SSAT Upper Level Quantitative (Math)

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SSAT Upper Level Quantitative › SSAT Upper Level Quantitative (Math)

Questions 1 - 10

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 .

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

$\frac{5}{6}\div \frac{1}{12}=?$

$\frac{5}{72}$

$\frac{1}{2}$

Explanation

Turn the second fraction upside down to find its reciprocal, and then multiply the first fraction by the reciprocal of the second fraction to divide the two fractions. When multiplying fractions, you can multiply across, finding the products of the numbers being multiplied in the numerator and in the denominator.

$\frac{5}{6}\div \frac{1}{12}=\frac{5}{6}\times\frac{12}{1}=\frac{60}{6}=10$

Define a function as follows:

$g(x)=\left | x^{5}-x^{2}\right |$

Evaluate .

Explanation

$g(x)=\left | x^{5}-x^{2}\right |$

$g(2)=\left | 2^{5}-2^{2}\right | = \left | 32-4\right | = \left | 28\right | = 28$

$g(-2)=\left |( -2)^{5}-\left (-2 \right )^{2}\right | = \left | -32-4\right | = \left | -36\right | = 36$

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.

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

The area of a right triangle is $50\textup{ in}^2$ . If the base of the triangle is $10\textup{ in}$ , what is the length of the height, in inches?

Explanation

To find the height, plug what is given in the question into the formula used to find the area of a triangle.

$\text{Area}=\frac{base\times height}{2}$

Use the information given in the question:

$\text{Base}=10$

$\text{Area}=50$

$50=\frac{10\times height}{2}$

Now, solve for the height.

$10\times height=100$

Order the following fractions from least to greatest:

$\frac{1}{2}, \frac{9}{10}, \frac{3}{4}, \frac{5}{7}$

$\frac{1}{2}, \frac{5}{7}, \frac{3}{4}, \frac{9}{10}$

$\frac{9}{10}, \frac{1}{2}, \frac{3}{4}, \frac{5}{7}$

$\frac{1}{2}, \frac{5}{7}, \frac{9}{10}, \frac{3}{4}$

$\frac{3}{4}, \frac{9}{10}, \frac{1}{2}, \frac{5}{7}$

Explanation

Convert the fractions so that they all share the same denominator.

$\frac{1}{2}\cdot \frac{70}{70}=\frac{70}{140}$

$\frac{3}{4}\cdot \frac{35}{35}=\frac{105}{140}$

$\frac{5}{7}\cdot \frac{20}{20}=\frac{100}{140}$

$\frac{9}{10}\cdot \frac{14}{14}=\frac{126}{140}$

Now, look at the numerators and order them from the least to the greatest:

$\frac{1}{2}, \frac{5}{7}, \frac{3}{4}, \frac{9}{10}$

A right triangle has a hypotenuse of and one leg has a length of . What is the length of the other leg?

$3\sqrt{7}$

$7\sqrt{3}$

Explanation

When calculating the lengths of sides of a right triangle, we can use the Pythagorean Theorem as follows:

$a^{2}+b^{2}=c^{2}$ , where and are legs of the triangle and is the hypotenuse.

Plugging in our given values:

$9^{2}+b^{2}=12^{2}$

Subtracting from each side of the equation:

$b^{2}=12^{2}-9^{2}$

$b^{2}=144-81$

$b^{2}=63$

Taking the square root of each side of the equation:

$b=\sqrt{63}$

Simplifying the square root:

$b=\sqrt{9\times 7}$

$b=3\sqrt{7}$

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