Equivalent Fractions and Simplifying: Additional Practice (Set 1)

The greatest common factor of $40$ and $64$ is $8$, since $40 = 8 \times 5$ and $64 = 8 \times 8$.

$\frac{40}{64} = \frac{40 \div 8}{64 \div 8} = \frac{5}{8}$

Since $5$ and $8$ share no factor above $1$, $\frac{5}{8}$ is in lowest terms.

The greatest common factor of $75$ and $100$ is $25$, since $75 = 25 \times 3$ and $100 = 25 \times 4$.

$\frac{75}{100} = \frac{75 \div 25}{100 \div 25} = \frac{3}{4}$

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

Rewrite both over the common denominator $18$.

$\frac{4}{9} = \frac{8}{18}, \qquad \frac{1}{2} = \frac{9}{18}$

With the same denominator, the larger numerator wins: $9 > 8$, so $\frac{1}{2} > \frac{4}{9}$.

The greatest common factor of $44$ and $55$ is $11$, since $44 = 11 \times 4$ and $55 = 11 \times 5$.

$\frac{44}{55} = \frac{44 \div 11}{55 \div 11} = \frac{4}{5}$

Since $4$ and $5$ share no factor above $1$, $\frac{4}{5}$ is in lowest terms.

The greatest common factor of $90$ and $108$ is $18$, since $90 = 18 \times 5$ and $108 = 18 \times 6$.

$\frac{90}{108} = \frac{90 \div 18}{108 \div 18} = \frac{5}{6}$

Since $5$ and $6$ share no factor above $1$, $\frac{5}{6}$ is in lowest terms.

The denominator went from $9$ to $72$, which is multiplying by $8$ since $9 \times 8 = 72$. Multiply the numerator by the same $8$.

$\frac{5}{9} = \frac{5 \times 8}{9 \times 8} = \frac{40}{72}$

So the missing numerator is $40$.

Rewrite both over the common denominator $24$.

$\frac{7}{8} = \frac{21}{24}, \qquad \frac{5}{6} = \frac{20}{24}$

With the same denominator, the larger numerator wins: $21 > 20$, so $\frac{7}{8} > \frac{5}{6}$.

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

$\frac{16}{24} = \frac{2}{3}, \quad \frac{21}{28} = \frac{3}{4}, \quad \frac{18}{27} = \frac{2}{3}$

But $8$ and $15$ share only the factor $1$, so $\frac{8}{15}$ is already in lowest terms.

The denominator went from $4$ to $32$, which is multiplying by $8$ since $4 \times 8 = 32$. Multiply the numerator by the same $8$.

$\frac{9}{4} = \frac{9 \times 8}{4 \times 8} = \frac{72}{32}$

So the missing numerator is $72$. (This is an improper fraction, which is fine.)

The greatest common factor of $96$ and $144$ is $48$, since $96 = 48 \times 2$ and $144 = 48 \times 3$.

$\frac{96}{144} = \frac{96 \div 48}{144 \div 48} = \frac{2}{3}$

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

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

$14 \times 27 = 378 \qquad 18 \times 21 = 378$

The cross products are equal, so $\frac{14}{18} = \frac{21}{27}$. (Both also simplify to $\frac{7}{9}$.)

Each equivalent of $\frac{3}{4}$ is $\frac{3k}{4k}$, and its numerator and denominator add to $3k + 4k = 7k$. Set this equal to $35$.

$7k = 35 \;\Rightarrow\; k = 5$

So the fraction is $\frac{3 \times 5}{4 \times 5} = \frac{15}{20}$, whose parts $15$ and $20$ add to $35$.