Set up the following equation.

$\displaystyle \dpi{100} \frac{1}{3}x=2\cdot 4$

$\displaystyle \dpi{100} \frac{1}{3}x=8$

$\displaystyle \dpi{100} 3\cdot \frac{1}{3}x=8\cdot 3$

$\displaystyle \dpi{100} x=24$

All of these choices are equal.

$\displaystyle \dpi{100} \frac{3}{4}$ is .75 in fraction form, and .75 is $\displaystyle \dpi{100} \frac{3}{4}$ in decimal form.

$\displaystyle \dpi{100} \frac{1}{2}$ of 1.5 is the same as $\displaystyle \dpi{100} \frac{1}{2}\cdot \frac{3}{2}=\frac{3}{4}$.

a) The square root of $\displaystyle 121$ is $\displaystyle 11$, because $\displaystyle 11\cdot 11=121$.

b) $\displaystyle \frac{1}{20}$ of $\displaystyle 200$ is $\displaystyle 10$, because $\displaystyle 200\div 20=10$.

c) The average of $\displaystyle 5$ and $\displaystyle 15$ is $\displaystyle 10$, because $\displaystyle \frac{5+15}{2}=10$.

Therefore (b) and (c) are equal, and they are both smaller than (a).

a) $\displaystyle 50$ percent of $\displaystyle 96$ is $\displaystyle 48$ because $\displaystyle 96\cdot .50=48$.

b) $\displaystyle 30$ percent of $\displaystyle 120$ is $\displaystyle 40$ because $\displaystyle 120\cdot .30=40$.

c) $\displaystyle 60$ percent of $\displaystyle 110$ is $\displaystyle 66$ because $\displaystyle 111\cdot .60=66$.

Therefore (b) is smaller than (a) which is smaller than (c).

a)  ^{$\displaystyle 2^{(2+3)}$ $\displaystyle = 2^{5} = 2\cdot 2\cdot 2\cdot 2\cdot 2 = 32$}

b) $\displaystyle 2(2+3)$ $\displaystyle = 2\cdot 5 = 10$

c) $\displaystyle \frac{2}{(2+3)}$ $\displaystyle =\frac{2}{5}= .4$

Therefore (a) is larger than (b) which is larger than (c).

a) $\displaystyle 4^{2} = 4\cdot 4 =16$

b) $\displaystyle 2^{4} = 2\cdot 2\cdot 2\cdot 2 = 16$

c) $\displaystyle 2^{2} + 4 = 2\cdot 2 + 4 = 4+4 = 8$

Therefore (a) and (b) are equal, and they are larger than (c).

This question tests your understanding of the order of operations. First complete operations in parentheses, then multiplication and division, and finally addition and subtraction.

a) $\displaystyle 5+\frac{6+8}{2} = 5+\frac{14}{2} = 5+7 = 12$

b) $\displaystyle \frac{5(6+8)}{2}=\frac{5\cdot 14}{2}=\frac{70}{2}=35$ 

c) $\displaystyle \frac{5}{2}\cdot6+8=2.5\cdot6+8=15+8=23$ 

Therefore (a) is smaller than (c) which is smaller than (b).

a) $\displaystyle \sqrt{64}$ $\displaystyle = 8$

b) $\displaystyle 10$ percent of $\displaystyle 64$ $\displaystyle = 64\cdot.10=6.4$

c) $\displaystyle \frac{64}{8}$$\displaystyle =64\div8=8$

Therefore (a) and (c) are the same, and they are both larger than (b).

a) $\displaystyle 85$ percent of $\displaystyle 15$ percent of $\displaystyle 400$

$\displaystyle .85\cdot.15\cdot400=51$

b) $\displaystyle 400$ percent of $\displaystyle 85$ percent of $\displaystyle 15$

$\displaystyle 4.00\cdot.85\cdot15=51$

c) $\displaystyle 15$ percent of $\displaystyle 400$ percent of $\displaystyle 85$

$\displaystyle .15\cdot4.00\cdot85=51$

Therefore (a), (b), and (c) are all equal.

In each of these scenarios, if the percentage increases, the number decreases by the same factor. All cases are the same value:

a) $\displaystyle 42$ percent of $\displaystyle 500$

$\displaystyle 500\cdot.42=210$

b) $\displaystyle 21$ percent of $\displaystyle 1000$

$\displaystyle 1000\cdot.21=210$

c) $\displaystyle 84$ percent of $\displaystyle 250$

$\displaystyle 250\cdot.84=210$