Posts Tagged ‘calculation’

Don’t multiply yet!

Suppose you are trying to solve the following problem:

If $p \cdot 12 \cdot 13 \cdot 14=q \cdot 24 \cdot 26 \cdot 28$ , what is the value of $\displaystyle \frac{p}{q}$ ?

Here is one (bad) option for solving the problem:

$p \cdot 12 \cdot 13 \cdot 14=q \cdot 24 \cdot 26 \cdot 28$

$2184p=17472q$ (Multiply everything together)

$\displaystyle p=\frac{17472q}{2184}$ (Divide both sides by 2184)

$\displaystyle \frac{p}{q}=\frac{17472}{2184}$ (Divide both sides by $q$ )

$\displaystyle \frac{p}{q}=8$ (Simplify.)

This method produces the correct answer, 8, but it requires a lot of complex arithmetic, which wastes time and increases the likelihood of mistakes.

Here is a better way to solve it:

$\displaystyle p \cdot 12 \cdot 13 \cdot 14=q \cdot 24 \cdot 26 \cdot 28$

$\displaystyle p=\frac{q \cdot 24 \cdot 26 \cdot 28}{12 \cdot 13 \cdot 14}$ (Divide both sides by $12 \cdot 13 \cdot 14$ }

$\displaystyle \frac{p}{q}=\frac{24 \cdot 26 \cdot 28}{12 \cdot 13 \cdot 14}$ (Divide both sides by $q$ )

$\displaystyle \frac{p}{q}=\frac{24 \cdot 26 \cdot 28}{12 \cdot 13 \cdot 14}=\frac{2 \cdot 2 \cdot 2}{1 \cdot 1 \cdot 1}=8$ (Simplify.)

Notice that in the last line of this solution, I divided the bottom numbers into the top numbers first instead of multiplying everything together. Dividing 24 by 12, 26 by 13, and 28 by 14 is a whole lot easier than multiplying 12 by 13 by 14 and 24 by 26 by 28 and then dividing 17,472 by 2,184. The principle that we are exploiting is simple: smaller numbers are easier to work with. And, generally, division produces smaller numbers than multiplication. Hence, the title of this entry. Don’t be in a rush to multiply everything together. Keep the numbers in factored form, and simplify anything you can with division first. Don’t multiply until the end.