SAT Problem Solving Framework • Topic 2 of 4

Identifying Hidden Variables

Many problems hinge on a quantity the question never names directly — a total, a rate, an original price, the “before” amount. The move is to name it with a variable even though the problem didn’t. If a price rises 20% to $60, the hidden variable is the original price p, and 1.2p = 60 gives p = 50. If a question gives a part and a percent, the whole is hidden. Ask: “what unknown would let me write an equation?” Name it, relate it to the given numbers, and solve. Surfacing the hidden quantity is usually the entire difficulty of the problem.

✅ Solved examples

1. After a 20% rise a price is $60. Find the original price.
Let p = original; 1.2p = 60 → p = 50.
2. 15 is 25% of what number?
Hidden whole w: 0.25w = 15 → w = 60.
3. A class is 40% girls and has 12 girls. Class size?
Hidden total t: 0.40t = 12 → t = 30.
4. After a 10% discount an item costs $90. Original?
Let p; 0.9p = 90 → p = 100.

✏️ Practice — try these, take hints as needed

1. After a 25% rise a price is $50. Original price?
1.25p = 50.
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$40.
2. 18 is 30% of what number?
0.3w = 18.
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60.
3. A team is 20% rookies and has 4 rookies. Team size?
0.2t = 4.
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20.
4. After a 50% discount an item is $30. Original?
0.5p = 30.
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$60.
5. 9 is 75% of what number?
0.75w = 9.
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12.
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