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ISEE Middle Level Quantitative : Operations

Study concepts, example questions & explanations for ISEE Middle Level Quantitative

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All ISEE Middle Level Quantitative Resources

6 Diagnostic Tests 110 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Multiply Variables

Mulitply:

(x-7 )(x+5)

Possible Answers:

$x^{2} -35$

$x^{2} -2x-35$

$x^{2} +2x-35$

$x^{2} -35x-12$

$x^{2} -12x-35$

Correct answer:

$x^{2} -2x-35$

Explanation:

The FOIL method works nicely here:

$(x-7 )(x+5) = x \cdot x + x \cdot5 + (-7) \cdot x + (-7) \cdot 5 = x^{2} +5x -7x -35 = x^{2} - 2x -35$

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Example Question #2 : How To Multiply Variables

Which quantity is greater?

$(a)\ (2)^{-3}$

$(b)\ (-2)^{3}$

Possible Answers:

(b) is greater

(a) and (b) are equal

(a) is greater

It is not possible to tell from the information given

Correct answer:

(a) is greater

Explanation:

$(a) =(2)^{-3} = \frac{1}{(2)^{3}}=\frac{1}{8}$

$(b)= (-2)^{3}=(-2)(-2)(-2)=-8$

(a) is the greater amount.

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Example Question #3 : How To Multiply Variables

Which quantity is greater?

$(a)\ (-\frac{1}{2})^{4}$

$(b)\ (-\frac{1}{2})^{-4}$

Possible Answers:

(b) is greater

(a) and (b) are equal

(a) is greater

It is not possible to tell from the information given

Correct answer:

(b) is greater

Explanation:

$(a)= (-\frac{1}{2})^{4} = (-\frac{1}{2})(-\frac{1}{2})(-\frac{1}{2})(-\frac{1}{2})=\frac{1}{16}$

$(b) =(-\frac{1}{2})^{-4}=\frac{1}{(-\frac{1}{2})^{4}}=\frac{1}{(-\frac{1}{2})(-\frac{1}{2})(-\frac{1}{2})(-\frac{1}{2})}=\frac{1}{\frac{1}{16}}=16$

(b) is the greater amount.

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Example Question #1 : How To Multiply Variables

Which quantity is greater?

$(a)\ (3)^{-2}$

$(b)\ (\frac{1}{3})^{2}$

Possible Answers:

It is not possible to tell from the information given

(b) is greater

(a) is greater

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

$(a)= (3)^{-2}=\frac{1}{3^{2}}=\frac{1}{9}$

$(b)=(\frac{1}{3})^{2}=(\frac{1}{3})(\frac{1}{3})=\frac{1}{9}$

(a) and (b) are equal.

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Example Question #2 : How To Multiply Variables

Which quantity is greater?

$(a)\ -4(x^{2}+2x-10)$

$(b)\ -4x^2-8x-40$

Possible Answers:

(a) is greater

It is not possible to tell from the information given

(b) is greater

(a) and (b) are equal

Correct answer:

(a) is greater

Explanation:

$(a)= -4(x^{2}+2x-10)$

Use the distributive property to expand.

(a)= (-4)x^2+(-4)(2x)-(-4)(10)

(a)=-4x^2-8x+40

(b) =-4x^2-8x-40

To compare the expressions, we can remove common factors. (-4x^2) and (-8x) are common in both expressions, so they can be removed:

(a)=(-4x^2-8x)+40

(b)=(-4x^2-8x)-40

40>-40

-4x^2-8x+40>-4x^2-8x-40

(a)>(b)

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Example Question #6 : How To Multiply Variables

Which quantity is greater?

$(a)\ -3(x^3+1)$

$(b)\ 3(-x^3+1)$

Possible Answers:

It is not possible to tell from the information given

(b) is greater

(a) is greater

(a) and (b) are equal

Correct answer:

(b) is greater

Explanation:

Use the distributive property to evaluate (a) and (b) .

(a)= -3(x^3+1)=-3x^3-3

(b)= 3(-x^3+1)=-3x^3+3

Since 3>-3 , we get:

-3x^3+3>-3x^3-3

(b)>(a)

(b) is the greater quantity.

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Example Question #7 : How To Multiply Variables

Which quantity is greater?

$(a)\ -2(x^4+x^2-7)$

$(b)\ 2(-x^4-x^2+6)$

Possible Answers:

It is not possible to tell from the information given

(b) is greater

(a) is greater

(a) and (b) are equal

Correct answer:

(a) is greater

Explanation:

Use the distributive property to evaluate each quantity.

$(a) =-2(x^{4}+x^2-7)$

(a)= (-2)x^4+(-2)(x^2)-(-2)(7)

(a)=-2x^4-2x^2+14

(b)= 2(-x^4-x^2+6)

(b)= -(2)x^4-(2)(x^2)+(2)(6)

(b)= -2x^4-2x^2+12

(-2x^4) and (-2x^2) are common in both expressions, so we can write:

14>12

-2x^4-2x^2+14>-2x^4-2x^2+12

(a)>(b)

(a) is the greater quantity

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Example Question #8 : How To Multiply Variables

Which quantity is greater?

$(a)\ (9)^{-2}$

$(b)\ (-3)^{-4}$

Possible Answers:

It is not possible to tell from the information given

(b) is greater

(a) is greater

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

$(a)\ (9)^{-2}=\frac{1}{9^2}=\frac{1}{81}$

$(b)\ (-3)^{-4}=\frac{1}{(-3)^4}=\frac{1}{(-3)(-3)(-3)(-3)}=\frac{1}{81}$

(a) and (b) are equal.

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Example Question #3 : How To Multiply Variables

Which quantity is greater?

$(a)\ (-4)^{-2}$

$(b)\ (-4)^{2}$

Possible Answers:

(b) is greater

(a) and (b) are equal

(a) is greater

It is not possible to tell from the information given

Correct answer:

(b) is greater

Explanation:

$(a) =(-4)^{-2} = \frac{1}{(-4)^{2}}=\frac{1}{(-4)(-4)}=\frac{1}{16}$

$(b)= (-4)^{2}=(-4)(-4)=16$

(b) is the greater amount.

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Example Question #10 : How To Multiply Variables

Which of the following is equivalent to 20y ?

Possible Answers:

4 (5y)

5(3y + y)

4 (3y + 2y)

All four of the expressions given in the other responses are correct.

5 (4y)

Correct answer:

All four of the expressions given in the other responses are correct.

Explanation:

We demonstrate that all four of the expressions are equivalent to 20y .

By the associative property of multiplication,

$4 (5y) = (4 \cdot 5) \cdot y = 20y$

and

$5 (4y) = (5\cdot 4) \cdot y = 20y$

By combining like terms, and applying the distribution property:

$5(3y + y) = 5(3y + 1 y) = 5[ (3 + 1 ) y] = 5 \left ( 4 y \right )= 20y$

$4 (3y + 2y) = 4\left [ (3 + 2) y \right ] = 4\left ( 5y\right ) = 20y$

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All ISEE Middle Level Quantitative Resources

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ISEE Middle Level Quantitative : Operations

Study concepts, example questions & explanations for ISEE Middle Level Quantitative

All ISEE Middle Level Quantitative Resources

Example Questions

Example Question #1 : How To Multiply Variables

Example Question #2 : How To Multiply Variables

Example Question #3 : How To Multiply Variables

Example Question #1 : How To Multiply Variables

Example Question #2 : How To Multiply Variables

Example Question #6 : How To Multiply Variables

Example Question #7 : How To Multiply Variables

Example Question #8 : How To Multiply Variables

Example Question #3 : How To Multiply Variables

Example Question #10 : How To Multiply Variables

All ISEE Middle Level Quantitative Resources

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