Squeezed in

Answer: $92$

Using mixed numbers

$\frac{71}{7} = 10\frac 17$ and $\frac{113}{11} = 10\frac{3}{11}$

$10$ and some ninths

$\frac19\lt\frac17$

$\frac29\gt\frac17$, is $\frac29\lt\frac3{11}$?

$\frac29=\frac{22}{99}$ and $\frac3{11}=\frac{27}{99}$ so yes

$10\frac29=\frac{92}9$

Using decimals

$\frac x9$ is either a whole number, or a recurring decimal like $_.11\dot{1}$ of $_.22\dot2$ etc.

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$10.22\dot2$ is between $10.14...$ and $10.27...$

$10.22\dot2=10\frac29=\frac{92}9$

Using inequalities

$\frac{71}{7} = \frac{71\times11}{77}=\frac{770+11}{77}$

$\frac{113}{11} = \frac{113\times7}{77}=\frac{770+21}{77}$

$\therefore \frac{71}7\lt\frac x9\lt\frac{113}{11}$

• $\frac x9 \gt \frac{71}{7}$, so $x \gt \frac{9 \times 71}7 = \frac{639}7 = 91\frac 27$.
  
   
• $\frac x9 \lt \frac{113}{11}$, so $x \lt \frac{9 \times 113}{11} = \frac{1017}{11} = 92 \frac{5}{11}$.

Therefore $x = 92$, as $x$ is an integer.