How to divide fractions

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ISEE Middle Level Quantitative Reasoning › How to divide fractions

Questions 1 - 10

is a positive number. Which is the greater quantity?

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

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

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

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

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.

Divide the following:

$\frac{3}{4} \div \frac{1}{2}$

$\frac{3}{2}$

$\frac{1}{2}$

$\frac{3}{8}$

$\frac{1}{2}$

$\frac{1}{6}$

Explanation

To divide fractions, we will use the following steps:

Leave the first fraction alone.
Change the division sign to a multiplication sign.
Replace the second fraction with it's reciprocal.
Multiply.

So, given the problem

$\frac{3}{4} \div \frac{1}{2}$

we will follow the steps:

Leave the first fraction alone.

$\frac{3}{4} \div \frac{1}{2}$ 2. Change the division sign to a multiplication sign.

$\frac{3}{4} \cdot \frac{1}{2}$ 3. Replace the second fraction with it's reciprocal. To find the reciprocal, the numerator becomes the denominator, and the denominator becomes the numerator. In other words, we will flip the fraction.

$\frac{3}{4} \cdot \frac{2}{1}$ 4. Multiply.

$\frac{3 \cdot 2}{4 \cdot 1}$

$\frac{6}{4}$

$\frac{3}{2}$

One Mexican peso is equal to about 7.4 cents. To the nearest whole number, for how many pesos can a tourist to Mexico expect to exchange $500?

Explanation

One peso is equivalent to 7.4 cents, or $0.074, so divide $500 dollars by this conversion factor. $500 is equivalent to

$500 \div 0.074 \approx 6,757$ pesos.

One euro is worth approximately $1.27. For how many euros (nearest whole) can an American tourist expect to exchange $1,000?

Explanation

One Euro is equivalent to $1.27, so divide the number of dollars by this conversion factor. $1,000 is equivalent to

$1,000 \div 1.27\approx 787$ euros

Divide the following:

$10 \div \frac{1}{2}$

$\frac{2}{10}$

$\frac{1}{10}$

$\frac{1}{2}$

Explanation

To divide fractions, follow these steps:

Leave the first fraction alone.
Change the division sign to a multiplication sign.
Replace the second fraction with it's reciprocal.
Multiply.

So, given the problem

$10 \div \frac{1}{2}$

we will follow the steps.

Leave the first fraction alone. In this case, we need to write 10 as a fraction. We know that whole numbers can be written as fractions over 1. So, we get

$\frac{10}{1} \div \frac{1}{2}$ 2. Change the division sign to a multiplication sign.

$\frac{10}{1} \cdot \frac{1}{2}$ 3. Replace the second fraction with it's reciprocal. To write the reciprocal, the numerator becomes the denominator, and the denominator becomes the numerator. In other words, we will flip the second fraction.

$\frac{10}{1} \cdot \frac{2}{1}$ 4. Multiply. We will multiply straight across.

$\frac{10 \cdot 2}{1 \cdot 1}$

$\frac{20}{1}$

is a positive integer.

Which is the greater quantity?

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

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

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

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

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,

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

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

Divide the following:

$\frac{8}{9} \div \frac{1}{2}$

$\frac{16}{9}$

$\frac{4}{9}$

$\frac{1}{2}$

$\frac{8}{4.5}$

$\frac{8}{18}$

Explanation

To divide fractions, we will take the first fraction and multiply it by the reciprocal of the second fraction. To find the reciprocal, the numerator will become the denominator, and the denominator will become the numerator. In other words, we will flip the fraction. So, we get

$\frac{8}{9} \div \frac{1}{2}$

$\frac{8}{9} \cdot \frac{2}{1}$

$\frac{8 \cdot 2}{9 \cdot 1}$

$\frac{16}{9}$

Simplify the following expression:

$\frac{12}{35}\div\frac{5}{42}$

$\frac{72}{25}$

$\frac{95}{6}$

Explanation

Simplify the following expression:

$\frac{12}{35}\div\frac{5}{42}$

To divide fractions, we need to multiply by the reciprocal. This means that we flip the second fraction and then multiply:

$\frac{12}{35}\div\frac{5}{42}=\frac{12}{35}\ast\frac{42}{5}$

Before multiplying, simplify the two fractions by factoring out a seven from the 42 and the 35. We can do this, because they are on opposite sides of the fraction.

$\frac{12}{35}\ast\frac{42}{5}=\frac{12}{5}*\frac{6}{5}=\frac{72}{25}$

So we get an answer of:

$\frac{72}{25}$

Divide the following:

$\frac{4}{5} \div \frac{1}{2}$

$\frac{8}{5}$

$\frac{2}{5}$

$\frac{4}{10}$

$\frac{2}{10}$

$\frac{5}{8}$

Explanation

To divide fractions, we will take the first fraction and multiply it by the reciprocal of the second fraction.

To find the reciprocal, the numerator will become the denominator, and the denominator will become the numerator. In other words, we will flip the fraction.

So, we get

$\frac{4}{5} \div \frac{1}{2}$

$\frac{4}{5} \cdot \frac{2}{1}$

$\frac{4 \cdot 2}{5 \cdot 1}$

$\frac{8}{5}$

Divide the following:

$\frac{8}{9} \div \frac{1}{4}$

$\frac{32}{9}$

$\frac{4}{9}$

$\frac{8}{3}$

$\frac{2}{3}$

Explanation

To divide, we will multiply the first fraction by the reciprocal of the second fraction. To find the reciprocal, the numerator becomes the denominator, and the denominator becomes the numerator (in other words, we will flip the fraction).

We get

$\frac{8}{9} \div \frac{1}{4}$

$\frac{8}{9} \cdot \frac{4}{1}$

$\frac{8 \cdot 4}{9 \cdot 1}$

$\frac{32}{9}$

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