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SSAT Upper Level Math : Algebra

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Example Questions

Example Question #1 : Ssat Upper Level Quantitative (Math)

Find the sum of this infinite geometric series:

$4 + \frac{7}{2} + \frac{49}{16} + ... + 4 \cdot \left ( \frac {7}{8} \right ) ^{n}+...$ $\displaystyle 4 + \frac{7}{2} + \frac{49}{16} + ... + 4 \cdot \left ( \frac {7}{8} \right ) ^{n}+...$

Possible Answers:

$\displaystyle 32$

$\displaystyle 64$

$\displaystyle 48$

$\displaystyle 96$

$\displaystyle 80$

Correct answer:

$\displaystyle 32$

Explanation:

The sum of an infinite series with first term $\displaystyle a$ and common ratio $\displaystyle r$ is

$S = \frac{a}{1-r}$ $\displaystyle S = \frac{a}{1-r}$

Set $a=4,r= \frac{7}{8}$ $\displaystyle a=4,r= \frac{7}{8}$, and evaluate:

$S = \frac{a}{1-r} = \frac{4}{1-\frac{7}{8}} = \frac{4}{\left ( \frac{1}{8}\right )} = 4 \cdot 8=32$ $\displaystyle S = \frac{a}{1-r} = \frac{4}{1-\frac{7}{8}} = \frac{4}{\left ( \frac{1}{8}\right )} = 4 \cdot 8=32$

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Example Question #1 : Patterns

Give the next number in the following sequence:

$\displaystyle 3, 5, 10, 12, 24, 26, 52, 54,$ _____

Possible Answers:

108 $\displaystyle 108$

$\displaystyle 56$

$\displaystyle 58$

102 $\displaystyle 102$

104 $\displaystyle 104$

Correct answer:

108 $\displaystyle 108$

Explanation:

The sequence is generated by alternately adding 2, then multiplying by 2:

$\displaystyle 3 + 2= 5$

$5 \times 2 = 10$ $\displaystyle 5 \times 2 = 10$

$\displaystyle 10+2 = 12$

$12 \times 2 = 24$ $\displaystyle 12 \times 2 = 24$

$\displaystyle 24 + 2 = 26$

$26 \times 2 = 52$ $\displaystyle 26 \times 2 = 52$

$\displaystyle 52 + 2 = 54$

$54 \times 2 = \underline{108}$ $\displaystyle 54 \times 2 = \underline{108}$, which is the correct choice.

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Example Question #2 : Patterns

Define an operation on the real numbers as follows:

$a \bigtriangleup b = 3a - 2b$ $\displaystyle a \bigtriangleup b = 3a - 2b$

Find the value of $\displaystyle x$ that makes this statement true:

$x\bigtriangleup 5 = 32$ $\displaystyle x\bigtriangleup 5 = 32$

Possible Answers:

$x = \frac{2}{3}$ $\displaystyle x = \frac{2}{3}$

x= 14 $\displaystyle x= 14$

$x = 20 \frac{2}{3}$ $\displaystyle x = 20 \frac{2}{3}$

$x = 7\frac{1}{3}$ $\displaystyle x = 7\frac{1}{3}$

There is no such value of $\displaystyle x$.

Correct answer:

x= 14 $\displaystyle x= 14$

Explanation:

Replace $\displaystyle a=x, b=5$ in the defintiion:

$a \bigtriangleup b = 3a - 2b$ $\displaystyle a \bigtriangleup b = 3a - 2b$

$x\bigtriangleup 5 =3x - 2 \cdot 5$ $\displaystyle x\bigtriangleup 5 =3x - 2 \cdot 5$

$x\bigtriangleup 5 =3x - 10$ $\displaystyle x\bigtriangleup 5 =3x - 10$

Now, set this equal to 32 and solve for $\displaystyle x$:

$x\bigtriangleup 5 = 32$ $\displaystyle x\bigtriangleup 5 = 32$

$\displaystyle 3x - 10 = 32$

$\displaystyle 3x - 10 + 10= 32+ 10$

3x = 42 $\displaystyle 3x = 42$

$3x \div 3= 42 \div 3$ $\displaystyle 3x \div 3= 42 \div 3$

x= 14 $\displaystyle x= 14$

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Example Question #2 : Patterns

What number replaces the circle?

$1, 1, 2, 4, 6, 18, 21, 84, 88, \square, \bigcirc...$ $\displaystyle 1, 1, 2, 4, 6, 18, 21, 84, 88, \square, \bigcirc...$

Possible Answers:

440 $\displaystyle 440$

352 $\displaystyle 352$

465 $\displaystyle 465$

445 $\displaystyle 445$

357 $\displaystyle 357$

Correct answer:

445 $\displaystyle 445$

Explanation:

The sequence is formed by alternately multiplying by a number, then adding the same number; the number incrementally increases every other term.

$1 \times 1 = 1$ $\displaystyle 1 \times 1 = 1$

$\displaystyle 1 + 1 = 2$

$2 \times 2 = 4$ $\displaystyle 2 \times 2 = 4$

$\displaystyle 4 + 2 = 6$

$6 \times 3 = 18$ $\displaystyle 6 \times 3 = 18$

$\displaystyle 18 + 3 = 21$

$21 \times4 = 84$ $\displaystyle 21 \times4 = 84$

$\displaystyle 84 + 4 = 88$

$88 \times 5 = 440$ $\displaystyle 88 \times 5 = 440$, the number which replaces the square.

$\displaystyle 440 + 5 = 445$, the number which replaces the circle.

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Example Question #3 : Patterns

$\left \lfloor N \right \rfloor$ $\displaystyle \left \lfloor N \right \rfloor$ is defined as the greatest integer less than or equal to $\displaystyle N$.

Evaluate the expression for x = 0.7 $\displaystyle x = 0.7$:

$\left \lfloor 2x +0.5 \right \rfloor + \left \lfloor 3x +0.2\right \rfloor$ $\displaystyle \left \lfloor 2x +0.5 \right \rfloor + \left \lfloor 3x +0.2\right \rfloor$

Possible Answers:

$\displaystyle 5$

$\displaystyle 4$

4.2 $\displaystyle 4.2$

$\displaystyle 3$

5.2 $\displaystyle 5.2$

Correct answer:

$\displaystyle 3$

Explanation:

Substitute:

$\left \lfloor 2x +0.5 \right \rfloor + \left \lfloor 3x +0.2\right \rfloor$ $\displaystyle \left \lfloor 2x +0.5 \right \rfloor + \left \lfloor 3x +0.2\right \rfloor$

$= \left \lfloor 2 \cdot 0.7 +0.5 \right \rfloor + \left \lfloor 3 \cdot 0.7 +0.2\right \rfloor$ $\displaystyle = \left \lfloor 2 \cdot 0.7 +0.5 \right \rfloor + \left \lfloor 3 \cdot 0.7 +0.2\right \rfloor$

$= \left \lfloor 1.4 +0.5 \right \rfloor + \left \lfloor 2.1 +0.2\right \rfloor$ $\displaystyle = \left \lfloor 1.4 +0.5 \right \rfloor + \left \lfloor 2.1 +0.2\right \rfloor$

$= \left \lfloor 1.9\right \rfloor + \left \lfloor 2.3\right \rfloor$ $\displaystyle = \left \lfloor 1.9\right \rfloor + \left \lfloor 2.3\right \rfloor$

$\displaystyle =1+2 = 3$

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Example Question #2 : Algebra

$\left \lceil N\right \rceil$ $\displaystyle \left \lceil N\right \rceil$ is defined as the least integer greater than or equal to $\displaystyle N$.

Evaluate for x = 0.8 $\displaystyle x = 0.8$ :

$\left \lceil 4 x -0.4 \right \rceil + 5x+ 0.3$ $\displaystyle \left \lceil 4 x -0.4 \right \rceil + 5x+ 0.3$

Possible Answers:

6.3 $\displaystyle 6.3$

$\displaystyle 9$

7.3 $\displaystyle 7.3$

$\displaystyle 7$

$\displaystyle 8$

Correct answer:

7.3 $\displaystyle 7.3$

Explanation:

Substitute x = 0.8 $\displaystyle x = 0.8$:

$\left \lceil 4 x -0.4 \right \rceil + 5x+ 0.3$ $\displaystyle \left \lceil 4 x -0.4 \right \rceil + 5x+ 0.3$

$= \left \lceil 4 \cdot 0.8 -0.4 \right \rceil + 5 \cdot 0.8 + 0.3$ $\displaystyle = \left \lceil 4 \cdot 0.8 -0.4 \right \rceil + 5 \cdot 0.8 + 0.3$

$= \left \lceil 3.2 -0.4 \right \rceil + 4 + 0.3$ $\displaystyle = \left \lceil 3.2 -0.4 \right \rceil + 4 + 0.3$

$= \left \lceil 2.8 \right \rceil + 4 + 0.3$ $\displaystyle = \left \lceil 2.8 \right \rceil + 4 + 0.3$

$\displaystyle =3 + 4 + 0.3 = 7.3$

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Example Question #4 : Patterns

What number replaces the circle in this sequence?

$\bigcirc , \square , 8, 6, 12, 10, 20, 18, 36....$ $\displaystyle \bigcirc , \square , 8, 6, 12, 10, 20, 18, 36....$

Possible Answers:

$\displaystyle 4$

$\displaystyle 0$

$\displaystyle 2$

$\displaystyle 6$

$\displaystyle 8$

Correct answer:

$\displaystyle 6$

Explanation:

This sequence is alternately generated by subtracting two and multiplying by two.

We can therefore find the numbers that replace the square and the circle by reversing the pattern - alternately dividing by two and adding two beginning at 36:

$36 \div 2 = 18$ $\displaystyle 36 \div 2 = 18$

$\displaystyle 18 + 2 = 20$

$20 \div 2 =10$ $\displaystyle 20 \div 2 =10$

$\displaystyle 10 + 2 = 12$

$12 \div 2 = 6$ $\displaystyle 12 \div 2 = 6$

$\displaystyle 6 + 2 = 8$

$8 \div 2 = 4$ $\displaystyle 8 \div 2 = 4$ - this replaces the circle

4+2=6 $\displaystyle 4+2=6$ - this replaces the square

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Example Question #2 : Algebra

In the following number sequence, what number goes in place of the circle?

$1, 9, 25, 49, 81, 121, \bigcirc...$ $\displaystyle 1, 9, 25, 49, 81, 121, \bigcirc...$

Possible Answers:

141 $\displaystyle 141$

169 $\displaystyle 169$

144 $\displaystyle 144$

196 $\displaystyle 196$

161 $\displaystyle 161$

Correct answer:

169 $\displaystyle 169$

Explanation:

The sequence comprises the squares of the odd integers, in order:

$1^{2} = 1$ $\displaystyle 1^{2} = 1$

$3^{2} = 9$ $\displaystyle 3^{2} = 9$

...

$11^{2} = 121$ $\displaystyle 11^{2} = 121$

The next number, which replaces the circle, is

$13^{2} = 169$ $\displaystyle 13^{2} = 169$.

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Example Question #7 : Patterns

Define an operation on the set of real numbers as follows:

$a\; \S\; b = a^{2} + b ^{2}$ $\displaystyle a\; \S\; b = a^{2} + b ^{2}$

Evaluate:

$-4\; \S\; \left ( -5 \right )$ $\displaystyle -4\; \S\; \left ( -5 \right )$

Possible Answers:

$\displaystyle 81$

$\displaystyle -81$

The expression is undefined.

$\displaystyle 41$

$\displaystyle -41$

Correct answer:

$\displaystyle 41$

Explanation:

$a\; \S\; b = a^{2} + b ^{2}$ $\displaystyle a\; \S\; b = a^{2} + b ^{2}$

$-4\; \S\; \left ( -5 \right ) = \left ( -4 \right )^{2} + \left ( -5\right ) ^{2}$ $\displaystyle -4\; \S\; \left ( -5 \right ) = \left ( -4 \right )^{2} + \left ( -5\right ) ^{2}$

$= \left ( -4 \right )\times \left ( -4 \right ) + \left ( -5\right ) \times \left ( -5 \right )$ $\displaystyle = \left ( -4 \right )\times \left ( -4 \right ) + \left ( -5\right ) \times \left ( -5 \right )$

$\displaystyle = 16 +25 = 41$

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Example Question #2 : Ssat Upper Level Quantitative (Math)

In the following number sequence, what number goes in place of the circle?

$20, 10, 16, 8, 14, 7, \square, \bigcirc...$ $\displaystyle 20, 10, 16, 8, 14, 7, \square, \bigcirc...$

Possible Answers:

$11\frac{1}{2}$ $\displaystyle 11\frac{1}{2}$

$7\frac{1}{2}$ $\displaystyle 7\frac{1}{2}$

$9\frac{1}{2}$ $\displaystyle 9\frac{1}{2}$

$10\frac{1}{2}$ $\displaystyle 10\frac{1}{2}$

$6\frac{1}{2}$ $\displaystyle 6\frac{1}{2}$

Correct answer:

$6\frac{1}{2}$ $\displaystyle 6\frac{1}{2}$

Explanation:

The sequence is generated by alternately dividing by 2 and adding 6:

$20 \div 2 = 10$ $\displaystyle 20 \div 2 = 10$

$\displaystyle 10 + 6 = 16$

$16 \div 2 = 8$ $\displaystyle 16 \div 2 = 8$

$\displaystyle 8 + 6 = 14$

$14 \div 2 = 7$ $\displaystyle 14 \div 2 = 7$

$\displaystyle 7 + 6 = 13$ - This number replaces the square.

$13\div 2 = 6\frac{1}{2}$ $\displaystyle 13\div 2 = 6\frac{1}{2}$ - This number replaces the circle.

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All SSAT Upper Level Math Resources

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SSAT Upper Level Math : Algebra

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

All SSAT Upper Level Math Resources

Example Questions

Example Question #1 : Ssat Upper Level Quantitative (Math)

Example Question #1 : Patterns

Example Question #2 : Patterns

Example Question #2 : Patterns

Example Question #3 : Patterns

Example Question #2 : Algebra

Example Question #4 : Patterns

Example Question #2 : Algebra

Example Question #7 : Patterns

Example Question #2 : Ssat Upper Level Quantitative (Math)

All SSAT Upper Level Math Resources

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