Chapter 1, Problem 1PCT

Write $\frac{7}{8}$ and $\frac{9}{20}$ as equivalent fractions with the least common denominator.

To determine

To write: The equivalent fractions of $\frac{7}{8}$ and $\frac{9}{20}$ with the least common denominator.

Answer to Problem 1PCT

The equivalent fractions of $\frac{7}{8}$ and $\frac{9}{20}$ with the least common denominator is $\frac{35}{40}$ and $\frac{18}{40}$ respectively.

Explanation of Solution

Procedure used:

“Step 1: Find the least common denominator of the fractions.

Step 2: Rewrite each fraction with the least common denominator”.

Property used:

“If a, b, and c are whole numbers, then $\frac{a}{b}=\frac{a\cdot c}{b\cdot c}$ where $b\ne 0$, $c\ne 0$”.

Calculation:

The given fractions are $\frac{7}{8}$ and $\frac{9}{20}$.

Recall that, the least common denominator is the least common multiple of the denominators.

Step 1: Find the least common denominator of the fractions.

The denominators of the fractions are 8 and 20.

Obtain the least common multiple of 8 and 20.

The number 8 can be written as the product of 2, 2, and 2.

Therefore, the prime factorization of 8 is $8=2\cdot 2\cdot 2$.

The number 20 can be written as the product of 2. 2, and 5.

Therefore, the prime factorization of 20 is $20=2\cdot 2\cdot 5$.

Now align the common factors vertically and write down the common factors and then write down the remaining factors as follows.

$\begin{array}{cccccc}8& =& 2\cdot & 2\cdot & 2\cdot & \\ 20& =& & 2\cdot & 2\cdot & 5\\ & & \downarrow & \downarrow & \downarrow & \downarrow \\ & & 2& 2& 2& 5\end{array}$

Multiply the above listed factors.

$2\cdot 2\cdot 2\cdot 5=40$.

Thus, the least common denominator is 40.

Step2: Write each fractions with the least common denominator.

The number 8 becomes 40 when multiplied with 5.

That is, $8\cdot 5=40$.

By the above property,

$\begin{array}{c}\frac{7}{8}=\frac{7\cdot 5}{8\cdot 5}\\ =\frac{35}{40}\end{array}$

Thus, the equivalent fraction of $\frac{7}{8}$ with denominator 40 is $\frac{35}{40}$.

The number 20 becomes 40 when multiplied by 2.

That is, $20\cdot 2=40$.

By the above property,

$\begin{array}{c}\frac{9}{20}=\frac{9\cdot 2}{20\cdot 2}\\ =\frac{18}{40}\end{array}$

Thus, the equivalent fraction of $\frac{9}{20}$ with denominator 40 is $\frac{18}{40}$

Therefore, the equivalent fractions of $\frac{7}{8}$ and $\frac{9}{20}$ with the least common denominator is $\frac{35}{40}$ and $\frac{18}{40}$ respectively.