Fraction bounds extreme value Practice Questions (Grade 9)

Warm-upmedium

Given that $1 \le m \le 5$ , $3 \le n \le 7$ , and $8 \le p \le 12$ , what is the greatest possible value of $\dfrac{m}{n} \cdot \dfrac{1}{p}$ ?

Maximize $\dfrac{m}{n}$ with $m = 5$ and $n = 3$ , and maximize $\dfrac{1}{p}$ with $p = 8$ : $\dfrac{5}{3} \cdot \dfrac{1}{8} = \dfrac{5}{24}$ .

Mid-gamemedium

Given that $1 \le m \le 5$ , $3 \le n \le 7$ , and $8 \le p \le 12$ , what is the least possible value of $\dfrac{m}{n} \cdot \dfrac{1}{p}$ ?

Minimize $\dfrac{m}{n}$ with $m = 1$ and $n = 7$ , and minimize $\dfrac{1}{p}$ with $p = 12$ : $\dfrac{1}{7} \cdot \dfrac{1}{12} = \dfrac{1}{84}$ .

Mid-gamemedium

Given that $2 \le m \le 6$ , $1 \le n \le 4$ , and $5 \le p \le 10$ , what is the greatest possible value of $\dfrac{m + n}{p}$ ?

Maximize the numerator with $m = 6$ and $n = 4$ , and minimize the denominator with $p = 5$ : $\dfrac{6 + 4}{5} = 2$ .

Mid-gamemedium

Given that $2 \le m \le 6$ , $1 \le n \le 4$ , and $5 \le p \le 10$ , what is the least possible value of $\dfrac{m + n}{p}$ ?

Minimize the numerator with $m = 2$ and $n = 1$ , and maximize the denominator with $p = 10$ : $\dfrac{2 + 1}{10} = \dfrac{3}{10}$ .

Mid-gamemedium

Given that $3 \le m \le 8$ , $1 \le n \le 2$ , and $4 \le p \le 6$ , what is the greatest possible value of $\dfrac{m}{n + p}$ ?

Maximize $m$ with $m = 8$ , and minimize $n + p$ with $n = 1$ and $p = 4$ : $\dfrac{8}{1 + 4} = \dfrac{8}{5}$ .

Mid-gamemedium

Given that $3 \le m \le 8$ , $1 \le n \le 2$ , and $4 \le p \le 6$ , what is the least possible value of $\dfrac{m}{n + p}$ ?

Minimize $m$ with $m = 3$ , and maximize $n + p$ with $n = 2$ and $p = 6$ : $\dfrac{3}{2 + 6} = \dfrac{3}{8}$ .

Mid-gamemedium

Given that $2 \le m \le 5$ and $3 \le n \le 8$ , what is the greatest possible value of $\dfrac{1}{m} + \dfrac{1}{n}$ ?

Each term $\dfrac{1}{m}$ and $\dfrac{1}{n}$ is greatest when its denominator is least: $\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{5}{6}$ .

Buzzer beatermedium

Given that $2 \le m \le 5$ and $3 \le n \le 8$ , what is the least possible value of $\dfrac{1}{m} + \dfrac{1}{n}$ ?

Each term $\dfrac{1}{m}$ and $\dfrac{1}{n}$ is least when its denominator is greatest: $\dfrac{1}{5} + \dfrac{1}{8} = \dfrac{13}{40}$ .

Fraction bounds extreme value (ACT). Game on.