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SSAT Upper Level Math : Basic Addition, Subtraction, Multiplication and Division

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Example Questions

Example Question #11 : How To Subtract

Define a function $\displaystyle f$ on the real numbers as follows:

$f(x) = 100 -\left ( \sqrt{x} - \sqrt[4]{x} \right )$ $\displaystyle f(x) = 100 -\left ( \sqrt{x} - \sqrt[4]{x} \right )$

Evaluate f(-1) $\displaystyle f(-1)$.

Possible Answers:

100 $\displaystyle 100$

$\displaystyle 98$

102 $\displaystyle 102$

The expression is undefined.

The correct answer is not among the other responses.

Correct answer:

The expression is undefined.

Explanation:

$f(x) = 100 -\left ( \sqrt{x} - \sqrt[4]{x} \right )$ $\displaystyle f(x) = 100 -\left ( \sqrt{x} - \sqrt[4]{x} \right )$

$f(-1) = 100 -\left ( \sqrt{-1} - \sqrt[4]{-1} \right )$ $\displaystyle f(-1) = 100 -\left ( \sqrt{-1} - \sqrt[4]{-1} \right )$

Since even-numbered roots of negative numbers are not defined for real-valued functions, the expression is undefined.

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Example Question #22 : Basic Addition, Subtraction, Multiplication And Division

Define an operation $\bigtriangleup$ $\displaystyle \bigtriangleup$ on the real numbers as follows:

For all real numbers a, b $\displaystyle a, b$:

$a \bigtriangleup b = \sqrt[3]{a} - \sqrt[4]{b}$ $\displaystyle a \bigtriangleup b = \sqrt[3]{a} - \sqrt[4]{b}$.

Evaluate $(-1) \bigtriangleup (-1)$ $\displaystyle (-1) \bigtriangleup (-1)$.

Possible Answers:

$\displaystyle 1$

$\displaystyle -2$

$\displaystyle 2$

$\displaystyle -1$

The expression is undefined on the real numbers.

Correct answer:

The expression is undefined on the real numbers.

Explanation:

$a \bigtriangleup b = \sqrt[3]{a} - \sqrt[4]{b}$ $\displaystyle a \bigtriangleup b = \sqrt[3]{a} - \sqrt[4]{b}$

$(-1) \bigtriangleup (-1) = \sqrt[3]{-1} - \sqrt[4]{-1}$ $\displaystyle (-1) \bigtriangleup (-1) = \sqrt[3]{-1} - \sqrt[4]{-1}$

However, $\sqrt[4]{-1}$ $\displaystyle \sqrt[4]{-1}$ is undefined in the real numbers; subsequently, so is $(-1) \bigtriangleup (-1)$ $\displaystyle (-1) \bigtriangleup (-1)$.

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Example Question #31 : Number Concepts And Operations

$\dpi{100} \frac{\frac{1}{2}\times \frac{1}{3}}{\frac{1}{9}}=$ $\displaystyle \dpi{100} \frac{\frac{1}{2}\times \frac{1}{3}}{\frac{1}{9}}=$

Possible Answers:

$\dpi{100} \frac{3}{2}$ $\displaystyle \dpi{100} \frac{3}{2}$

$\dpi{100} \frac{1}{9}$ $\displaystyle \dpi{100} \frac{1}{9}$

$\dpi{100} \frac{1}{54}$ $\displaystyle \dpi{100} \frac{1}{54}$

$\dpi{100} \frac{2}{3}$ $\displaystyle \dpi{100} \frac{2}{3}$

Correct answer:

$\dpi{100} \frac{3}{2}$ $\displaystyle \dpi{100} \frac{3}{2}$

Explanation:

First multiply the fraction in the numerator.

$\dpi{100} \frac{1}{2}\times \frac{1}{3}=\frac{1\times 1}{2\times 3}=\frac{1}{6}$ $\displaystyle \dpi{100} \frac{1}{2}\times \frac{1}{3}=\frac{1\times 1}{2\times 3}=\frac{1}{6}$

Now we have $\dpi{100} \frac{\frac{1}{6}}{\frac{1}{9}}$ $\displaystyle \dpi{100} \frac{\frac{1}{6}}{\frac{1}{9}}$

Never divide fractions. We multiply the numerator by the reciprocal of the denominator.

$\dpi{100} \frac{\frac{1}{6}}{\frac{1}{9}}=\frac{1}{6}\div \frac{1}{9}=\frac{1}{6}\times \frac{9}{1}=\frac{1\times 9}{6\times 1}=\frac{9}{6}=\frac{3}{2}$ $\displaystyle \dpi{100} \frac{\frac{1}{6}}{\frac{1}{9}}=\frac{1}{6}\div \frac{1}{9}=\frac{1}{6}\times \frac{9}{1}=\frac{1\times 9}{6\times 1}=\frac{9}{6}=\frac{3}{2}$

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Example Question #32 : Number Concepts And Operations

Which of these expressions is the greatest?

Possible Answers:

One fourth of 0.2 $\displaystyle 0.2$

One fifth of 0.25 $\displaystyle 0.25$

All of these expressions are equivalent

Twenty-five percent of one fifth

Twenty percent of one fourth

Correct answer:

All of these expressions are equivalent

Explanation:

The easiest way to see that all four are equivalent is to convert each to a decimal product, noting that twenty percent and one-fifth are equal to 0.2 $\displaystyle 0.2$, and twenty-five percent and one fourth are equal to 0.25 $\displaystyle 0.25$.

One fourth of 0.2: $\frac{1}{4}*0.2=\frac{1}{4}*\frac{2}{10}=\frac{2}{40}=\frac{1}{20}$ $\displaystyle \frac{1}{4}*0.2=\frac{1}{4}*\frac{2}{10}=\frac{2}{40}=\frac{1}{20}$

One fifth of 0.25: $\frac{1}{5}*0.25=\frac{1}{5}*\frac{25}{100}=\frac{25}{500}=\frac{5}{100}=\frac{1}{20}$ $\displaystyle \frac{1}{5}*0.25=\frac{1}{5}*\frac{25}{100}=\frac{25}{500}=\frac{5}{100}=\frac{1}{20}$

Twenty-five percent of one fifth: $*\frac{1}{5}=\frac{25}{100}*\frac{1}{5}=\frac{25}{500}=\frac{1}{20}$ $\displaystyle *\frac{1}{5}=\frac{25}{100}*\frac{1}{5}=\frac{25}{500}=\frac{1}{20}$

Twenty percent of one fourth: $*\frac{1}{4}=\frac{20}{100}*\frac{1}{4}=\frac{20}{400}=\frac{1}{20}$ $\displaystyle *\frac{1}{4}=\frac{20}{100}*\frac{1}{4}=\frac{20}{400}=\frac{1}{20}$

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Example Question #21 : Basic Addition, Subtraction, Multiplication And Division

Write .007341 in scientific notation.

Possible Answers:

$7.341\times10^{^{-3}}$ $\displaystyle 7.341\times10^{^{-3}}$

$7.341\times10^{-2}$ $\displaystyle 7.341\times10^{-2}$

$7.341\times10^{^{3}}$ $\displaystyle 7.341\times10^{^{3}}$

$7.341\times10^{-1}$ $\displaystyle 7.341\times10^{-1}$

$73.41\times10^{^{-2}}$ $\displaystyle 73.41\times10^{^{-2}}$

Correct answer:

$7.341\times10^{^{-3}}$ $\displaystyle 7.341\times10^{^{-3}}$

Explanation:

The answer is $7.341\times10^{^{-3}}$ $\displaystyle 7.341\times10^{^{-3}}$.

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Example Question #22 : Basic Addition, Subtraction, Multiplication And Division

If X, Y, Z $\displaystyle X, Y, Z$ are consecutive negative numbers, which of the following is false?

Possible Answers:

$\left ( xyz^{}\right )^{2}>0$ $\displaystyle \left ( xyz^{}\right )^{2}>0$

xyz>0 $\displaystyle xyz>0$

x+y+z<0 $\displaystyle x+y+z< 0$

xy>0 $\displaystyle xy>0$

xyz<0 $\displaystyle xyz< 0$

Correct answer:

xyz>0 $\displaystyle xyz>0$

Explanation:

When three negative numbers are multiplied together, the product will be negative as well. All the other expressions are true.

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Example Question #27 : Basic Addition, Subtraction, Multiplication And Division

Fill in the circle to yield a true statement:

$5 \times \bigcirc \equiv 3 \mod 6$ $\displaystyle 5 \times \bigcirc \equiv 3 \mod 6$

Possible Answers:

$\displaystyle 3$

$\displaystyle 2$

$\displaystyle 1$

$\displaystyle 5$

$\displaystyle 4$

Correct answer:

$\displaystyle 3$

Explanation:

The problem is asking for a number whose product with 5 yields a number congruent to 3 in modulo 6 arithmetic - that is, a number which, when divided by 6, yields remainder 3. We multiply 5 by each choice and look for a product with this characteristic.

$5 \times 1 = 5$ $\displaystyle 5 \times 1 = 5$; $5 \div 6 = 0 \textrm{ R } 5$ $\displaystyle 5 \div 6 = 0 \textrm{ R } 5$

$5 \times 2 = 10$ $\displaystyle 5 \times 2 = 10$; $10 \div 6 = 1 \textrm{ R } 4$ $\displaystyle 10 \div 6 = 1 \textrm{ R } 4$

$5 \times 3= 15$ $\displaystyle 5 \times 3= 15$; $15 \div 6 = 2 \textrm{ R } 3$ $\displaystyle 15 \div 6 = 2 \textrm{ R } 3$

$5 \times 4=20$ $\displaystyle 5 \times 4=20$; $20 \div 6 = 3 \textrm{ R } 2$ $\displaystyle 20 \div 6 = 3 \textrm{ R } 2$

$5 \times 5 = 25$ $\displaystyle 5 \times 5 = 25$; $25 \div 6 = 4 \textrm{ R } 1$ $\displaystyle 25 \div 6 = 4 \textrm{ R } 1$

The only choice whose product with 5 yields a number congruent to 3 modulo 6 is 3, so this is the correct choice.

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Example Question #28 : Basic Addition, Subtraction, Multiplication And Division

You are asked to fill in all three circles in the statement

$\bigcirc \times \bigcirc \times \bigcirc \equiv 9 \mod 10$ $\displaystyle \bigcirc \times \bigcirc \times \bigcirc \equiv 9 \mod 10$

with the same number from the set

$\left \{ 0, 1, 2, 3,...9\right \}$ $\displaystyle \left \{ 0, 1, 2, 3,...9\right \}$

to make a true statement.

How many ways can you do this?

Possible Answers:

Two

One

Five

Four

Three

Correct answer:

One

Explanation:

The problem is asking for a number whose cube is a number congruent to 9 in modulo 10 arithmetic - that is, a number whose cube, when divided by 10, yields remainder 9. If the quotient of a number and 10 has remainder 9, then it is an integer that ends with the digit "9". Since this makes the cube odd, the number that is cubed must also be odd, so we need only test the five odd integers:

$1 ^{3} = 1$ $\displaystyle 1 ^{3} = 1$

$3^{3}= 27$ $\displaystyle 3^{3}= 27$

$5^{3} = 125$ $\displaystyle 5^{3} = 125$

$7 ^{3} = 343$ $\displaystyle 7 ^{3} = 343$

$9^{3} = 729$ $\displaystyle 9^{3} = 729$

Only 9 fits the criterion, so "one" is the correct response.

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Example Question #4 : How To Multiply

You are asked to fill in both circles in the statement

$\bigcirc \times \bigcirc \equiv 1 \mod 12$ $\displaystyle \bigcirc \times \bigcirc \equiv 1 \mod 12$

with the same number from the set

$\left \{ 0, 1, 2, 3,...11\right \}$ $\displaystyle \left \{ 0, 1, 2, 3,...11\right \}$

to make a true statement.

How many ways can you do this?

Possible Answers:

Two

None of the other responses is correct.

Four

None

Six

Correct answer:

Four

Explanation:

The problem is asking for a number whose square is a number congruent to 1 in modulo 12 arithmetic - that is, a number whose square, when divided by 12, yields remainder 1. This square must be odd, so the number squared must also be odd. Therefore, we need only test the odd integers. We see that:

$1^{2} = 1 ; 1 \div 12 = 0 \textrm{ R }1$ $\displaystyle 1^{2} = 1 ; 1 \div 12 = 0 \textrm{ R }1$

$3^{2} = 9 ; 9 \div 12 = 0 \textrm{ R }9$ $\displaystyle 3^{2} = 9 ; 9 \div 12 = 0 \textrm{ R }9$

$5^{2} = 25 ; 25 \div 12 = 2 \textrm{ R }1$ $\displaystyle 5^{2} = 25 ; 25 \div 12 = 2 \textrm{ R }1$

$7^{2} =49 ; 49 \div 12 = 4 \textrm{ R }1$ $\displaystyle 7^{2} =49 ; 49 \div 12 = 4 \textrm{ R }1$

$9^{2} = 81 ; 81 \div 12 = 8 \textrm{ R }9$ $\displaystyle 9^{2} = 81 ; 81 \div 12 = 8 \textrm{ R }9$

$11^{2} = 121; 121 \div 12 = 10 \textrm{ R }1$ $\displaystyle 11^{2} = 121; 121 \div 12 = 10 \textrm{ R }1$

Four of the integers have squares congruent to 1 in modulo 12 arithmetic.

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Example Question #5 : How To Multiply

Multiply:

$6 \times 2 \textrm{ lbs } 5 \textrm{ oz }$ $\displaystyle 6 \times 2 \textrm{ lbs } 5 \textrm{ oz }$

Possible Answers:

None of the other responses is correct.

$13 \textrm{ lbs } 14 \textrm{ oz }$ $\displaystyle 13 \textrm{ lbs } 14 \textrm{ oz }$

$14 \textrm{ lbs } 8 \textrm{ oz }$ $\displaystyle 14 \textrm{ lbs } 8 \textrm{ oz }$

$14 \textrm{ lbs } 6 \textrm{ oz }$ $\displaystyle 14 \textrm{ lbs } 6 \textrm{ oz }$

$14 \textrm{ lbs }$ $\displaystyle 14 \textrm{ lbs }$

Correct answer:

$13 \textrm{ lbs } 14 \textrm{ oz }$ $\displaystyle 13 \textrm{ lbs } 14 \textrm{ oz }$

Explanation:

We can write 2 pounds, 5 ounces as just ounces as follows:

$2 \textrm{ lbs } 5 \textrm{ oz } =\left ( 2 \times 16 + 5 \right ) \textrm{ oz } = 37 \textrm{ oz }$ $\displaystyle 2 \textrm{ lbs } 5 \textrm{ oz } =\left ( 2 \times 16 + 5 \right ) \textrm{ oz } = 37 \textrm{ oz }$

Multiply:

$6 \times 2 \textrm{ lbs } 5 \textrm{ oz }$ $\displaystyle 6 \times 2 \textrm{ lbs } 5 \textrm{ oz }$

$= 6 \times 37 \textrm{ oz }$ $\displaystyle = 6 \times 37 \textrm{ oz }$

$= 222 \textrm{ oz }$ $\displaystyle = 222 \textrm{ oz }$

Divide by 16, noting quotient and remainder, to get pounds and ounces:

$222 \div 16 = 13 \textrm{ R } 14$ $\displaystyle 222 \div 16 = 13 \textrm{ R } 14$

Therefore, the correct response is 13 pounds, 14 ounces.

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SSAT Upper Level Math : Basic Addition, Subtraction, Multiplication and Division

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

All SSAT Upper Level Math Resources

Example Questions

Example Question #11 : How To Subtract

Example Question #22 : Basic Addition, Subtraction, Multiplication And Division

Example Question #31 : Number Concepts And Operations

Example Question #32 : Number Concepts And Operations

Example Question #21 : Basic Addition, Subtraction, Multiplication And Division

Example Question #22 : Basic Addition, Subtraction, Multiplication And Division

Example Question #27 : Basic Addition, Subtraction, Multiplication And Division

Example Question #28 : Basic Addition, Subtraction, Multiplication And Division

Example Question #4 : How To Multiply

Example Question #5 : How To Multiply

All SSAT Upper Level Math Resources

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