Fractions

1. The denominators are $3$ and $4$. Find the least common denominator (LCD):

$\text{LCD of } 3 \text{ and } 4 = 12$

Convert each fraction:

$\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$

$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$

Add:

$\frac{8}{12} + \frac{3}{12} = \frac{11}{12}$

Maria uses $\mathbf{\frac{11}{12}}$ cup of sugar in total.

2. Find $\frac{7}{8} - \frac{1}{3}$. The LCD of $8$ and $3$ is $24$.

Convert each fraction:

$\frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24}$

$\frac{1}{3} = \frac{1 \times 8}{3 \times 8} = \frac{8}{24}$

Subtract:

$\frac{21}{24} - \frac{8}{24} = \frac{13}{24}$

Tomás needs to hike $\mathbf{\frac{13}{24}}$ of a mile farther.

3. Division expression: $3 \div 4$.

A fraction is another way to write division:

$3 \div 4 = \frac{3}{4}$

Each friend gets $\mathbf{\frac{3}{4}}$ of a pizza.

Check: $4 \times \frac{3}{4} = \frac{12}{4} = 3$ pizzas. ✓

4. Multiply the length by the width:

$\frac{3}{4} \times \frac{2}{5}$

Multiply numerators: $3 \times 2 = 6$.

Multiply denominators: $4 \times 5 = 20$.

$\frac{3}{4} \times \frac{2}{5} = \frac{6}{20}$

Simplify by dividing numerator and denominator by their GCF ($2$):

$\frac{6}{20} = \frac{3}{10}$

The area of the garden is $\mathbf{\frac{3}{10}}$ square meter.

5. (a) Estimate using benchmarks:

• $\frac{5}{6}$ is close to $1$ (just a little less).
• $\frac{1}{4}$ is exactly $0.25$.

Since $\frac{5}{6} + \frac{1}{4}$ is roughly $1 + 0.25 = 1.25$, yes — they sold more than one whole box.

(b) Find $\frac{5}{6} + \frac{1}{4}$. The LCD of $6$ and $4$ is $12$.

Convert each fraction:

$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$

$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$

Add:

$\frac{10}{12} + \frac{3}{12} = \frac{13}{12} = 1\frac{1}{12}$

They sold $\mathbf{\frac{13}{12}}$ (or $1\frac{1}{12}$) of their candy bars in total.