Dividing Fractions

Dividing fractions is a fundamental mathematical operation that may appear complex initially, yet practice makes it manageable. Mastering this skill is crucial as it plays a vital role in mathematics and finds practical application across various real-world scenarios.

What are Fractions?

Fractions are a method used to represent parts of a whole. Imagine you have a pizza cut into 8 equal slices. If you decide to share one slice with a friend, you can express this as 1/8 of the pizza. The numerator (top number) indicates how many slices you have, while the denominator (bottom number) shows the total number of slices the pizza was divided into.

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What is meant by dividing fractions?

Dividing fractions involves determining how many times one fraction can fit into another. For instance, if you have 2/3 of a pizza and want to know how many times 1/4 of the pizza fits into it, you perform a division of these two fractions.

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How to Divide Fractions

1. Invert the second fraction: Flip the fraction you're dividing by (the divisor).
2. Multiply the fractions: Multiply the first fraction (the dividend) by the inverted second fraction.
3. Simplify the result: If possible, simplify the resulting fraction.

For example, to find out how many times 1/4 of a pizza fits into 2/3 of a pizza, invert 1/4 to get 4/1, then multiply:

(2/3) ÷ (1/4) = (2/3) × (4/1) = 8/3

Thus, 1/4 of a pizza fits into 2/3 of a pizza approximately 2.67 times.

Also Check: Cosine Function

Dividing Fraction by a Fraction

When dividing one fraction by another, follow the same steps as above. Invert the second fraction and then multiply.

For example, to find out how many times 25 of a pizza fits into 34 of a pizza, you would invert the second fraction (25) to 52 and then multiply the two fractions:

(34) (25) = (34) (52) = 158

So, 25 of a pizza fits into 34 of a pizza 158 times.

Dividing Fraction by a Whole Number

To divide a fraction by a whole number, simply divide the numerator of the fraction by the whole number.

For example, to find out how many times 13 of a pizza fits into 4 slices, you would divide the top number (1) by 4:

13 4 = 13 14 = 112

So, 13 of a pizza fits into 4 slices 112 times.

Dividing Fraction by a Mixed Fraction

When dividing a fraction by a mixed fraction, convert the mixed fraction to an improper fraction first and then proceed with the division.

For example, to find out how many times 23 of a pizza fits into 312 slices, you would first convert 312 into an improper fraction:

312 = 72

And then follow the steps to divide the two fractions:

(23) (72) = (23) (27) = 421

So, 23 of a pizza fits into 312 slices 421 times.

Also Check: Continuity and Discontinuity

Dividing Decimals

Before we explore dividing fractions, it's essential to understand how to divide decimals. Dividing decimals follows the same steps as dividing whole numbers, with a few small differences. To divide decimals, place the decimal point in the quotient directly above the decimal point in the dividend. Then, divide the numbers as usual.

Examples

Let's review some examples to understand how to divide decimals better.

Example 1: Divide 0.6 by 2

1. Divide the numbers as usual: 6 ÷ 2 = 3.
2. Place the decimal point directly above the decimal point in the dividend: 0.6 ÷ 2 = 3.
3. The quotient’s decimal point should be above the dividend's decimal point: 0.6 ÷ 2 = 0.3.

Example 2: Divide 1.2 by 0.6

1. Divide the numbers as usual: 1.2 ÷ 0.6 = 2.
2. Place the decimal point directly above the decimal point in the dividend: 1.2 ÷ 0.6 = 2.
3. The quotient’s decimal point should be above the dividend's decimal point: 1.2 ÷ 0.6 = 2.0.

Solved Examples

Now that we understand dividing decimals, let's learn about dividing fractions. Dividing fractions involves a different method. We convert the division problem into a multiplication problem by using the reciprocal of the divisor. The reciprocal of a fraction is simply the fraction flipped upside down.

Examples

Let's review some examples to understand how to divide fractions better.

Example 1: Divide 12 by 13

1. Find the reciprocal of the divisor (13): The reciprocal of 13 is 1/3.
2. Multiply the dividend (12) by the reciprocal of the divisor: 12 × 1/3 = 4.
3. Simplify the fraction if possible: 4 is already in its simplest form.

Example 2: Divide 2/3 by 3/4

1. Find the reciprocal of the divisor (3/4): The reciprocal of 3/4 is 4/3.
2. Multiply the dividend (2/3) by the reciprocal of the divisor: 2/3 × 4/3 = 8/9.
3. Simplify the fraction if possible: 8/9 is already in its simplest form.