Chapter 3.2, Problem 55PPS

To determine

To calculate: The least rational number with a denominator of 9 that is greater than $\frac{5}{7}$ , the greatest rational number with a denominator of 8 that is less than $\frac{5}{7}$ . Also, the inequality comparing the three numbers obtained.

Answer to Problem 55PPS

  $\frac{5}{8}<\frac{5}{7}<\frac{7}{9}$

Explanation of Solution

Fractions with the same denominator are called like fractions.

Convert $\frac{5}{8},\frac{5}{7},\frac{7}{9}$ to like fractions.

Denominator of $\frac{5}{7}=7$

Denominator of the other number = 9

LCM (7,9) = 63

So,

  $\frac{5}{7}=\frac{5\times 9}{7\times 9}=\frac{45}{63}$

Let the other rational number be $\frac{x}{9}=\frac{x\times 7}{9\times 7}=\frac{7x}{63}$

According to question,

  $\begin{array}{l}\frac{5}{9}<\frac{x}{9}\\ \frac{45}{63}<\frac{7x}{63}\end{array}$

So, x can be 7 such that $\frac{45}{63}<\frac{7\times 7}{63}=\frac{49}{63}=\frac{7}{9}$

So, the least rational number with a denominator of 9 that is greater than $\frac{5}{7}$ is $\frac{7}{9}$

Denominator of $\frac{5}{7}=7$

Let the rational number with a denominator of 8 be $\frac{y}{8}$

Denominator of $\frac{y}{8}=8$

LCM (7,8) = 56

So,

  $\begin{array}{l}\frac{5}{7}=\frac{5\times 8}{7\times 8}=\frac{40}{56}\\ \frac{y}{8}=\frac{y\times 7}{7\times 8}=\frac{7y}{56}\end{array}$

According to question,

  $\begin{array}{l}\frac{y}{8}<\frac{5}{7}\\ \frac{7y}{56}<\frac{40}{56}\end{array}$

So, y can be 5 such that $\frac{7\times 5}{56}=\frac{35}{56}<\frac{40}{56}$

Hence, the greatest rational number with a denominator 8 is $\frac{35}{56}=\frac{5}{8}$ that is less than $\frac{5}{7}$

Comparison between numbers $\frac{5}{7},\frac{7}{9},\frac{5}{8}$

  $\frac{5}{8}<\frac{5}{7}<\frac{7}{9}$