Equivalent Fractions and Simplifying: Practice

The denominator went from $8$ to $24$, which is multiplying by $3$ since $8 \times 3 = 24$. Multiply the numerator by the same $3$.

$\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}$

So the missing numerator is $9$.

Cross-multiply: pair each numerator with the other fraction's denominator.

$4 \times 9 = 36 \qquad 6 \times 6 = 36$

The cross products are equal, so $\frac{4}{6} = \frac{6}{9}$.

Divide by the greatest common factor of $15$ and $20$, which is $5$.

$\frac{15}{20} = \frac{15 \div 5}{20 \div 5} = \frac{3}{4}$

Since $3$ and $4$ share no factor above $1$, $\frac{3}{4}$ is in lowest terms.

Divide by the greatest common factor of $18$ and $30$, which is $6$.

$\frac{18}{30} = \frac{18 \div 6}{30 \div 6} = \frac{3}{5}$

Since $3$ and $5$ share no factor above $1$, $\frac{3}{5}$ is in lowest terms.

Divide by the greatest common factor of $14$ and $21$, which is $7$.

$\frac{14}{21} = \frac{14 \div 7}{21 \div 7} = \frac{2}{3}$

Since $2$ and $3$ share no factor above $1$, $\frac{2}{3}$ is in lowest terms.

A fraction is in lowest terms when its numerator and denominator share no factor above $1$. Check each.

$\frac{6}{9} = \frac{2}{3}, \quad \frac{4}{8} = \frac{1}{2}, \quad \frac{10}{15} = \frac{2}{3}$

But $5$ and $8$ share only the factor $1$, so $\frac{5}{8}$ is already in lowest terms.

The denominator went from $7$ to $35$, which is multiplying by $5$ since $7 \times 5 = 35$. Multiply the numerator by the same $5$.

$\frac{2}{7} = \frac{2 \times 5}{7 \times 5} = \frac{10}{35}$

So the missing numerator is $10$.

Cross-multiply: pair each numerator with the other fraction's denominator.

$6 \times 15 = 90 \qquad 10 \times 9 = 90$

The cross products are equal, so $\frac{6}{10} = \frac{9}{15}$.

Divide by the greatest common factor of $24$ and $40$, which is $8$.

$\frac{24}{40} = \frac{24 \div 8}{40 \div 8} = \frac{3}{5}$

Since $3$ and $5$ share no factor above $1$, $\frac{3}{5}$ is in lowest terms.

The denominator went from $5$ to $20$, which is multiplying by $4$ since $5 \times 4 = 20$. Multiply the numerator by the same $4$.

$\frac{6}{5} = \frac{6 \times 4}{5 \times 4} = \frac{24}{20}$

So the missing numerator is $24$. (This is an improper fraction, which is fine: it is greater than $1$.)

The numerator went from $3$ to $15$, which is multiplying by $5$ since $3 \times 5 = 15$. Multiply the denominator by the same $5$.

$\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$

So the missing denominator is $20$.

A fraction is not in lowest terms when its numerator and denominator share a factor above $1$. Check each.

$8 \text{ and } 12 \text{ share the factor } 4: \quad \frac{8}{12} = \frac{2}{3}$

The pairs $4$ and $9$, $7$ and $12$, and $5$ and $9$ each share only $1$, so $\frac{8}{12}$ is the one not in lowest terms.