SSAT Upper Level Quantitative - SSAT Upper Level Quantitative (Math)

Define an operation $\Upsilon$ as follows:

For all real numbers ,

$a \Upsilon b = a^{5} - b^{4}$ .

Evaluate: $\frac{1}{2}; \Upsilon; \frac{1}{2}$ .

$-\frac{1}{32}$

$\frac{1}{32}$

$-\frac{1}{16}$

$\frac{1}{16}$

Explanation

$a \Upsilon b = a^{5} - b^{4}$

$\frac{1}{2}; \Upsilon; \frac{1}{2} =\left ( \frac{1}{2} \right )^{5} - \left ( \frac {1}{2} \right )^{4}$

$= \frac{1}{32} - \frac{1}{16}$

$= \frac{1}{32} - \frac{2}{32}$

$= -\frac{1}{32}$

Which of the following lines is parallel with the line $y=-\frac{1}{5}x+1$ ?

$y=-\frac{1}{5}x+9$

$y=\frac{1}{5}x-2$

Explanation

Parallel lines have the same slope. The slope of a line in slope-intercept form is the value of . So, the slope of the line $y=-\frac{1}{5}x+1$ is $-\frac{1}{5}$ . That means that for the two lines to be parallel, the slope of the second line must also be $-\frac{1}{5}$ .

Function 2

What equation is graphed in the above figure?

$y = \left \lfloor x\right \rfloor - 4$

$y = \left \lceil x\right \rceil - 4$

$y = \left \lfloor x\right \rfloor - 3$

$y = \left \lceil x\right \rceil - 3$

$y = \left \lfloor x\right \rfloor +3$

Explanation

The greatest integer function, or floor function, $y = \left \lfloor x\right \rfloor$ , pairs each value of with the greatest integer less than or equal to . Its graph is below.

Floor function

The given graph is the above graph shifted downward four units. The graph of any function shifted downward four units is , so the given graph corresponds to equation $y = \left \lfloor x\right \rfloor - 4$ .

What is the slope of a line which passes through coordinates $\dpi{100} \small (3,7)$ and $\dpi{100} \small (4,12)$ ?

$\dpi{100} \small 5$

$\dpi{100} \small \frac{1}{5}$

$\dpi{100} \small \frac{1}{2}$

$\dpi{100} \small 2$

$\dpi{100} \small 3$

Explanation

Slope is found by dividing the difference in the $\dpi{100} \small y$ -coordinates by the difference in the $\dpi{100} \small x$ -coordinates.

$\dpi{100} \small \frac{(12-7)}{(4-3)}=\frac{5}{1}=5$

The lengths of the hypotenuses of ten similar right triangles form an arithmetic sequence. The smallest triangle has legs of lengths 3 and 4 inches; the second-smallest triangle has a hypotenuse of length one foot.

Which of the following responses comes closest to the area of the largest triangle?

8 square feet

7 square feet

6 square feet

9 square feet

5 square feet

Explanation

The hypotenuse of the smallest triangle can be calculated using the Pythagorean Theorem:

$c = \sqrt{a^{2}+b^{2}} = \sqrt{3^{2}+4^{2}} = \sqrt{9+16} = \sqrt{25}=5$ inches.

Let $c_{n}$ be the lengths of the hypotenuses of the triangles in inches. $c_{1} = 5$ and $c_{2} = 12$ , so their common difference is

$d = c_{2} - c_{1} = 12 - 5= 7$

The arithmetic sequence formula is

$c_{n} = c_{1} + (n-1)d$

The length of the hypotenuse of the largest triangle - the tenth triangle - can be found by substituting $c_{1} = 5, n= 10, d = 7$ :

$c_{10} =5+ (10-1) 7 = 5 + 9 \cdot 7 = 5 + 63 = 68$ inches.

The largest triangle has hypotenuse of length 68 inches. Since the triangles are similar, corresponding sides are in proportion. If we let and be the lengths of the legs of the largest triangle, then

$\frac{a}{3} = \frac{68}{5}$

$\frac{a}{3} \cdot 3 = \frac{68}{5} \cdot 3$

Similarly,

$\frac{b}{4} = \frac{68}{5}$

$\frac{b}{4} \cdot 4 = \frac{68}{5} \cdot 4$

The area of a right triangle is half the product of its legs:

$A = \frac{ab}{2}= \frac{40.8 \cdot 54.4}{2} = 1,109.76$ square inches.

Divide this by 144 to convert to square feet:

$1,109.76 \div 144 \approx 7.7$

Of the given responses, 8 square feet is the closest, and is the correct choice.

Convert $\frac{7}{9}$ to a percent.

Explanation

Percent also means out of one hundred. Set up the following ratio to find the percent value.

$\frac{7}{9}=\frac{x}{100}$

Now, solve for , which will be the percent value because in the ratio we set up, is the numerator of a fraction with a denominator of .

$x=\frac{700}{9}=77.8$

Find the slope of the line perpendicular to the line that has the equation .

$-\frac{1}{2}$

$\frac{1}{2}$

Explanation

Perpendicular lines have slopes that are negative reciprocals of each other. In other words, change the sign and flip the fraction around.

$\frac{2}{1}\rightarrow-\frac{1}{2}$

Which of the following responses comes closest to the area of the largest triangle?

8 square feet

7 square feet

6 square feet

9 square feet

5 square feet

Explanation

The hypotenuse of the smallest triangle can be calculated using the Pythagorean Theorem:

$c = \sqrt{a^{2}+b^{2}} = \sqrt{3^{2}+4^{2}} = \sqrt{9+16} = \sqrt{25}=5$ inches.

Let $c_{n}$ be the lengths of the hypotenuses of the triangles in inches. $c_{1} = 5$ and $c_{2} = 12$ , so their common difference is

$d = c_{2} - c_{1} = 12 - 5= 7$

The arithmetic sequence formula is

$c_{n} = c_{1} + (n-1)d$

The length of the hypotenuse of the largest triangle - the tenth triangle - can be found by substituting $c_{1} = 5, n= 10, d = 7$ :

$c_{10} =5+ (10-1) 7 = 5 + 9 \cdot 7 = 5 + 63 = 68$ inches.

The largest triangle has hypotenuse of length 68 inches. Since the triangles are similar, corresponding sides are in proportion. If we let and be the lengths of the legs of the largest triangle, then

$\frac{a}{3} = \frac{68}{5}$

$\frac{a}{3} \cdot 3 = \frac{68}{5} \cdot 3$

Similarly,

$\frac{b}{4} = \frac{68}{5}$

$\frac{b}{4} \cdot 4 = \frac{68}{5} \cdot 4$

The area of a right triangle is half the product of its legs:

$A = \frac{ab}{2}= \frac{40.8 \cdot 54.4}{2} = 1,109.76$ square inches.

Divide this by 144 to convert to square feet:

$1,109.76 \div 144 \approx 7.7$

Of the given responses, 8 square feet is the closest, and is the correct choice.

There are 500 students at the high school. There are only two menu options: chicken or fish. If 15% of students ordered fish, how many students ordered chicken?

$425\ students$