uses the scale 3 cm = 120 km. Two cities are 8 cm apart on the map. What is the actual distance?

Question 5 of 5

If 8 pencils cost $6.00, how much do 14 pencils cost?