How to divide fractions - ISEE Middle Level Quantitative Reasoning

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Question

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?

Answer

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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Question

Which is the greater quantity?

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

(b) The reciprocal of

Answer

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 :

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

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

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

making (b) greater

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Question

Which is the greater quantity?

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

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

Answer

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 .

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

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

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

making (b) greater

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Question

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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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Question

is a positive integer.

Which is the greater quantity?

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

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

Answer

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,

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

or

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

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Question

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

(a) The reciprocal of

(b)

Answer

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 is $- \frac{5}{2}$ , which is less than 0.

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Question

is the reciprocal of . Which of the following is the reciprocal of in terms of ?

Answer

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

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Question

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

(a) The reciprocal of

(b) The reciprocal of

Answer

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

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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Question

is a positive number. Which is the greater quantity?

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

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

Answer

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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Question

is a negative number. Which is the greater quantity?

(a) The reciprocal of

(b) The reciprocal of

Answer

is negative, so we will let , making positive. It follows that .

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

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

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

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