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Bracketing a product

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To bound $38 \times 71 \times 19$ , round every factor down to a multiple of $10$ for the low end and up to a multiple of $10$ for the high end. What is the ratio of the high end to the low end? Give a fraction in lowest terms.

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Hint

Compute both ends separately, then divide.

Solution

Rounding down gives the low end $30 \times 70 \times 10 = 21000$ .
Rounding up gives the high end $40 \times 80 \times 20 = 64000$ .
The exact product $38 \times 71 \times 19 = 51262$ lies between them, as it must.
The ratio is $\frac{64000}{21000} = \frac{64}{21}$ .

Answer: $\frac{64}{21}$

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