Problem 7 · 1991 AJHSME
Hard
Arithmetic & Operations
estimationfactoring
The value of (487,000)(12,027,300) + (9,621,001)(487,000)(19,367)(.05) is closest to
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Answer: D — About 10,000,000,000.
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Hint 1 of 3
The question says "closest to," and the choices jump by factors of TEN — so don't multiply anything exactly. The size (number of digits) is all that matters. But first, do the two products on top share something you can factor out?
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Hint 2 of 3
Both terms on top are (something) × 487,000. Factor out 487,000 so the messy top becomes 487,000 × (one sum). Then round every number to a single digit times a power of ten and just count zeros.
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Hint 3 of 3
Round: 487,000 ≈ 5 × 10⁵, the bracket sum ≈ 2 × 10⁷, and on the bottom 19,367 × 0.05 ≈ 10³. Track only the powers of ten.
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Approach: factor out the shared term, then size everything with powers of ten
- Spot the shared factor on top: both products are (…) × 487,000, so the numerator = 487,000 × (12,027,300 + 9,621,001). Factoring first means one rounding instead of two.
- Now estimate magnitudes only. 487,000 ≈ 5 × 10⁵; the bracket ≈ 21,648,301 ≈ 2 × 10⁷. Their product ≈ 10 × 10¹² = 10¹³.
- Bottom: 19,367 ≈ 2 × 10⁴, and × 0.05 gives ≈ 10³.
- So the value ≈ 10¹³ ÷ 10³ = 10¹⁰ = about 10,000,000,000.
- Why this transfers: when answers differ by powers of ten and the question says "closest to," estimating is the intended shortcut — round each factor to one significant digit, then just add and subtract exponents. Exact multiplication here would be a waste of time and a chance to slip.
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