Answer
a) $\frac{5}{6}$
b) $\frac{19}{12}$
c) $\frac{16}{3}$
d) $8$
Work Step by Step
a) Simplify the following:
$$\frac{1}{3} + \frac{1}{2}$$
Put $\frac{1}{3} + \frac{1}{2}$ over the common denominator 6. $\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6}$:
$$\frac{2}{6} + \frac{3}{6}$$
$\frac{2}{6} + \frac{3}{6} = \frac{2 + 3}{6}$:
$$\frac{2 + 3}{6}$$
$2 + 3 = 5$:
Answer:
$$\frac{5}{6}$$
b) Simplify the following:
$$2 - \frac{2}{3} + \frac{1}{4}$$
Put $2 - \frac{2}{3} + \frac{1}{4}$ over the common denominator $12$. $2 - \frac{2}{3} + \frac{1}{4} = \frac{12 \times 2}{12} + \frac{4(-2)}{12} + \frac{3}{12}$:
$$\frac{12 \times 2}{12} + \frac{4(-2)}{12} + \frac{3}{12}$$
$12 \times 2 = 24$:
$$\frac{24}{12} + \frac{4(-2)}{12} + \frac{3}{12}$$
$4(-2) = -8$:
$$\frac{24}{12} + \frac{-8}{12} + \frac{3}{12}$$
$\frac{24}{12} - \frac{8}{12} + \frac{3}{12} = \frac{24 - 8 + 3}{12}$:
$$\frac{24 - 8 + 3}{12}$$
$24 + 3 = 27$:
$$\frac{27 - 8}{12}$$
$27 - 8 = 19$:
Answer:
$$\frac{19}{12}$$
c) Simplify the following:
$$4\left(2 - \frac{2}{3}\right)$$
Put $2 - \frac{2}{3}$ over the common denominator $3$. $2 - \frac{2}{3} = \frac{3 \times 2}{3} - \frac{2}{3}$:
$$4\left(\frac{3 \times 2}{3} - \frac{2}{3}\right)$$
$3 \times 2 = 6$:
$$4\left(\frac{6}{3} - \frac{2}{3}\right)$$
$\frac{6}{3} - \frac{2}{3} = \frac{6 - 2}{3}$:
$$4\left(\frac{6-2}{3}\right)$$
$6 - 2 = 4$:
$$4 \times \frac{4}{3}$$
$4 \times \frac{4}{3} = \frac{4 \times 4}{3}$:
$$ \frac{4 \times 4}{3} $$
$4 \times 4 = 16$:
Answer:
$$\frac{16}{3}$$
d) Simplify the following:
$$\frac{12}{\frac{4}{3} + \frac{1}{6}}$$
Put $\frac{4}{3} + \frac{1}{6}$ over the common denominator 6. $\frac{4}{3} + \frac{1}{6} = \frac{2 \times 4}{6} + \frac{1}{6}$:
$$\left(\frac{2 \times 4}{6} + \frac{1}{6}\right)$$
$2 \times 4 = 8$:
$$\frac{12}{\frac{8}{6} + \frac{1}{6}}$$
$\frac{8}{6} + \frac{1}{6} = \frac{8 + 1}{6}$:
$$\frac{12}{\left(\frac{8 + 1}{6}\right)}$$
$8 + 1 = 9$:
$$\frac{12}{\frac{9}{6}}$$
Multiply the numerator of $\frac{12}{\frac{9}{6}}$ by the reciprocal of the denominator $\frac{12}{\frac{9}{6}} = \frac{12 \times 6}{9}$:
$$\frac{12 \times 6}{9}$$
The gcd of $12$ and $9$ is $3$, so $\frac{12 \times 6}{9} = \frac{(3 \times 4)6}{3 \times 3} = \frac{3}{3} \times \frac{4 \times 6}{3} = \frac{4 \times 6}{3}$:
$$\frac{4 \times 6}{3}$$
$\frac{6}{3} = \frac{3 \times 2}{3} = 2$:
$$4 \times 2$$
$4 \times 2 = 8$:
Answer:
$$8$$
