Question

I tell you these facts about a mystery number, $c$: $\bullet$ $1.5 < c < 2$ $\bullet$ $c$ can be written as a fraction with one digit for the numerator and one digit for the denominator. $\bullet$ Both $c$ and $1/c$ can be written as finite (non-repeating) decimals. What is this mystery number?

Answer 1

Answer:

Possible answer: $\displaystyle c = \frac{16}{10} = \frac{8}{5} = 1.6$.

Step-by-step explanation:

Rewrite the bounds of $c$ as fractions:

The simplest fraction for $1.5$ is $\displaystyle \frac{3}{2}$. Write the upper bound $2$ as a fraction with the same denominator:

$\displaystyle 2 = 2 \times 1 = 2 \times \frac{2}{2} = \frac{4}{2}$.

Hence the range for $c$ would be:

$\displaystyle \frac{3}{2} < c < \frac{4}{2}$.

If the denominator of $c$ is also $2$, then the range for its numerator (call it $p$) would be $3 < p < 4$. Apparently, no whole number could fit into this interval. The reason is that the interval is open, and the difference between the bounds is less than $2$.

To solve this problem, consider scaling up the denominator. To make sure that the numerator of the bounds are still whole numbers, multiply both the numerator and the denominator by a whole number (for example, 2.)

$\displaystyle \frac{3}{2} = \frac{2 \times 3}{2 \times 2} = \frac{6}{4}$.

$\displaystyle \frac{4}{2} = \frac{2\times 4}{2 \times 2} = \frac{8}{4}$.

At this point, the difference between the numerators is now $2$. That allows a number ($7$ in this case) to fit between the bounds. However, $\displaystyle \frac{1}{c} = \frac{4}{7}$ can't be written as finite decimals.

Try multiplying the numerator and the denominator by a different number.

$\displaystyle \frac{3}{2} = \frac{3 \times 3}{3 \times 2} = \frac{9}{6}$.

$\displaystyle \frac{4}{2} = \frac{3\times 4}{3 \times 2} = \frac{12}{6}$.

$\displaystyle \frac{3}{2} = \frac{4 \times 3}{4 \times 2} = \frac{12}{8}$.

$\displaystyle \frac{4}{2} = \frac{4\times 4}{4 \times 2} = \frac{16}{8}$.

$\displaystyle \frac{3}{2} = \frac{5 \times 3}{5 \times 2} = \frac{15}{10}$.

$\displaystyle \frac{4}{2} = \frac{5\times 4}{5 \times 2} = \frac{20}{10}$.

It is important to note that some expressions for $c$ can be simplified. For example, $\displaystyle \frac{16}{10} = \frac{2 \times 8}{2 \times 5} = \frac{8}{5}$ because of the common factor $2$.

Apparently $\displaystyle c = \frac{16}{10} = \frac{8}{5}$ works. $c = 1.6$ while $\displaystyle \frac{1}{c} = \frac{5}{8} = 0.625$.

Answer 2

Answer: 1.6

This answer is copied and pasted from an online site (aops) and this is what their explanation was, so please do not report for plagiarism!

Step-by-step explanation:

Because $c$ can be written as a finite decimal, we know it can be written as a fraction whose numerator is an integer and whose denominator is a power of $10$. Thus, after simplification, the denominator must still be a divisor of some power of $10$. That is, it must be factorable into $2$s and $5$s.

Since this denominator is a single digit, our choices are $1,$ $2,$ $4,$ $5,$ or $8$. We have the same options for the numerator, since we know $1/c$ also has a finite decimal. From here we could just test all the possibilities to see if they're between $1.5$ and $2,$ but with a little cleverness we can eliminate some of the remaining possibilities. If we don't use $5$ as the numerator or denominator, then $c$ is forced to be a power of $2$, so it can't be between $1.5$ and $2$. So, we must use $5$, and our only plausible choices are $5/2$ (which is $2.5$), $5/4$ (which is $1.25$), and $8/5$ (which is $1.6$). Of these, only $c=\boxed{8/5}$ works.