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.


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