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ISEE Middle Level Quantitative : How to multiply variables

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 #31 : Operations

Mulitply:

$\displaystyle (x-7 )(x+5)$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The FOIL method works nicely here:

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

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Example Question #2 : How To Multiply Variables

Which quantity is greater?

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

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

Possible Answers:

(a) $\displaystyle (a)$ is greater

It is not possible to tell from the information given

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

(b) $\displaystyle (b)$ is greater

Correct answer:

(a) $\displaystyle (a)$ is greater

Explanation:

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

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

(a) $\displaystyle (a)$ is the greater amount.

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Example Question #31 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which quantity is greater?

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

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

Possible Answers:

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

(b) $\displaystyle (b)$ is greater

(a) $\displaystyle (a)$ is greater

It is not possible to tell from the information given

Correct answer:

(b) $\displaystyle (b)$ is greater

Explanation:

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

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

(b) $\displaystyle (b)$ is the greater amount.

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Example Question #34 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which quantity is greater?

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

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

Possible Answers:

(b) $\displaystyle (b)$ is greater

It is not possible to tell from the information given

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

(a) $\displaystyle (a)$ is greater

Correct answer:

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

Explanation:

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

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

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal.

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Example Question #35 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which quantity is greater?

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

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

Possible Answers:

(a) $\displaystyle (a)$ is greater

It is not possible to tell from the information given

(b) $\displaystyle (b)$ is greater

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

Correct answer:

(a) $\displaystyle (a)$ is greater

Explanation:

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

Use the distributive property to expand.

$\displaystyle (a)= (-4)x^2+(-4)(2x)-(-4)(10)$

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

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

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

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

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

40>-40 $\displaystyle 40>-40$

$\displaystyle -4x^2-8x+40>-4x^2-8x-40$

(a)>(b) $\displaystyle (a)>(b)$

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Example Question #36 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which quantity is greater?

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

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

Possible Answers:

(a) $\displaystyle (a)$ is greater

(b) $\displaystyle (b)$ is greater

It is not possible to tell from the information given

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

Correct answer:

(b) $\displaystyle (b)$ is greater

Explanation:

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

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

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

Since 3>-3 $\displaystyle 3>-3$, we get:

$\displaystyle -3x^3+3>-3x^3-3$

(b)>(a) $\displaystyle (b)>(a)$

(b) $\displaystyle (b)$ is the greater quantity.

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Example Question #37 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which quantity is greater?

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

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

Possible Answers:

(a) $\displaystyle (a)$ is greater

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

(b) $\displaystyle (b)$ is greater

It is not possible to tell from the information given

Correct answer:

(a) $\displaystyle (a)$ is greater

Explanation:

Use the distributive property to evaluate each quantity.

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

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

$\displaystyle (a)=-2x^4-2x^2+14$

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

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

$\displaystyle (b)= -2x^4-2x^2+12$

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

14>12 $\displaystyle 14>12$

$\displaystyle -2x^4-2x^2+14>-2x^4-2x^2+12$

(a)>(b) $\displaystyle (a)>(b)$

(a) $\displaystyle (a)$ is the greater quantity

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Example Question #2 : How To Multiply Variables

Which quantity is greater?

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

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

Possible Answers:

(a) $\displaystyle (a)$ is greater

It is not possible to tell from the information given

(b) $\displaystyle (b)$ is greater

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

Correct answer:

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

Explanation:

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

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

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal.

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Example Question #39 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which quantity is greater?

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

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

Possible Answers:

It is not possible to tell from the information given

(a) $\displaystyle (a)$ is greater

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

(b) $\displaystyle (b)$ is greater

Correct answer:

(b) $\displaystyle (b)$ is greater

Explanation:

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

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

(b) $\displaystyle (b)$ is the greater amount.

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Example Question #40 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which of the following is equivalent to 20y $\displaystyle 20y$ ?

Possible Answers:

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

$\displaystyle 5(3y + y)$

4 (5y) $\displaystyle 4 (5y)$

5 (4y) $\displaystyle 5 (4y)$

$\displaystyle 4 (3y + 2y)$

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 $\displaystyle 20y$.

By the associative property of multiplication,

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

and

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

By combining like terms, and applying the distribution property:

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

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

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All ISEE Middle Level Quantitative Resources

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ISEE Middle Level Quantitative : How to multiply variables

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

All ISEE Middle Level Quantitative Resources

Example Questions

Example Question #31 : Operations

Example Question #2 : How To Multiply Variables

Example Question #31 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Example Question #34 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Example Question #35 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Example Question #36 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Example Question #37 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Example Question #2 : How To Multiply Variables

Example Question #39 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Example Question #40 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

All ISEE Middle Level Quantitative Resources

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