$\displaystyle A \cup B$ denotes the union of $\displaystyle A$ and $\displaystyle B$, the set of all elements that are in $\displaystyle A$, $\displaystyle B$, or both. To find this set, first, collect the elements in both sets:

$\displaystyle A= \left \{ \textup{a, b, f, j, k, m, p} \right \}$

$\displaystyle B = \left \{ \textup{c, d, i, k, m, n, p, z}\right \}$

$\displaystyle A \cup B = \left \{ \textup{a, b, f, j, k, m, p, c, d, i, k, m, n, p, z } \right \}$

Arranging:

$\displaystyle A \cup B = \left \{ \textup{a, b, c, d, f, i, j, k, k, m, m, n, p, p, z } \right \}$

Removing duplicates:

$\displaystyle A \cup B = \left \{ \textup{a, b, c, d, f, i, j, k, m, n, p, z } \right \}$

There are 12 elements in this set.

$\displaystyle A \cap B$ denotes the intersection of $\displaystyle A$ and $\displaystyle B$, the set of all elements the two sets have in common. These elements are underlined below:

$\displaystyle A= \left \{ \textup{a, b, f, j, \underline{k}, \underline{m}, \underline{p}} \right \}$

$\displaystyle B = \left \{ \textup{c, d, i, \underline{k}, \underline{m}, n, \underline{p}, z}\right \}$

$\displaystyle A \cap B= \left \{ \textup{k,m,p} \right \}$

$\displaystyle A \cap B$ is a set with three elements.

Add the decimals vertically by aligning the decimal points, and adding the digits without regard to the decimal point:

$\displaystyle \begin{matrix} 0.4287\\ \underline{0.2398 } \\0.6685 \end{matrix}$

Round to the nearest hundredth (second place after the decimal point) by examining the next digit to the right. This digit is an 8, so round the hundredths digit up to 7. This makes the correct choice 0.67.

First convert 1 mile per minute into miles per hour by multiplying by 60, then simply convert 60 miles per hour to km per hour by 1.6 to get 96.6, rounded up to 97.

Fluid ounces are the only provided measure of volume. Volume is measured in liquid displaced by an object. Kilometers are a measure of distance, and pounds are a measure of weight, which is related to, but not the same as, volume.

Joules are units of measurement used to describe energy (kinetic or potential). While "electricity" may be a tempting option here, "energy" is more correct; while electrical energy can be measured in Joules, not all energy measured in Joules is electrical in nature.

This is a basic, clear geometric rule that all students must know: for a standard triangle the sum of the sides is equal to the perimeter. No other answer was acceptable.

To solve for $\displaystyle x$ in the equation

$\displaystyle h+ 3= 12$,

subtract 3 from both sides to reverse the addition of 3:

$\displaystyle h+ 3-3= 12 - 3$

$\displaystyle h = 9$

The height of a triangle is $\displaystyle h = 9$, and the length of its base is $\displaystyle b= 12$. The area of a triangle with base $\displaystyle b$ and height $\displaystyle h$ is

$\displaystyle A = \frac{1}{2}bh$,

so, substituting these values,

$\displaystyle A = \frac{1}{2} (12)(9) = 6(9) = 54$,

the correct response.