Set up and solve ratio, rate, and proportion problems — scaling a recipe, reading a map, mixing a solution. The setup decides the answer; the cross-multiplication at the end is the easy part. Every question comes with a written breakdown of how the proportion was set up.
If the first ratio is written as boys:girls, the second one has to stay boys:girls too — flipping the order on one side breaks the whole equation even though the numbers might still look reasonable.
In a class with a 2:3 ratio of boys to girls, boys aren't 2/3 of the class — they're 2/5, since the ratio's two parts add up to the whole.
A rate given in different units on each side of a proportion — minutes on one side, hours on the other — needs to be converted to matching units before cross-multiplying, not after.
Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.
A recipe uses a ratio of 2 cups of flour to 3 cups of sugar. If you use 10 cups of flour, how many cups of sugar are needed?
A — 15. Set up the proportion keeping flour over sugar on both sides: 2/3 = 10/x. Cross-multiplying gives 2x = 30, so x = 15. 6.7 (B) comes from inverting the ratio — setting up sugar over flour on one side and flour over sugar on the other, which mismatches the proportion: 3/2 = 10/x gives x ≈ 6.7.
A car travels 180 miles in 3 hours. At the same rate, how many minutes will it take to travel 60 miles?
A — 60. Find the unit rate first: 180 miles ÷ 3 hours = 60 miles per hour. At that rate, 60 miles takes 60/60 = 1 hour, which is 60 minutes. 20 (B) comes from dividing the 60 miles by the original 3 hours instead of by the computed rate of 60 mph — reusing a number from the setup instead of the actual rate.
Two alloys are mixed: Alloy A is 40% copper and Alloy B is 70% copper. How many kilograms of Alloy B must be mixed with 6 kg of Alloy A to produce an alloy that is 60% copper?
A — 12. Balance total copper against total weight: 0.4(6) + 0.7x = 0.6(6 + x). That gives 2.4 + 0.7x = 3.6 + 0.6x, so 0.1x = 1.2 and x = 12 kg. 6 (C) comes from assuming equal parts of each alloy give the average of the two percentages — (40% + 70%)/2 = 55%, which isn't 60%, so equal amounts can't be the answer, but it's the shortcut a rushed guess reaches for.
Ratio and proportion problems are almost entirely about the setup — once the proportion is written correctly, cross-multiplying rarely goes wrong.
Write what each number represents next to it — flour, sugar, hours, copper — before building the proportion. Untimed practice is where that labeling habit sticks.
Move to timed sessions once setup is fast and consistent. Ratio problems run a bit longer since there's a real-world scenario to parse first.
Pair ratio and proportion with word problems in a mock — both are translation-heavy skills before any arithmetic starts.
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