Chapter 4, Problem 1A

Express $\frac{1}{3},\frac{2}{5},\frac{5}{6},$and $\frac{4}{9}$as equivalent fractions having the lowest common denominator.

To determine

Evaluate the lowest common denominator and its equivalent fraction.

Answer to Problem 1A

The equivalent fractions are $\frac{30}{90}$, $\frac{36}{90}$, $\frac{75}{90}$ and $\frac{40}{90}$.

Explanation of Solution

Given:

Fractions are $\frac{1}{3}$, $\frac{2}{5}$, $\frac{5}{6}$ and $\frac{4}{9}$.

Concept used:

Lowest common denominator is the smallest number which divides all the denominators of the fractions with zero remainder. Divide the lowest common number by denominator and then obtained value is multiplied with each fraction on numerator and denominator.

Calculation:

Prime factorizations of the denominators are given as follows:

  $\begin{array}{c}3=1\times 3\\ 5=1\times 5\\ 6=1\times 2\times 3\\ 9=1\times 3^2\end{array}$

The lowest common denominator is given as follows:

  $\begin{array}{l}\Rightarrow 1\times 3^2\times 2\times 5\\ \Rightarrow 90\end{array}$

Divide $90$ by first fraction denominator $3$.

  $\begin{array}{l}\Rightarrow 90÷3\\ \Rightarrow 30\end{array}$

Multiply $30$ with first fraction $\frac{1}{3}$ on numerator and denominator.

  $\begin{array}{c}\frac{1}{3}=\frac{1\times 30}{3\times 30}\\ =\frac{30}{90}\end{array}$

Divide $90$ by second fraction denominator $5$.

  $\begin{array}{l}\Rightarrow 90÷5\\ \Rightarrow 18\end{array}$

Multiply $18$ with second fraction $\frac{2}{5}$ on numerator and denominator.

  $\begin{array}{c}\frac{2}{5}=\frac{2\times 18}{5\times 18}\\ =\frac{36}{90}\end{array}$

Divide $90$ by third fraction denominator $6$.

  $\begin{array}{l}\Rightarrow 90÷6\\ \Rightarrow 15\end{array}$

Multiply $15$ with third fraction $\frac{5}{6}$ on numerator and denominator.

  $\begin{array}{c}\frac{5}{6}=\frac{5\times 15}{6\times 15}\\ =\frac{75}{90}\end{array}$

Divide $90$ by fourth fraction denominator $9$.

  $\begin{array}{l}\Rightarrow 90÷9\\ \Rightarrow 10\end{array}$

Multiply $10$ with fourth fraction $\frac{4}{9}$ on numerator and denominator.

  $\begin{array}{c}\frac{4}{9}=\frac{4\times 10}{9\times 10}\\ =\frac{40}{90}\end{array}$

Thus, the equivalent fractions are $\frac{30}{90}$, $\frac{36}{90}$, $\frac{75}{90}$ and $\frac{40}{90}$.

Conclusion:

The equivalent fractions are $\frac{30}{90}$, $\frac{36}{90}$, $\frac{75}{90}$ and $\frac{40}{90}$.