Henry Ernest Dudeney/Puzzles and Curious Problems/173 - Short Cuts

by : $173$

 * Short Cuts
 * We have from time to time given various short cuts in mental arithmetic.
 * Here is an example that will interest those who are unfamiliar with the process.


 * Can you multiply $993$ by $879$ mentally?
 * It is remarkable that any two numbers of two figures each,
 * where the tens are the same, and the sum of the units make ten, can always be multiplied thus:
 * $97 \times 93 = 9021$
 * Multiply the $7$ by $3$ and set it down,
 * then add the $1$ to the $9$ and multiply by the other $9$, $9 \times 10 = 90$.


 * This is very useful for squaring any number ending in $5$, as $85^2 = 7225$.
 * With two fractions, when we have the whole numbers the same and the sum of the fractions equal unity,
 * we get an easy rule for multiplying them.
 * Take $7 \tfrac 1 4 \times 7 \tfrac 3 4 = 56 \tfrac 3 {16}$.
 * Multiply the fractions $= \tfrac 3 {16}$, add $1$ to one of the $7$'s, and multiply by the other, $7 \times 8 = 56$.