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ISEE Middle Level Quantitative : How to divide fractions

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Example Question #11 : How To Divide Fractions

is a positive number, and $B = A \div \frac{1}{2 }$ . Which is the greater quantity?

(a) The reciprocal of

(b) The reciprocal of

Possible Answers:

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

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

$B = A \div \frac{1}{2 } = A \cdot 2 = 2A$ .

The reciprocal of is therefore $\frac{1}{B} = \frac{1}{2A} = \frac{1}{2} \cdot \frac{1}{A}$ - that is, $\frac{1}{2}$ times the reciprocal of . The reciprocal of a positive number must be positive, and $1 > \frac{1}{2}$ , so

$1 \cdot \frac{1}{A} > \frac{1}{2} \cdot \frac{1}{A}$

$\frac{1}{A} > \frac{1}{B}$

The reciprocal of is the greater number.

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Example Question #12 : How To Divide Fractions

is a positive number. Which is the greater quantity?

(a) The reciprocal of $Y - \frac{1}{2}$

(b) The reciprocal of $Y + \frac{1}{2}$

Possible Answers:

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

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

Correct answer:

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

Explanation:

We examine two scenarios to demonstrate that the given information is insufficient.

Case 1: $Y = 2\frac{1}{2}$

$Y - \frac{1}{2} = 2 \frac{1}{2 }- \frac{1}{2} = 2$

The reciprocal of this is $\frac{1}{2}$ .

$Y + \frac{1}{2} = 2 \frac{1}{2 }+ \frac{1}{2} = 3$

The reciprocal of this is $\frac{1}{3}$ .

$\frac{1}{2} > \frac{1}{3}$ , so $Y - \frac{1}{2}$ has the greater reciprocal.

Case 2: $Y = \frac{1}{4}$ .

$Y - \frac{1}{2} = \frac{1}{4 }- \frac{1}{2} = \frac{1}{4 }- \frac{2}{4} = - \frac{1}{4 }$

The reciprocal of this is .

$Y + \frac{1}{2} = \frac{1}{4 }+ \frac{1}{2} = \frac{1}{4 }+ \frac{2}{4} = \frac{3}{4 }$

The reciprocal of this is $\frac{4 }{3}$ .

$\frac{4 }{3} > -4$ , so $Y + \frac{1}{2}$ has the greater reciprocal.

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Example Question #13 : How To Divide Fractions

is a negative number. Which is the greater quantity?

(a) The reciprocal of

(b) The reciprocal of Y - 1

Possible Answers:

(a) is the greater quantity

(a) and (b) are equal

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:

is negative, so we will let X = -Y , making positive. It follows that Y = -X .

The reciprocal of is $\frac{1}{Y} = \frac{1}{ -X} = - \frac{1}{X}$ ;

the reciprocal of Y - 1 is $\frac{1}{Y-1} = \frac{1}{ -X-1 } = - \frac{1}{X+1}$ .

and X+ 1 are both positive; X+1 > X , so $\frac{1}{X+1}< \frac{1}{X }$ , and $- \frac{1}{X+1}> -\frac{1}{X }$ . Therefore, the reciprocal of Y-1 is the greater quantity.

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ISEE Middle Level Quantitative : How to divide fractions

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

All ISEE Middle Level Quantitative Resources

Example Questions

Example Question #11 : How To Divide Fractions

Example Question #12 : How To Divide Fractions

Example Question #13 : How To Divide Fractions

All ISEE Middle Level Quantitative Resources

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