Improper Fractions

An improper fraction is a fraction in which the numerator is greater than the denominator. Thus, an improper fraction is always greater than 1.

To check whether a fraction is improper or not, we have to compare between the numerator and denominator. If the numerator is greater, the fraction is improper. If the denominator is bigger, the fraction is proper. If they are equal, the fraction is equal to 1.

Identify if $\frac {8}{7}$ is an improper fraction or not.

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Since $8>7,$ $\frac {8}{7}$ is improper. $_\square$

Identify if $\frac {6}{11}$ is an improper fraction or not.

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Since $6<11,$ $\frac {6}{11}$ is proper. $_\square$

Find the value(s) of positive integers $a<13$ for which the fraction $\frac{a}{9}$ is improper.

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Since we want the fraction $\frac{a}{9}$ to be improper, we must have $9<a.$ It is given that $a<13,$ so $9<a<13,$ which implies $a\in\{10,11,12\}.$ $_\square$

Find the value(s) of integers $a>3$ for which the fraction $\frac{9}{a}$ is improper.

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Since we want the fraction $\frac{9}{a}$ to be improper, we must have $a<9.$ It is given that $a>3,$ so $3<a<9,$ which implies $a\in\{4,5,6,7,8\}.$ $_\square$