Ratios Rates Proportional Relationships and Units Pattern - Variable Ratios

Variable Ratios

This pattern gives you a proportional relationship using variables and asks you to write an algebraic expression. Instead of computing with specific numbers, you set up the relationship with a variable and simplify.

Worked Examples

Example 1. A recipe serves $4$ people and uses $3$ cups of flour. How many cups of flour are needed for $p$ people?

A) $\dfrac{3p}{4}$
B) $\dfrac{4p}{3}$
C) $3p + 4$
D) $12p$

Set up a proportion: $\dfrac{3 \text{ cups}}{4 \text{ people}} = \dfrac{x \text{ cups}}{p \text{ people}}$
Cross-multiply: $x = \dfrac{3p}{4}$
Gotcha: Option B flips the ratio. Option C adds instead of multiplying. In proportional relationships, you always multiply (or divide), never add.
The answer is A.

Example 2. A machine assembles $x$ components in $8h$ minutes. What is the rate in components per minute?

Rate $= \dfrac{x}{8h}$

Example 3. A field produces $50$ kilograms per $A$ square meters. How many kilograms from $200$ square meters?

$\dfrac{50}{A} \times 200 = \dfrac{10{,}000}{A}$
Or equivalently: $\dfrac{50 \times 200}{A} = \dfrac{10{,}000}{A}$.

Example 4. Gas costs $$d$ per gallon. How many gallons can you buy with $$40$?

Gallons $= \dfrac{40}{d}$
Gotcha: Don't multiply $40 \times d$ — that gives you the cost of $40$ gallons, not the number of gallons you can buy.

What to Do on Test Day

• Proportional = multiply or divide. Never add or subtract in a proportion problem. If you see $+$ or $-$ in the answer choices, those are usually wrong.
• Set up the unit rate first with numbers, then scale by the variable.
• Check with a specific number. If $p = 8$ (double the original $4$), the flour should double from $3$ to $6$. Does $\dfrac{3(8)}{4} = 6$? ✓
• "Per" means divide. Components per minute $= \dfrac{\text{components}}{\text{minutes}}$.