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ISEE Middle Level Quantitative : Numbers and Operations

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Example Questions

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:

(b) is the greater quantity

(a) is the greater quantity

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

(a) and (b) are equal

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 : 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:

(a) is the greater quantity

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

(b) is the greater quantity

(a) and (b) are equal

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 : Fractions

is a negative number. Which is the greater quantity?

(a) The reciprocal of

(b) The reciprocal of Y - 1

Possible Answers:

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

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

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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Example Question #1 : How To Add Fractions

Express the sum as a fraction in lowest terms:

$3 \frac{1}{2} + 5 \frac{1}{3} + 4 \frac{1}{4}$

Possible Answers:

$12 \frac{1}{9}$

$13 \frac{1}{9}$

$12 \frac{11}{12}$

$13 \frac{1}{12}$

$12 \frac{1}{3}$

Correct answer:

$13 \frac{1}{12}$

Explanation:

Rewrite the fractions in terms of their least common denominator, 12.

(LCD(2,3,4)=12)

Add, then rewrite as a mixed fraction:

$\frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{6}{12} + \frac{4}{12} + \frac{3}{12} =\frac{13}{12} = 1 \frac{1}{12}$

Add the integers:

3 + 5 + 4 = 12

Now add the sums:

$12 + 1 \frac{1}{12} = 13 \frac{1}{12}$

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Example Question #2 : How To Add Fractions

$\small \frac{3}{4}+\frac{1}{3}$

Possible Answers:

$\small \frac{4}{7}$

$\small \frac{4}{12}$

$\small \frac{1}{3}$

$\small \frac{13}{12}$

$\small \frac{4}{3}$

Correct answer:

$\small \frac{13}{12}$

Explanation:

In order to add fractions we must find a common denominator. Since $\small 12$ is a multiple of both $\small 3$ and $\small 4$ , we must multiply the numerator and denominator of each fraction by a number to get a denomintor of $\small 12$ .

Since $\small 4$ times $\small 3$ is $\small 12$ , we can multiply the numerator and denominator of the first fraction by $\small 3$ .

$\small \frac{(3\cdot 3)}{(4\cdot 3)}= \frac{9}{12}$

Since $\small 3$ times $\small 4$ is $\small 12$ , we can multiply the numerator and demonimator of the second fraction by $\small 4$ .

$\small \frac{(1\cdot 4)}{(3\cdot 4)}=\frac{4}{12}$

Now we add together the numerators.

$\small \frac{9}{12}+\frac{4}{12}=\frac{13}{12}$

The answer is $\small \frac{13}{12}$ .

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Example Question #3 : How To Add Fractions

If a rectangle has a length of $\small \frac{1}{2}$ and a width of $\small \frac{1}{8}$ what is the perimeter of the rectangle, in simplest form?

Possible Answers:

$\small \frac{4}{20}$

$\small \small \frac{6}{8}$

$\small \frac{4}{8}$

$\small \frac{1}{5}$

$\frac{5}{4}$

Correct answer:

$\frac{5}{4}$

Explanation:

In order to find the perimeter of a rectangle, you add together all the sides. In this particular case, however, you must first find a common denominator for all of the fractions. Luckily, $\small 8$ is a multiple of $\small 2$ , so we can multiply the numerator and denominator of $\small \frac{1}2{}$ by $\small 4$ to get a denominator of $\small 8$ .

$\small \frac{(1\cdot 4)}{(2\cdot 4)}=\frac{4}{8}$

Now we simply add all four sides.

$\small \frac{4}{8}+\frac{4}{8}+\frac{1}{8}+\frac{1}{8}=\frac{10}{8}$

Since $\small \frac{10}{8}$ can be reduced by dividing the numerator and denominator by $\small 2$ , we must simplify.

$\small \frac{(10/2)}{(8/2)}=\frac{5}{4}$

The perimeter of the rectangle is $\small \frac{5}{4}$ .

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Example Question #4 : How To Add Fractions

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

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

Which is the greater quantity?

(a) x + y

(b)

Possible Answers:

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

Correct answer:

(a) and (b) are equal

Explanation:

Add both sides of the two equations:

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

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

$x + y = \frac{1}{2} + \frac{1}{3}+ \frac{1}{2} + \frac{2}{3}$

$x + y = \frac{1}{2} + \frac{1}{2}+ \frac{1}{3} + \frac{2}{3}$

x + y =1+1

x + y =2

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Example Question #5 : How To Add Fractions

t - 4.5 = 8.9

$y -6 \frac{1}{10} = 7 \frac{2}{5}$

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(a) is greater

Correct answer:

(b) is greater

Explanation:

(a) t - 4.5 = 8.9

t - 4.5 + 4.5 = 8.9 + 4.5

t = 13.4

(b) $y -6 \frac{1}{10} = 7 \frac{2}{5}$

$y -6 \frac{1}{10} + 6 \frac{1}{10} = 7 \frac{2}{5} + 6 \frac{1}{10}$

$y = 7 \frac{4}{10} + 6 \frac{1}{10} = 13 \frac {5}{10} = 13 \frac{1}{2}$

$13 \frac{1}{2} = 13.5 > 13.4$

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Example Question #6 : How To Add Fractions

Column A Column B

$\frac{2}{5}+\frac{1}{2}$ $\frac{1}{9}+\frac{7}{8}$

Possible Answers:

The quantities in each column are equal.

The quantity in Column B is greater.

There is no way to determine the relationship between the quantities in the columns.

The quantity in Column A is greater.

Correct answer:

The quantity in Column B is greater.

Explanation:

First, you must add the fractions in each column. When adding fractions, find the common denominator. The common denominator for Column A is 10. Then, change the numerators to reflect changing the denominators to give you $\frac{4}{10}+\frac{5}{10}$ . Combie the numerators to give you $\frac{9}{10}.$ Then, add the fractions in Column B. The common denominator for those fractions is 72. Therefore, you get $\frac{8}{72}+\frac{63}{72}$ . Combine the numerators to get $\frac{71}{72}$ . Compare those two fractions. Think of them as slices of pizza. There would be way more of Column B. Therefore, it is greater. Also, a little to trick to comparing fractions is cross-multiply. The side that has the biggest product is the greatest.

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Example Question #7 : How To Add Fractions

Which is the greater quantity?

(A) $\frac{3}{5} + \frac{1}{4}$

(B) 0.4 + 0.44

Possible Answers:

(A) is greater

(B) is greater

(A) and (B) are equal

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

Correct answer:

(A) is greater

Explanation:

$\frac{3}{5} = 3 \div 5 = 0.6$ and $\frac{1}{4 } = 1 \div 4 = 0.25$ , so

$\frac{3}{5} + \frac{1}{4} = 0.6 + 0.25 = 0.85$ , the decimal equivalent of (A).

0.4 + 0.44 = 0.84 , the value of (B).

(A) is the greater.

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All ISEE Middle Level Quantitative Resources

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ISEE Middle Level Quantitative : Numbers and Operations

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 : Fractions

Example Question #13 : Fractions

Example Question #1 : How To Add Fractions

Example Question #2 : How To Add Fractions

Example Question #3 : How To Add Fractions

Example Question #4 : How To Add Fractions

Example Question #5 : How To Add Fractions

Example Question #6 : How To Add Fractions

Example Question #7 : How To Add Fractions

All ISEE Middle Level Quantitative Resources

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