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ISEE Middle Level Quantitative : Variables

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

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All ISEE Middle Level Quantitative Resources

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Example Questions

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Example Question #8 : How To Subtract Variables

5 < A < 7

5 < B < 7

5 <C < 7

Which is the greater quantity?

(a) A - B + C

(b) 9

Possible Answers:

(a) and (b) are equal

(a) is the greater quantity

It is impossible to determine which is greater from the information given

(b) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

5 < A < 7

5 <C < 7

also, since 5 < B < 7 , it follows that

-7 < -B < -5

A - B + C = A + (-B)+ C , and by the inequality properties,

A + (-B)+ C < 7 + (-5) + 7

A -B + C < 7 -5 + 7

A -B + C < 9

making 9 the greater quantity.

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Example Question #1 : How To Divide Variables

What is the value of $9\div0.1$ ?

Possible Answers:

4.5

Correct answer:

Explanation:

$9\div 0.1=\frac{9}{0.1}=\frac{90}{1}=90$

(The numerator and the denominator are both multiplied by 10 in order to convert the fraction to whole numbers.)

Therefore, 90 is the correct answer.

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Example Question #2 : How To Divide Variables

What is the value of $6\div0.2$ ?

Possible Answers:

300

Correct answer:

Explanation:

$6\div 0.2=\frac{6}{0.2}=\frac{60}{2}=30$

(The numerator and the denominator are both multiplied by 10 in order to convert the fraction to whole numbers.)

Therefore, 30 is the correct answer.

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Example Question #1 : How To Divide Variables

Simplify:

$\frac{4x^2y}{12xy^3z}$

Possible Answers:

$\frac{xy^2z}{3}$

$\frac{4x^3}{12y^4z}$

$\frac{1x}{3y^2z}$

3xy^3z

$\frac{1x^2y}{4xy^3z}$

Correct answer:

$\frac{1x}{3y^2z}$

Explanation:

Since all of the variables are positive powers, this is easy. Start by reducing the numeric coefficient:

$\frac{4x^2y}{12xy^3z} = \frac{1x^2y}{3xy^3z}$

Next, cancel out the variables. Subtract the smaller power from the larger one. Remember that if there is no power listed, it is 1:

$\frac{1x^2y}{3xy^3z} = \frac{1x}{3y^2z}$

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Example Question #1 : How To Divide Variables

Simplify:

$\frac{45a^2tz^2}{3tz^3}$

Possible Answers:

$\frac{15a^2}{z}$

$\frac{a^2}{15tz}$

$\frac{45a^2t}{a^2z^2}$

$\frac{15a^2t}{z^2}$

$\frac{15z^2}{at}$

Correct answer:

$\frac{15a^2}{z}$

Explanation:

Since all of the variables are positive powers, this is easy. Start by reducing the numeric coefficient:

$\frac{45a^2tz^2}{3tz^3} = \frac{15a^2tz^2}{tz^3}$

Next, cancel out the variables. Subtract the smaller power from the larger one. Remember that if there is no power listed, it is 1:

$\frac{15a^2tz^2}{tz^3} = \frac{15a^2}{z}$

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Example Question #21 : Variables

Simplify:

$\frac{5a^{-2}x}{ax^2}$

Possible Answers:

$\frac{1}{5a^3x}$

$\frac{a}{25a^2x}$

$\frac{5}{a^3x}$

$\frac{5a}{x}$

$\frac{x^2}{25a}$

Correct answer:

$\frac{5}{a^3x}$

Explanation:

The easiest way to begin with questions like this one is to "flip" the negative exponents to the top or bottom of the fraction. When you do this, you make the exponent's sign positive:

$\frac{5a^{-2}x}{ax^2} = \frac{5x}{a^2ax^2}$

Now, since all of the variables are positive powers, this is easy. Normally, you would begin by reducing the numeric coefficient. This is not necessary since there is only a 5 in the numerator. Therefore, combine the like variables first:

$\frac{5x}{a^2ax^2} = \frac{5x}{a^3x^2}$

Next, cancel out the variables. Subtract the smaller power from the larger one. Remember that if there is no power listed, it is 1:

$\frac{5x}{a^3x^2} = \frac{5}{a^3x}$

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Example Question #1 : How To Divide Variables

Simplify:

$\frac{35x^{-3}y^2z^{-1}}{105ax^2y^{-2}{}z^{-5}}$

Possible Answers:

$\frac{xy}{3az^4}$

$\frac{y^4z^4}{3ax^5}$

$\frac{y^2z^4}{35ax^5z^1}$

$\frac{xyz^4}{3a}$

$\frac{3y^4z^4}{35ax^5}$

Correct answer:

$\frac{y^4z^4}{3ax^5}$

Explanation:

The easiest way to begin with questions like this one is to "flip" the negative exponents to the top or bottom of the fraction. When you do this, you make the exponent's sign positive:

$\frac{35x^{-3}y^2z^{-1}}{105ax^2y^{-2}z^{-5}} = \frac{35y^2y^2z^5}{105ax^3x^2z^1}$

Next, go ahead and reduce the numeric coefficient:

$\frac{35y^2y^2z^5}{105ax^3x^2z^1} = \frac{1y^2y^2z^5}{3ax^3x^2z^1}$

Then, combine the like variables first:

$\frac{1y^2y^2z^5}{3ax^3x^2z^1} = \frac{1y^4z^5}{3ax^5z^1}$

Finally, cancel out the variables. Subtract the smaller power from the larger one. Remember that if there is no power listed, it is 1:

$\frac{1y^4z^5}{3ax^5z^1} = \frac{1y^4z^4}{3ax^5} = \frac{y^4z^4}{3ax^5}$

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Example Question #1 : How To Divide Variables

Solve for :

ax+4a = 3a^2

$a \ne 0$

Possible Answers:

3-4a

3a^2+4a-x

4-3a

3a-4

3a^2 - 4a

Correct answer:

3a-4

Explanation:

To start, notice that the left side of the equation has a common factor of . You can factor this out:

ax+4a = 3a^2

becomes

a(x+4) = 3a^2

Next, you can divide both sides by since you know that it does not equal :

$x+4 = \frac{3a^2}{a}$

Simplifying the right side, you get:

x+4 = 3a

Finally, subtract from both sides:

x = 3a-4

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Example Question #1 : How To Divide Variables

Solve for :

5xb-25x=450x^3

$x \ne 0$

Possible Answers:

90x^3+5x

$\frac{450x^3}{5x-25}$

$\frac{-45x^2}{2}$

No possible answer

90x^2+5

Correct answer:

90x^2+5

Explanation:

To start, notice that the left side of the equation has common factors of and . You can factor these out:

5xb-25x=450x^3

becomes...

5x(b-5)=450x^3

Next, you can divide both sides by since you know that it does not equal :

$(b-5)=\frac{450x^3}{5x}$

Simplify the right side by canceling the and reducing the :

(b-5)=90x^2

Finally, add to both sides:

b=90x^2+5

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Example Question #9 : How To Divide Variables

is the multiplicative inverse of . Which is the greater quantity?

(a) $\frac{a}{b}$

(b)

Possible Answers:

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(a) is the greater quantity

Correct answer:

It is impossible to determine which is greater from the information given

Explanation:

is the multiplicative inverse of , so, by definition, ab= 1 . We show that this is insufficient information, however, with two cases.

Case 1: $a = 2, b= \frac{1}{2}$

$ab = 2 \cdot \frac{1}{2} = 1$

$\frac{a}{b} = a \div b = 2 \div \frac{1}{2} = 2 \cdot 2 = 4 > 1$

Case 2: $a= \frac{1}{2}, b = 2$

$ab = \frac{1}{2} \cdot 2 = 1$

$\frac{a}{b} = a \div b = \frac{1}{2} \div 2 = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} < 1$

Therefore,it is not clear which, if either, is the greater, $\frac{a}{b}$ or 1.

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ISEE Middle Level Quantitative : Variables

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

All ISEE Middle Level Quantitative Resources

Example Questions

Example Question #8 : How To Subtract Variables

Example Question #1 : How To Divide Variables

Example Question #2 : How To Divide Variables

Example Question #1 : How To Divide Variables

Example Question #1 : How To Divide Variables

Example Question #21 : Variables

Example Question #1 : How To Divide Variables

Example Question #1 : How To Divide Variables

Example Question #1 : How To Divide Variables

Example Question #9 : How To Divide Variables

All ISEE Middle Level Quantitative Resources

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