## Task

Malia found a "short cut" to find the decimal representation of the fraction $\frac{117}{250}$. Rather than use long division she noticed that because $250 \times 4 = 1000$,
$$\frac{117}{250} = \frac{117 \times 4}{250 \times 4} = \frac{468}{1000} = 0.468.$$

- For which of the following fractions does Malia's strategy work to find the decimal representation?
$$\frac{1}{3}, \frac{3}{4}, -\frac{6}{25}, \frac{18}{7}, \frac{13}{8} \;\; {\rm and} \;\; -\frac{113}{40}.$$
For each one for which the strategy does work, use it to find the decimal representation.
- For which denominators can Malia's strategy work?

## IM Commentary

This task is most suitable for instruction. The purpose of the task is to get students to reflect on the definition of decimals as fractions (or sums of fractions), at a time when they are seeing them primarily as an extension of the base-ten number system and may have lost contact with the basic fraction meaning. Students also have their understanding of equivalent fractions and factors reinforced.

If students need help connecting this method with that of long division, they can be asked to perform long division when the denominator is a power of ten.

The denominators identified in the second part, namely numbers which are factors of powers of ten (or, equivalently, numbers for which $2$ and $5$ are the only prime factors) are in fact the only ones whose decimal expansions terminate when the fraction is in reduced form. While the problem does not ask for this fact, it should be shared and can be explained readily: A terminating decimal is equal to $\frac{a}{10^n}$. In reduced form, the denominator must be a quotient - and thus a factor - of $10^n$.

## Comments

Log in to comment## tthompson says:

about 6 yearsThis is a fantastic problem that really gets at students' understanding of equivalent fractions. Just one note: the solution to the -113/40 problem should be -2.825.