Question 3 · Aspect ratio: $1.\overline{7}$ and $\tfrac{16}{9}$
Show, without a calculator, that a repeating decimal is a fraction.
Maximum score · 1 pointsPaula's neighbour told her that her television screen has an aspect ratio of $1.\overline{7}$ and that it is the same as Paula's. At the shop they told Paula hers had an aspect ratio of $\dfrac{16}{9}$ and she does not understand how they can be the same. Show it without using a calculator.
Step-by-step solution
Let $x = 1.777\ldots$ Multiply by $10$ so the repeating part stays the same:
$$10x = 17{,}777\ldots \qquad x = 1{,}777\ldots$$
Subtract the two equalities: the decimals cancel.
$$10x - x = 17{,}777\ldots - 1{,}777\ldots \;\Longrightarrow\; 9x = 16 \;\Longrightarrow\; x = \dfrac{16}{9}$$
Result: $1.\overline{7} = \dfrac{16}{9}$: it is the same number written in two ways.