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ISEE Middle Level Quantitative : Numbers and Operations

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

Example Question #1 : Fractions

Which is the greater quantity?

(a) The reciprocal of $- \frac{5}{6}$

(b) The reciprocal of $-\frac{6}{7}$

Possible Answers:

(a) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

Correct answer:

(b) is greater

Explanation:

The reciprocal of any fraction can be found by switching numerator and denominator. Since both numbers are negative, both reciprocals will be negative.

(a) $-\frac{5}{6}$ will have reciprocal $-\frac{6}{5}$

(b) $-\frac{6}{7}$ will have reciprocal $-\frac{7}{6}$

We can compare these by writing them both with common denominator LCM (5,6) = 30 .

$-\frac{6}{5}= -\frac{6 \times 6}{5\times 6} = -\frac{36}{30}$

$-\frac{7}{6}= -\frac{5 \times 7}{5\times 6} = -\frac{35}{30}$

$-\frac{6}{5} = -\frac{36}{30} < -\frac{35}{30}-\frac{7}{6}$

making (b) greater

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Example Question #1 : How To Divide Fractions

Which is the greater quantity?

(a) The reciprocal of $\frac{3}{4}$

(b) The reciprocal of 0.7

Possible Answers:

(a) and (b) are equal

It cannot be determined from the information given

(a) is greater

(b) is greater

Correct answer:

(b) is greater

Explanation:

The reciprocal of any fraction can be found by switching numerator and denominator.

(a) $\frac{3}{4}$ will have reciprocal $\frac{4}{3}$

(b) $0.7 = \frac{7}{10}$ and will have reciprocal $\frac{10}{7}$

We can compare these by writing them both with common denominator LCM (3,7) = 21 :

$\frac{4}{3}= \frac{4\times 7}{3\times 7} = \frac{28}{21}$

$\frac{10}{7}= \frac{3\times10}{3\times 7} = \frac{30}{21}$

$\frac{4}{3} = \frac{28}{21} <\frac{30}{21} = \frac{10}{7}$

making (b) greater

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Example Question #2 : Numbers And Operations

$3\div \frac{1}{2} =$

Possible Answers:

$\frac{2}{3}$

$\frac{1}{2}$

$\frac{3}{2}$

Correct answer:

Explanation:

When dealing with math problems that involve fractions and whole numbers or mixed numbers, you should first convert the non-fraction into a fraction:

$3\rightarrow \frac{3}{1}$

The problem should look like this:

$\frac{3}{1}\div \frac{1}{2}$

When dividing with fractions, you need to find the reciprocal of the second fraction before you can do anything else. In other words, flip the second number.

$\frac{1}{2} \rightarrow \frac{2}{1}$

When you find the reciprocal of the second number, change the problem from division to multiplication. The new problem should look like this:

$\frac{3}{1}\div\frac{1}{2}=\frac{3}{1} \times \frac{2}{1} =$ $\frac{6}{1}$

The answer is 6.

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Example Question #3 : Numbers And Operations

$\frac{1}{2} \div \frac{1}{8} =$

Possible Answers:

$\frac{1}{16}$

$\frac{1}{6}$

$\frac{1}{4}$

Correct answer:

Explanation:

When dividing fractions, you must first find the reciprocal of the second number in the operation. In other words, flip the second fraction.

$\frac{1}{8}\rightarrow\frac{8}{1}$

When you do this, the operation also changes from division to multiplication. The problem should now look like this:

$\frac{1}{2} \times \frac{8}{1} =$

Then multiply both the numerators and denominators.

$\frac{1}{2} \times \frac{8}{1} =\frac{1\times8}{2\times1}=\frac{8}{2}$

When possible, always reduce the fraction. In this case, both 2 and 8 are divisible by 2.

$\tfrac{8}{2}\rightarrow\tfrac{4}{1}\rightarrow4$

The result is your answer.

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Example Question #3 : Numbers And Operations

$\frac{1}{3}\div\frac{3}{5} =$

Possible Answers:

$\frac{3}{5}$

$\frac{1}{5}$

$\frac{5}{9}$

$\frac{3}{2}$

$\frac{1}{15}$

Correct answer:

$\frac{5}{9}$

Explanation:

When dividing fractions, you must first find the reciprocal of the second fraction. In other words, flip the second fraction.

$\frac{3}{5} \rightarrow\frac{5}{3}$

When you do this, the operation also changes from division to multiplication. The problem should now look like this:

$\frac{1}{3}\times\frac{5}{3} =$

Multiply the numerators and denominators. The result is your answer.

$\frac{1}{3}\times\frac{5}{3} =\frac{1\times5}{3\times3}=\frac{5}{9}$

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Example Question #1 : How To Divide Fractions

$4\div\tfrac{1}{4} =$

Possible Answers:

$\frac{1}{16}$

$\frac{1}{4}$

Correct answer:

Explanation:

When dealing with fractions and whole numbers, first convert the whole number to a fraction. This is easily done by putting the whole number over 1.

$4\rightarrow\tfrac{4}{1}$

When dividing fractions, you must first find the reciprocal of the second fraction in the operation. In other words, flip the second fraction.

$\tfrac{1}{4}\rightarrow \tfrac{4}{1}$

When you do this, the operation changes from division to multiplication. The problem should now look like this:

$\tfrac{4}{1}\times\tfrac{1}{4}=$

Solve the multiplication by multiplying the numerators and the denominators.

$\tfrac{4}{1}\times\tfrac{4}{1} =\tfrac{4\times4}{1\times1}= \tfrac{16}{1}\rightarrow16$

Since the result is $\tfrac{16}{1}$ , it reduces to 16 as any fraction with a denominator of 1 is equal to the value of its numerator.

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Example Question #5 : Numbers And Operations

If Candy has $\small \frac{3}{4}$ a cup of trail mix and she wants to divide it evenly among herself and seven of her friends, how much does each person get?

Possible Answers:

$\small \frac{3}{32}$

$\small \frac{24}{4}$

$\small \frac{3}{34}$

$\small \frac{1}{32}$

$\small \frac{1}{16}$

Correct answer:

$\small \frac{3}{32}$

Explanation:

Candy has $\small \frac{3}{4}$ cups of trail mix. If she divides it amoung herself and seven of her friends, she is dividing it by $\small \frac{8}{1}$ . In order to solve this problem, we simply find the reciprocal of the second fraction and multiply.

To find the reciprical of a fraction, we switch the numerator and denominator.

$\small \frac{8}{1}$ becomes $\small \frac{1}{8}$ .

Now we multiply $\small \frac{3}{4}$ by $\small \frac{1}{8}$ .

$\small \frac{3}{4}\cdot \frac{1}{8}=\frac{3}{32}$

Each person gets $\small \small \frac{3}{32}$ of the trail mix.

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Example Question #1 : Numbers And Operations

is a positive integer.

Which is the greater quantity?

(a) $A \div \frac{1}{3}$

(b) $A \cdot \frac{1}{3}$

Possible Answers:

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

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

(a) is the greater quantity

Explanation:

Division by a fraction is equivalent to multiplication by its reciprocal, so

$A \div \frac{1}{3}$ can be rewritten as $A \cdot \frac{3}{1} = A \cdot 3$

$3 > \frac{1}{3}$ , and is positive, so, by the multiplication property of inequality,

$A \cdot 3 >A \cdot \frac{1}{3}$

$A \div \frac{1}{3}>A \cdot \frac{1}{3}$ .

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Example Question #8 : Fractions

is the reciprocal of . Which of the following is the reciprocal of X + 1 in terms of ?

Possible Answers:

$\frac{Y} {Y+1}$

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

$\frac{Y+1}{Y}$

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

Correct answer:

$\frac{Y} {Y+1}$

Explanation:

If is the reciprocal of , then

$X = \frac{1}{Y}$

$X + 1 = \frac{1}{Y} + 1 = \frac{1}{Y} + \frac{Y}{Y} = \frac{Y+1}{Y}$

The reciprocal of X + 1 is $\frac{Y} {Y+1}$ .

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Example Question #8 : Numbers And Operations

The reciprocal of is $1 \frac{2}{3}$ . Which is the greater quantity?

(a) The reciprocal of Y - 1

(b)

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

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

(a) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

The reciprocal of is $1 \frac{2}{3} = \frac{1 \times 3+ 2}{3} = \frac{5}{3}$ , so is the reciprocal of this, or

$Y = \frac{3}{5}$

$Y - 1 = \frac{3}{5} - 1 = \frac{3}{5} - \frac{5}{5} = - \frac{2}{5}$

The reciprocal of Y - 1 is $- \frac{5}{2}$ , which is less than 0.

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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 #1 : Fractions

Example Question #1 : How To Divide Fractions

Example Question #2 : Numbers And Operations

Example Question #3 : Numbers And Operations

Example Question #3 : Numbers And Operations

Example Question #1 : How To Divide Fractions

Example Question #5 : Numbers And Operations

Example Question #1 : Numbers And Operations

Example Question #8 : Fractions

Example Question #8 : Numbers And Operations

All ISEE Middle Level Quantitative Resources

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