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SSAT Upper Level Math : SSAT Upper Level Quantitative (Math)

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Example Questions

Example Question #2 : Algebraic Word Problems

Barbara went to the store and bought a shirt for $\$88$ dollars. It had been discounted by 20%. What is the original price of the shirt?

Possible Answers:

$\$ 108$

$\$$17.60$

$\$$88$

$\$70.40$

$\$ 110$

Correct answer:

$\$ 110$

Explanation:

To get the final price, you use as the variable for the original price. So,

$0.8x = \$ $88$ , and divide by 0.8 on both sides of the equations, and you get

$\$110$ for the original price of the shirt.

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Example Question #2 : Algebraic Word Problems

Two inlet pipes lead into a large water tank. One pipe can fill the tank in 45 minutes; the other can fill it in 40 minutes. To the nearest tenth of a minute, how long would it take the two pipes together to fill the tank if both were opened at the same time?

Possible Answers:

$18.8\textrm{ min}$

$25.0\textrm{ min}$

$23.8\textrm{ min}$

$21.2 \textrm{ min}$

$26.2\textrm{ min}$

Correct answer:

$21.2 \textrm{ min}$

Explanation:

Look at the work rates as "tanks per minute", not "minutes per tank".

The two pipes can fill the tank up at $\frac{1}{45}$ tanks per minute and $\frac{1}{40}$ tanks per minute.

Let be the time it took, in minutes, to fill the tank up. Then, since rate multiplied by time is equal to work, then two pipes filled up $\frac{1}{45} t$ and $\frac{1}{40} t$ tanks; together, they filled up $\frac{1}{45} t + \frac{1}{40} t$ tank - one tank. This sets up the equation to be solved:

$\frac{1}{45} t + \frac{1}{40} t = 1$

$360 \cdot \left ( \frac{1}{45} t + \frac{1}{40} t \right ) = 360 \cdot 1$

$\frac{360}{45} t + \frac{360}{40} t = 360$

8 t +9 t = 360

17 t = 360

$t = 360 \div 17 \approx 21.2$ minutes

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Example Question #6 : Algebraic Word Problems

The inlet pipe to a large water tank can fill the tank in 25 minutes; the drain of the tank can empty it in 55 minutes.

Once, the drain was left open by mistake when the tank was being filled. The mistake was not caught until the tank was full. To the nearest tenth of a minute, how long did it take to fill the tank?

Possible Answers:

$45.8 \textrm{ min}$

$65.8 \textrm{ min}$

$54.2 \textrm{ min}$

$34.2 \textrm{ min}$

$75.8 \textrm{ min}$

Correct answer:

$45.8 \textrm{ min}$

Explanation:

Look at the work rates as "tanks per minute", not "minutes per tank".

The inlet pipe can fill the tank at a rate of $\frac{1}{25}$ tank per minute. The drain is emptying the tank at a rate of $\frac{1}{55}$ tank per minute.

Let be the time it took, in minutes, to fill the tank up. Then, since rate multiplied by time is equal to work, then the inlet pipe let in $\frac{1}{25} t$ tank of water, but the drain let out $\frac{1}{55} t$ tank. Since they were working against each other, the work of the latter is subtracted from that of the former, and we can set up and solve the equation:

$\frac{1}{25} t -\frac{1}{55} t = 1$

$275 \cdot \left ( \frac{1}{25} t -\frac{1}{55} t \right ) = 275 \cdot 1$

$\frac{275 }{25} t -\frac{275 }{55} t = 275$

11 t -5 t = 275

6t = 275

$t = 275 \div 6 \approx 45.8$ minutes

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Example Question #7 : Algebraic Word Problems

One angle of a pentagon measures $t ^{\circ }$ . The other four angles are congruent to one another. Give the measure of one of those angles in terms of .

Possible Answers:

$\left (72 - \frac{1}{5} t \right ) {^\circ}$

$\left (108 - \frac{1}{5} t \right ) {^\circ}$

$\left (135 - \frac{1}{4} t \right ) {^\circ}$

$\left (72 - \frac{1}{4} t \right ) {^\circ}$

$\left (135 - t \right ) {^\circ}$

Correct answer:

$\left (135 - \frac{1}{4} t \right ) {^\circ}$

Explanation:

The total of the measures of the angles of a pentagon is

$180 \cdot (5-2) = 540^{\circ }$ .

One angle measures $t ^{\circ }$ ; since the other four angles have the same measure as one another, we let $X^{\circ }$ be their common measure. Then we set up the equation and solve for :

4X + t = 540

4X + t - t = 540- t

4X= 540- t

$4X\div 4 = \left ( 540- t \right )\div 4$

$X =135 - \frac{1}{4} t$

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Example Question #3 : Algebraic Word Problems

John and Julie represented West High in a math contest. John outscored Julie by 16 points; as a team, they scored 80 points.

David and Dana represented East High in the same contest. Dana outscored David by 10 points; as a team, they scored 60 points.

Arrange the four students from highest score to lowest score.

Possible Answers:

John, Julie, Dana, David

Dana, John, David, Julie

John, Dana, David, Julie

Dana, John, Julie, David

John, Dana, Julie, David

Correct answer:

John, Dana, Julie, David

Explanation:

Let Julie's score be . Then John's score was x + 16 . Since the sum of their scores was 80,

x + x + 16 = 80

2x+16 = 80

2x+16- 16 = 80 - 16

2x = 64

$2x \div 2 = 64 \div 2$

x = 32

Julie scored 32, and John scored 16 higher, or 48.

Let David's score be . Then Dana's score was y + 10 . Since the sum of their scores was 60,

y + y + 10 = 60

2y + 10 = 60

2y + 10 - 10 = 60 - 10

2y = 50

$2y\div 2 = 50 \div 2$

y = 25

David's score was 25, and Dana's score was 10 higher, or 35.

In descending order of score, the students were John, Dana, Julie, David.

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Example Question #11 : Algebraic Word Problems

The mean of , , , and is 125; the mean of , , , and is 150. Which of the following gives the sum of and if the mean of , , , , , and is ?

Possible Answers:

275 - M

275 + 6M

275 - 6M

1,100 + 6M

1,100- 6M

Correct answer:

1,100- 6M

Explanation:

Since the mean of the four numbers , , , and is 125,

$\frac{a + b + c + d}{4} = 125$

$\frac{a + b + c + d}{4} \cdot 4 = 125\cdot 4$

a + b + c + d = 500

Similarly,

$\frac{ c + d+e+f}{4} = 150$

$\frac{ c + d+e+f}{4}\cdot 4 = 150\cdot 4$

c + d+e+f = 600

Add the two sums:

a + b + c + d + c + d + e + f= 500 + 600

a + b + c + d + e + f+ c + d= 1,100

The mean of , , , , , and is , so

$\frac{a + b + c + d + e + f }{6}= M$

$\frac{a + b + c + d + e + f }{6}\cdot 6= M \cdot 6$

a + b + c + d + e + f= 6M

So:

a + b + c + d + e + f+ c + d= 1,100

6M+ c + d= 1,100

6M+ c + d - 6M= 1,100- 6M

c + d = 1,100- 6M

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Example Question #11 : Algebraic Word Problems

Which of the following sentences is represented by the equation

$\sqrt{x + 5} = x + 6$ ?

Possible Answers:

The sum of five and the square root of a number is six less than the number.

The square root of the sum of a number and five is six greater than the number.

The square root of the sum of a number and five is six less than the number.

The sum of five and the square root of a number is six greater than the number.

None of the other responses are correct.

Correct answer:

The square root of the sum of a number and five is six greater than the number.

Explanation:

$\sqrt{x + 5}$ is the square root of x + 5 , which in turn can be written as "the sum of a number and five"; $\sqrt{x + 5}$ can subsequently be written as "the square root of the sum of a number and five". Since $\sqrt{x + 5} = x + 6$ , $\sqrt{x + 5}$ is six greater than , the number, so the equation can be stated as "The square root of the sum of a number and five is six greater than the number."

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Example Question #11 : Algebraic Word Problems

Which of the following sentences is represented by the equation

5x = 3 (x - 7) ?

Possible Answers:

The sum of five and a number is equal to three times the difference of the number and seven.

The product of five and a number is equal to seven less than three times the number.

The sum of five and a number is equal to seven less than three times the number.

The product of five and a number is equal to three times the difference of the number and seven.

The product of five and a number is equal to three times the difference of seven and the number.

Correct answer:

The product of five and a number is equal to three times the difference of the number and seven.

Explanation:

The expression can be written as "the product of five and a number".

The expression x -7 can be written as " the difference of the number and seven"; the expression 3 (x-7) can be written as "three times the difference of the number and seven".

Therefore, the equation

5x = 3 (x - 7)

can be written as

"The product of five and a number is equal to three times the difference of the number and seven".

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Example Question #13 : Algebraic Word Problems

In a math contest, the team from Jefferson High comprised Velma, Wendell, Yancy, and Zack. Velma outscored Zack by 12 points, Zack outscored Yancy by 22 points, and Wendell scored 98 points. The team score was 391. Order the four students from greatest score to least score.

Possible Answers:

Wendell, Velma, Zack, Yancy

Velma, Zack, Yancy, Wendell

The order cannot be determined by the information given.

Velma, Zack, Wendell, Yancy

Velma, Wendell, Zack, Yancy

Correct answer:

Velma, Zack, Wendell, Yancy

Explanation:

If we let be Yancy's xcore, then, since Zack outscored Yancy by 22, Zack scored Y + 22 . Velma outscored Zack by 12, so Velma scored Y + 22 + 12 = Y + 34 .

The sum of the four scores is the team score, or 391. Since Wendell scored 98, we can add the four expressions to get 391, then solve for :

Y + (Y+22)+ (Y + 34) + 98 = 391

3Y +154 = 391

3Y +154- 154 = 391 - 154

3Y = 237

$3Y\div 3 = 237 \div 3$

Y = 79

Yancy scored 79; Zack scored 22 more than Yancy, or 101; Velma scored 12 more than Zack, or 113. Wendell scored 98, so the correct ordering, greatest to least, is: Velma, Zack, Wendell, Yancy.

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Example Question #11 : Algebraic Word Problems

Four of the five numbers of a set are:

67, 83, 72, 80

If the average of the five numbers is , give the fifth number in terms of .

Possible Answers:

5 t -302

$5t -60 \frac{2}{5}$

4t -302

$4t -75 \frac{1}{2}$

$5t -75 \frac{1}{2}$

Correct answer:

5 t -302

Explanation:

Call the fifth number . If the average of the five numbers is , then is the sum of the numbers divided by five:

$\left (67+ 83+ 72+ 80 +N \right ) \div 5 = t$

Solve for in the equation:

$\left (302+N \right ) \div 5 = t$

$\left (302+N \right ) \div 5 \times 5 = t \times 5$

302+N = 5 t

302+N-302 = 5 t -302

N= 5 t -302

The fifth number is 5 t -302 .

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SSAT Upper Level Math : SSAT Upper Level Quantitative (Math)

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #2 : Algebraic Word Problems

Example Question #2 : Algebraic Word Problems

Example Question #6 : Algebraic Word Problems

Example Question #7 : Algebraic Word Problems

Example Question #3 : Algebraic Word Problems

Example Question #11 : Algebraic Word Problems

Example Question #11 : Algebraic Word Problems

Example Question #11 : Algebraic Word Problems

Example Question #13 : Algebraic Word Problems

Example Question #11 : Algebraic Word Problems

All SSAT Upper Level Math Resources

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