4.1.5 Multiplying by Fraction Close to 1

This trick – which is more like clever factoring – is used whenever you see a whole number being multiplied by a fraction close to 1.

The Method

Treat the fraction as $(1 - \text{small number})$ or $(1 + \text{small number})$:

Example: Below 1

$$14 \times \frac{15}{16} = 14 \times (1 - \frac{1}{16}) = 14 - \frac{14}{16} = 14 - \frac{7}{8} = 13 \frac{1}{8}$$

Example: Above 1

$$17 \times \frac{13}{11} = 17 \times (1 + \frac{2}{11}) = 17 + \frac{34}{11} = 17 + 3\frac{1}{11} = 20 \frac{1}{11}$$

Tip: Perform the fractional multiplication first as it might affect adding/subtracting values from the whole number portion. If the fractional part is improper, reduce it to a mixed number!

Note: If the whole number and the numerator are the same value, you can also apply the $a \times a/b$ trick (Section 1.3.9).

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Problem Set 4.1.5

Practice these problems. Type your answer and press Enter to check: