Question 21

We need to make the bottom numbers of the fractions the same. This makes their fractional units the same.

Step 1 — For $\frac{7}{2}$ and $\frac{3}{5}$

We find the smallest common bottom number. The bottom numbers are 2 and 5. The smallest number that both 2 and 5 can divide into is 10. So, we change both fractions to have 10 at the bottom.

Let us change $\frac{7}{2}$. We multiply the bottom number 2 by 5 to get 10. We must also multiply the top number 7 by 5.

$\frac{7}{2} = \frac{7 \times 5}{2 \times 5}$

$= \frac{35}{10}$

Let us change $\frac{3}{5}$. We multiply the bottom number 5 by 2 to get 10. We must also multiply the top number 3 by 2.

$\frac{3}{5} = \frac{3 \times 2}{5 \times 2}$

$= \frac{6}{10}$

The new fractions are $\frac{35}{10}$ and $\frac{6}{10}$. Their fractional unit is $\frac{1}{10}$.

$\boxed{\frac{35}{10} \text{ and } \frac{6}{10}}$

Step 2 — For $\frac{8}{3}$ and $\frac{5}{6}$

We find the smallest common bottom number. The bottom numbers are 3 and 6. The smallest number that both 3 and 6 can divide into is 6. So, we change both fractions to have 6 at the bottom.

Let us change $\frac{8}{3}$. We multiply the bottom number 3 by 2 to get 6. We must also multiply the top number 8 by 2.

$\frac{8}{3} = \frac{8 \times 2}{3 \times 2}$

$= \frac{16}{6}$

The fraction $\frac{5}{6}$ already has 6 at the bottom. So, we keep it as it is.

The new fractions are $\frac{16}{6}$ and $\frac{5}{6}$. Their fractional unit is $\frac{1}{6}$.

$\boxed{\frac{16}{6} \text{ and } \frac{5}{6}}$

Step 3 — For $\frac{3}{4}$ and $\frac{3}{5}$

We find the smallest common bottom number. The bottom numbers are 4 and 5. The smallest number that both 4 and 5 can divide into is 20. So, we change both fractions to have 20 at the bottom.

Let us change $\frac{3}{4}$. We multiply the bottom number 4 by 5 to get 20. We must also multiply the top number 3 by 5.

$\frac{3}{4} = \frac{3 \times 5}{4 \times 5}$

$= \frac{15}{20}$

Let us change $\frac{3}{5}$. We multiply the bottom number 5 by 4 to get 20. We must also multiply the top number 3 by 4.

$\frac{3}{5} = \frac{3 \times 4}{5 \times 4}$

$= \frac{12}{20}$

The new fractions are $\frac{15}{20}$ and $\frac{12}{20}$. Their fractional unit is $\frac{1}{20}$.

$\boxed{\frac{15}{20} \text{ and } \frac{12}{20}}$

Step 4 — For $\frac{6}{7}$ and $\frac{8}{5}$

We find the smallest common bottom number. The bottom numbers are 7 and 5. The smallest number that both 7 and 5 can divide into is 35. So, we change both fractions to have 35 at the bottom.

Let us change $\frac{6}{7}$. We multiply the bottom number 7 by 5 to get 35. We must also multiply the top number 6 by 5.

$\frac{6}{7} = \frac{6 \times 5}{7 \times 5}$

$= \frac{30}{35}$

Let us change $\frac{8}{5}$. We multiply the bottom number 5 by 7 to get 35. We must also multiply the top number 8 by 7.

$\frac{8}{5} = \frac{8 \times 7}{5 \times 7}$

$= \frac{56}{35}$

The new fractions are $\frac{30}{35}$ and $\frac{56}{35}$. Their fractional unit is $\frac{1}{35}$.

$\boxed{\frac{30}{35} \text{ and } \frac{56}{35}}$

Step 5 — For $\frac{9}{4}$ and $\frac{5}{2}$

We find the smallest common bottom number. The bottom numbers are 4 and 2. The smallest number that both 4 and 2 can divide into is 4. So, we change both fractions to have 4 at the bottom.

The fraction $\frac{9}{4}$ already has 4 at the bottom. So, we keep it as it is.

Let us change $\frac{5}{2}$. We multiply the bottom number 2 by 2 to get 4. We must also multiply the top number 5 by 2.

$\frac{5}{2} = \frac{5 \times 2}{2 \times 2}$

$= \frac{10}{4}$

The new fractions are $\frac{9}{4}$ and $\frac{10}{4}$. Their fractional unit is $\frac{1}{4}$.

$\boxed{\frac{9}{4} \text{ and } \frac{10}{4}}$

Step 6 — For $\frac{1}{10}$ and $\frac{2}{9}$

We find the smallest common bottom number. The bottom numbers are 10 and 9. The smallest number that both 10 and 9 can divide into is 90. So, we change both fractions to have 90 at the bottom.

Let us change $\frac{1}{10}$. We multiply the bottom number 10 by 9 to get 90. We must also multiply the top number 1 by 9.

$\frac{1}{10} = \frac{1 \times 9}{10 \times 9}$

$= \frac{9}{90}$

Let us change $\frac{2}{9}$. We multiply the bottom number 9 by 10 to get 90. We must also multiply the top number 2 by 10.

$\frac{2}{9} = \frac{2 \times 10}{9 \times 10}$

$= \frac{20}{90}$

The new fractions are $\frac{9}{90}$ and $\frac{20}{90}$. Their fractional unit is $\frac{1}{90}$.

$\boxed{\frac{9}{90} \text{ and } \frac{20}{90}}$

Step 7 — For $\frac{8}{3}$ and $\frac{11}{4}$

We find the smallest common bottom number. The bottom numbers are 3 and 4. The smallest number that both 3 and 4 can divide into is 12. So, we change both fractions to have 12 at the bottom.

Let us change $\frac{8}{3}$. We multiply the bottom number 3 by 4 to get 12. We must also multiply the top number 8 by 4.

$\frac{8}{3} = \frac{8 \times 4}{3 \times 4}$

$= \frac{32}{12}$

Let us change $\frac{11}{4}$. We multiply the bottom number 4 by 3 to get 12. We must also multiply the top number 11 by 3.

$\frac{11}{4} = \frac{11 \times 3}{4 \times 3}$

$= \frac{33}{12}$

The new fractions are $\frac{32}{12}$ and $\frac{33}{12}$. Their fractional unit is $\frac{1}{12}$.

$\boxed{\frac{32}{12} \text{ and } \frac{33}{12}}$

Step 8 — For $\frac{13}{6}$ and $\frac{1}{9}$

We find the smallest common bottom number. The bottom numbers are 6 and 9. The smallest number that both 6 and 9 can divide into is 18. So, we change both fractions to have 18 at the bottom.

Let us change $\frac{13}{6}$. We multiply the bottom number 6 by 3 to get 18. We must also multiply the top number 13 by 3.

$\frac{13}{6} = \frac{13 \times 3}{6 \times 3}$

$= \frac{39}{18}$

Let us change $\frac{1}{9}$. We multiply the bottom number 9 by 2 to get 18. We must also multiply the top number 1 by 2.

$\frac{1}{9} = \frac{1 \times 2}{9 \times 2}$

$= \frac{2}{18}$

The new fractions are $\frac{39}{18}$ and $\frac{2}{18}$. Their fractional unit is $\frac{1}{18}$.

$\boxed{\frac{39}{18} \text{ and } \frac{2}{18}}$

Answer

a. $\frac{35}{10}$ and $\frac{6}{10}$ b. $\frac{16}{6}$ and $\frac{5}{6}$ c. $\frac{15}{20}$ and $\frac{12}{20}$ d. $\frac{30}{35}$ and $\frac{56}{35}$ e. $\frac{9}{4}$ and $\frac{10}{4}$ f. $\frac{9}{90}$ and $\frac{20}{90}$ g. $\frac{32}{12}$ and $\frac{33}{12}$ h. $\frac{39}{18}$ and $\frac{2}{18}$