Problem 8 · 1988 AJHSME
Medium
Fractions, Decimals & Percents
count-decimal-places
Betty used a calculator to find the product 0.075 × 2.56. She forgot to enter the decimal points. The calculator showed 19200. If Betty had entered the decimal points correctly, the answer would have been
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Answer: B — .192.
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Hint 1 of 2
The calculator multiplied the same whole-number digits Betty wanted — the decimal point's only job is to fix where the point lands. So the digits 192 are already correct; you just have to place the point.
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Hint 2 of 2
The number of decimal places in a product equals the *total* number of decimal places in the two factors. Count them up, then shift the point in 19200 that many places left.
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Approach: the product's decimal places = sum of the factors' decimal places
- 0.075 has 3 digits after its point and 2.56 has 2, so the answer must have 3 + 2 = 5 decimal places. Slide the decimal in 19200 five places to the left.
- 19200 → 0.19200 = 0.192.
- Sanity check: 0.075 is a bit under one-tenth and 2.56 is about 2½, so the product should be around 0.2 — and 0.192 fits, while 1.92 or 0.0192 don't.
- Why this transfers: never re-multiply to fix a decimal point. Multiply the digits once, then count the combined decimal places — that count alone tells you where the point goes.
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