Fractions | IT

Question 21

Find equivalent fractions for the given pairs of fractions such that the fractional units are the same.

a. $\dfrac{7}{2}$ and $\dfrac{3}{5}$

b. $\dfrac{8}{3}$ and $\dfrac{5}{6}$

c. $\dfrac{3}{4}$ and $\dfrac{3}{5}$

d. $\dfrac{6}{7}$ and $\dfrac{8}{5}$

e. $\dfrac{9}{4}$ and $\dfrac{5}{2}$

f. $\dfrac{1}{10}$ and $\dfrac{2}{9}$

g. $\dfrac{8}{3}$ and $\dfrac{11}{4}$

h. $\dfrac{13}{6}$ and $\dfrac{1}{9}$

Solution

Understand the Question

A fractional unit represents one unit piece of a fraction, given by $\dfrac{1}{d}$ where $d$ is the denominator.
For two fractions to have the same fractional unit, they must have a common denominator (the Least Common Multiple of both denominators).
To convert a fraction to an equivalent one with a new denominator, multiply both the numerator and the denominator by the same number.

a $\dfrac{7}{2}$ and $\dfrac{3}{5}$

Step 1 · Convert to Common Denominator 10

The denominators are $2$ and $5$ . The least common multiple is $\text{LCM}(2, 5) = 10$ . Diagram 1

Convert $\dfrac{7}{2}$ :

\begin{aligned} \dfrac{7}{2} &= \dfrac{7 \times 5}{2 \times 5} \\[0.6em] &= \dfrac{35}{10} \end{aligned}

Convert $\dfrac{3}{5}$ :

\begin{aligned} \dfrac{3}{5} &= \dfrac{3 \times 2}{5 \times 2} \\[0.6em] &= \dfrac{6}{10} \end{aligned}

The common fractional unit is $\dfrac{1}{10}$ .

Answer

a $\dfrac{35}{10} \text{ and } \dfrac{6}{10}$

b $\dfrac{8}{3}$ and $\dfrac{5}{6}$

Step 1 · Convert to Common Denominator 6

The denominators are $3$ and $6$ . The least common multiple is $\text{LCM}(3, 6) = 6$ . Diagram 2

Convert $\dfrac{8}{3}$ :

\begin{aligned} \dfrac{8}{3} &= \dfrac{8 \times 2}{3 \times 2} \\[0.6em] &= \dfrac{16}{6} \end{aligned}

The fraction $\dfrac{5}{6}$ already has denominator $6$ .

The common fractional unit is $\dfrac{1}{6}$ .

Answer

b $\dfrac{16}{6} \text{ and } \dfrac{5}{6}$

c $\dfrac{3}{4}$ and $\dfrac{3}{5}$

Step 1 · Convert to Common Denominator 20

The denominators are $4$ and $5$ . The least common multiple is $\text{LCM}(4, 5) = 20$ . Diagram 3

Convert $\dfrac{3}{4}$ :

\begin{aligned} \dfrac{3}{4} &= \dfrac{3 \times 5}{4 \times 5} \\[0.6em] &= \dfrac{15}{20} \end{aligned}

Convert $\dfrac{3}{5}$ :

\begin{aligned} \dfrac{3}{5} &= \dfrac{3 \times 4}{5 \times 4} \\[0.6em] &= \dfrac{12}{20} \end{aligned}

The common fractional unit is $\dfrac{1}{20}$ .

Answer

c $\dfrac{15}{20} \text{ and } \dfrac{12}{20}$

d $\dfrac{6}{7}$ and $\dfrac{8}{5}$

Step 1 · Convert to Common Denominator 35

The denominators are $7$ and $5$ . The least common multiple is $\text{LCM}(7, 5) = 35$ . Diagram 4

Convert $\dfrac{6}{7}$ :

\begin{aligned} \dfrac{6}{7} &= \dfrac{6 \times 5}{7 \times 5} \\[0.6em] &= \dfrac{30}{35} \end{aligned}

Convert $\dfrac{8}{5}$ :

\begin{aligned} \dfrac{8}{5} &= \dfrac{8 \times 7}{5 \times 7} \\[0.6em] &= \dfrac{56}{35} \end{aligned}

The common fractional unit is $\dfrac{1}{35}$ .

Answer

d $\dfrac{30}{35} \text{ and } \dfrac{56}{35}$

e $\dfrac{9}{4}$ and $\dfrac{5}{2}$

Step 1 · Convert to Common Denominator 4

The denominators are $4$ and $2$ . The least common multiple is $\text{LCM}(4, 2) = 4$ . Diagram 5

The fraction $\dfrac{9}{4}$ already has denominator $4$ .

Convert $\dfrac{5}{2}$ :

\begin{aligned} \dfrac{5}{2} &= \dfrac{5 \times 2}{2 \times 2} \\[0.6em] &= \dfrac{10}{4} \end{aligned}

The common fractional unit is $\dfrac{1}{4}$ .

Answer

e $\dfrac{9}{4} \text{ and } \dfrac{10}{4}$

f $\dfrac{1}{10}$ and $\dfrac{2}{9}$

Step 1 · Convert to Common Denominator 90

The denominators are $10$ and $9$ . The least common multiple is $\text{LCM}(10, 9) = 90$ . Diagram 6

Convert $\dfrac{1}{10}$ :

\begin{aligned} \dfrac{1}{10} &= \dfrac{1 \times 9}{10 \times 9} \\[0.6em] &= \dfrac{9}{90} \end{aligned}

Convert $\dfrac{2}{9}$ :

\begin{aligned} \dfrac{2}{9} &= \dfrac{2 \times 10}{9 \times 10} \\[0.6em] &= \dfrac{20}{90} \end{aligned}

The common fractional unit is $\dfrac{1}{90}$ .

Answer

f $\dfrac{9}{90} \text{ and } \dfrac{20}{90}$

g $\dfrac{8}{3}$ and $\dfrac{11}{4}$

Step 1 · Convert to Common Denominator 12

The denominators are $3$ and $4$ . The least common multiple is $\text{LCM}(3, 4) = 12$ . Diagram 7

Convert $\dfrac{8}{3}$ :

\begin{aligned} \dfrac{8}{3} &= \dfrac{8 \times 4}{3 \times 4} \\[0.6em] &= \dfrac{32}{12} \end{aligned}

Convert $\dfrac{11}{4}$ :

\begin{aligned} \dfrac{11}{4} &= \dfrac{11 \times 3}{4 \times 3} \\[0.6em] &= \dfrac{33}{12} \end{aligned}

The common fractional unit is $\dfrac{1}{12}$ .

Answer

g $\dfrac{32}{12} \text{ and } \dfrac{33}{12}$

h $\dfrac{13}{6}$ and $\dfrac{1}{9}$

Step 1 · Convert to Common Denominator 18

The denominators are $6$ and $9$ . The least common multiple is $\text{LCM}(6, 9) = 18$ . Diagram 8

Convert $\dfrac{13}{6}$ :

\begin{aligned} \dfrac{13}{6} &= \dfrac{13 \times 3}{6 \times 3} \\[0.6em] &= \dfrac{39}{18} \end{aligned}

Convert $\dfrac{1}{9}$ :

\begin{aligned} \dfrac{1}{9} &= \dfrac{1 \times 2}{9 \times 2} \\[0.6em] &= \dfrac{2}{18} \end{aligned}

The common fractional unit is $\dfrac{1}{18}$ .

Answer

h $\dfrac{39}{18} \text{ and } \dfrac{2}{18}$

Common Mistakes

Multiplying Only Denominators: Multiplying only the denominator by a factor changes the fraction's value. Always multiply both numerator and denominator by the same number.
Using Product Instead of LCM: While multiplying the denominators together (e.g. $6 \times 9 = 54$ ) gives a common denominator, using the LCM (e.g. $18$ ) keeps the numbers simpler.

Question 21

More questions in IT

Puzzle!