Problem 1 · AMC 8 Stretch
Core
Fractions, Decimals & Percents
Arithmetic & Operations
visual-representationpattern-recognition
On a real national test, more than half of junior-high students missed this question: What is 75% of 12? That is surprising, because the numbers are so friendly! Show how to see the answer with a picture instead of just punching buttons. (Then try the deliberately unfriendly version: what is 74% of 13?)
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Answer: 9 (and 74% of 13 is about 9.6)
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Hint 1 of 4
The word 'percent' just means 'out of 100.' Is there a simple fraction that equals 75%?
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Hint 2 of 4
Draw 12 little squares. If you split them into 4 equal groups, how many squares are in each group?
Still stuck? Show hint 3 →
Hint 3 of 4
75% is the same as 3 out of every 4, which is the fraction \(\tfrac34\). Take \(\tfrac14\) of 12 first, then take 3 of those groups.
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Approach: See the percent as a friendly fraction and picture it
- The trick is to see 75% as the friendly fraction \(\tfrac34\), not to reach for a percent rule.
- Draw 12 squares in a 3-by-4 array and split them into 4 equal columns. Each column has 3 squares, so each column is \(\tfrac14\) of the whole.
- \(\tfrac14\) of 12 = 3 squares (one column), so \(\tfrac34\) of 12 = three columns = 3 + 3 + 3 = 9.
- So 75% of 12 = \(\tfrac34 \times 12 = 9\).
- The unfriendly twin 74% of 13 looks almost the same on paper, but 74% is not a clean fraction and 13 won't split into equal small groups, so there is no neat picture — you would just estimate \(0.74 \times 13 \approx 9.6\). The real lesson: grab the easy picture when the numbers are friendly.
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