Apply proportional reasoning in context (recipes, scaling, density).
Free, timed SHSAT practice sets on proportional word problems— every question includes a full worked explanation, and there's no account required.
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In a proportion, two quantities grow or shrink together at the same rate. Write two equal ratios with the same units in the same places, then cross-multiply.
Example: 6 cups of flour make 18 cookies. How much flour for 24 cookies? \(\dfrac{6 \text{ cups}}{18 \text{ cookies}} = \dfrac{x \text{ cups}}{24 \text{ cookies}}\), so \(x = \dfrac{6 \times 24}{18} = 8\) cups.
A unit rate also works: \(6 \div 18 = \tfrac{1}{3}\) cup per cookie, and \(\tfrac{1}{3} \times 24 = 8\).
Inverse proportion: when more of one thing means less of the other (more workers, fewer days), the product stays the same. If 6 workers take 10 days, the job is 60 worker-days, so 5 workers need \(60 \div 5 = 12\) days.
Traps:
\(\dfrac{6}{9} = \dfrac{x}{24}\), so \(x = \dfrac{6 \times 24}{9} = 16\) cups. (Inverting the ratio (\(\dfrac{9 \times 24}{6}\)) gives 36 cups. Adding the difference instead of scaling (\(6 + (24 - 9)\)) gives 21 cups. Not rescaling at all gives 6 cups.)
Density is constant: \(\dfrac{7}{42} = \dfrac{m}{60}\), so \(m = \dfrac{7 \times 60}{42} = 10\) grams. (Adding the difference instead of scaling (\(7 + (60 - 42)\)) gives 25 grams. Not rescaling at all gives 7 grams. Setting up \(\dfrac{7}{60} = \dfrac{x}{42}\) (\(\dfrac{7 \times 42}{60}\)) gives 4.9 grams.)
The product stays constant: \(3 \times 8 = 24\). So the new value is \(24 \div 6 = 4\) hours. (Setting up a direct proportion, which moves the wrong way, gives 16 hours. Changing \(8\) by the same amount as the count (\(8 - (6 - 3)\)) gives 5 hours. Keeping the hours the same as before gives 8 hours.)