Problem 21 · 2009 Math Kangaroo
Stretch
Fractions, Decimals & Percents
fraction-to-decimalplace-value
How many 0s must replace the ? in the decimal number 1.?1 to obtain a number that is smaller than 20092008 but bigger than 2000920008?
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Answer: C — 3
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Hint 1 of 2
The number is 1.00…0 1 — a 1 followed by some zeros and then a final 1, so it equals 1 + 10^−(k+1).
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Hint 2 of 2
Compare its extra part with 1/2008 and 1/20008.
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Approach: bound the small added part by the two fractions
- Writing k zeros for the '?', the number is 1 + 10^−(k+1).
- The conditions are 1/20008 < 10^−(k+1) < 1/2008.
- Since 1/2008 ≈ 0.000498 and 1/20008 ≈ 0.0000500, only 10^−4 = 0.0001 fits.
- That needs k + 1 = 4, so k = 3 zeros.
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