Proportional Reasoning Formula

A proportion is written as two equal ratios: $\frac{a}{b} = \frac{c}{d}$

The Formula

\frac{a}{b} = \frac{c}{d} \iff a \times d = b \times c

When to use: If 3 pizzas feed 12 people, how many feed 20? Think multiplication, not addition.

Recipe serves 4, need to serve 10. Scale factor: \frac{10}{4} = 2.5 Multiply all ingredients by 2.5.

A proportion is written as two equal ratios: \frac{a}{b} = \frac{c}{d}

Thinking about multiplicative relationships between quantities that scale together.

If 3 pizzas feed 12 people, how many feed 20? Think multiplication, not addition.

\frac{a}{b} = \frac{c}{d} \iff ad = bc \quad (b, d \neq 0)

easy

If 3 notebooks cost \$7.50, how much do 7 notebooks cost?

Answer

\$17.50

Proportional reasoning often starts by finding the unit rate, then scaling to the desired quantity.

medium

A recipe calls for 2 cups of flour for every 3 cups of sugar. If you use 9 cups of sugar, how many cups of flour do you need?

Using additive reasoning instead of multiplicative: 'add 4 to each ingredient' instead of 'multiply each ingredient by 2'
Cross-multiplying incorrectly when setting up a proportion: \frac{3}{4} = \frac{x}{12} gives x = 9, not x = 16
Forgetting that scaling affects all parts of a recipe or ratio, not just some

Foundation for percentages, geometric similarity, unit rates, and setting up algebraic equations.

Thinking about multiplicative relationships between quantities that scale together.

If 3 pizzas feed 12 people, how many feed 20? Think multiplication, not addition.

A proportion is written as two equal ratios: \frac{a}{b} = \frac{c}{d}

Foundation for percentages, geometric similarity, unit rates, and setting up algebraic equations.

Using additive thinking when multiplicative is needed: doubling a recipe means multiplying, not adding 2 cups.

Before studying the Proportional Reasoning formula, you should understand: ratios, multiplication.