Dividing Fractions
- Introduction
- What Are Fractions?
- Dividing Fractions by Fractions
- The Rule for Dividing Fractions
- Dividing Fractions by Whole Numbers
- Dividing Fractions by Decimals
- Solved Examples
- Practice Problems
- Frequently asked questions
Introduction
Operations with fractions can sometimes seem challenging but don't worry – dividing fractions is not as hard as it may seem. It can be quite fun once you understand the basics.
What Are Fractions?
Fractions are a way to represent parts of a whole or quantities that are not whole numbers. They consist of two parts:
`1`. The numerator: This is the top number in a fraction and tells you how many parts you have.
`2`. The denominator: This is the bottom number in a fraction and tells you how many equal parts the whole is divided into.
For example, in the fraction `3/4`:
- Numerator `= 3`
- Denominator `= 4`
Dividing Fractions by Fractions
Dividing one fraction by another is the same as multiplying the first fraction by the reciprocal of the second fraction. We get the reciprocal of a fraction by flipping the position of the numerator and the denominator. If a fraction is stated as `p/q`, its reciprocal will be `q/p`, meaning the position of the numerator and denominator are reversed.
The Rule for Dividing Fractions
Now, let's learn how to divide one fraction by another. The rule for dividing fractions is simple: "Keep, Change, Flip." Here's what it means:
- Keep the first fraction as it is.
- Change the division sign to multiplication.
- Flip (or reciprocate) the second fraction, which means swap the numerator and denominator.
`p/q ÷ r/s = p/q × s/r`
Example: Divide `1/2` by `1/4`.
- Keep the first fraction: `1/2`
`1/2 ÷ 1/4` - Change the division sign to multiplication.
`1/2 × 1/4` - Flip the second fraction, which is `1/4`, so it becomes `4/1`.
`1/2 × 4/1`
So, `1/2 ÷ 1/4 = 1/2 × 4/1 = 2`
Dividing Fractions by Whole Numbers
Now, let's learn how to divide a fraction by a whole number. The rule for dividing a fraction by a whole number is the same as dividing a fraction by a fraction: "Keep, Change, Flip." Here's what it means:
Before applying the rule, change the whole number into a fraction with denominator `1`.
- Keep the first fraction as it is.
- Change the division sign to multiplication.
- Flip (or reciprocate) the second fraction, which means swapping the numerator and denominator. The second number which is a whole number can be turned into a fraction with denominator `1`.
Example: Divide `4/6` by `12`.
- Keep the first fraction as it is.
`4/6 ÷ 12`
The second number,`12` which is a whole number is turned into a fraction by rewriting it as `12/1`.
`4/6 ÷ 12/1` - Change the division sign to multiplication.
`4/6 × 12/1` - Flip (or reciprocate) the second fraction, which means swapping the numerator and denominator.
`4/6 × 1/12`
So, `4/6 ÷ 12 = 4/6 ÷ 12/1= 4/6 × 1/12 = 4/72 = 1/18`
Dividing Fractions by Decimals
Now, let's learn how to divide a fraction by a decimal. The rule for dividing fractions is the same: "Keep, Change, Flip." Here's what it means:
Before applying the rule, change the decimal into its equivalent fraction.
- Keep the first fraction as it is.
- Change the division sign to multiplication.
- Flip (or reciprocate) the second fraction, which means swapping the numerator and denominator. The second number which is a decimal should be first turned into its equivalent fraction.
Example: Divide `3/7` by `0.7`.
- Keep the first fraction as it is.
`3/7 ÷ 0.7`
The second number, `0.7` which is a decimal is turned into its equivalent fraction `7/10`.
`3/7 ÷ 7/10` - Change the division sign to multiplication.
`3/7 × 7/10` - Flip (or reciprocate) the second fraction, which means swap the numerator and denominator.
`3/7 × 10/7`
So, `3/7 ÷ 0.7 = 3/7 ÷ 7/10 = 3/7 × 10/7 = 30/49`
Solved Examples
Dividing fractions may seem a little complicated at first, but with practice, you'll get the hang of it. Remember the "Keep, Change, Flip" rule, and you'll be able to divide fractions like a pro.
Example `1`: Divide `3/4` by `1/2`.
- Keep the first fraction: `3/4`.
- Change the division sign to multiplication.
- Flip the second fraction: `1/2` becomes `2/1`.
Now, we have: `3/4 × 2/1`
Multiply the numerators (top numbers) together: `3 × 2 = 6`.
Multiply the denominators (bottom numbers) together: `4 × 1 = 4`.
So, `3/4 ÷ 1/2 = 6/4`.
The simplified answer is `3/2`.
So, `3/4 ÷ 1/2 = 3/2`.
Example `2`: Divide `2/3` by `5`.
- Keep the first fraction: `2/3`.
- Change the division sign to multiplication.
- Flip the second fraction: `5/1` becomes `1/5`.
Now, we have: `2/3 × 1/5`
Multiply the numerators together: `2 × 1 = 2`.
Multiply the denominators together: `3 × 5 = 15`.
So, `2/3 ÷ 5 = 2/15`.
Example `3`: Divide `9/25` by `0.6`.
- Keep the first fraction: `9/25`.
- Change the division sign to multiplication.
- Flip the second fraction: `3/5` becomes `5/3`. (Remember to write `0.6` as `6/10` which can be simplified to `3/5`).
Now, we have: `9/25 × 5/3`.
Multiply the numerators together: `9 × 5 = 45`.
Multiply the denominators together: `25 × 3 = 75`.
So, `9/25 ÷ 0.6 = 45/75`.
The simplified answer is `3/5`.
So, `9/25 ÷ 0.6 = 3/5`.
Practice Problems
Q.`1`. What is the reciprocal of `57`?
- `57/1`
- `57/0`
- `1/57`
- `0/57`
Answer: c
Q.`2`. What is `3/16` divided by `1/2`?
- `1/8`
- `1/4`
- `1/2`
- `3/8`
Answer: d
Q.`3`. If you divide `2/3` by `3/5`, what is the result?
- `10/9`
- `2/5`
- `5/6`
- `3/2`
Answer: a
Q.`4`. What is `4/19` divided by `0.6`?
- `20/57`
- `2/5`
- `5/6`
- `3/2`
Answer: a
Q.`5`. What is `0.6` divided by `4/19`?
- `57/20`
- `5/2`
- `6/5`
- `2/3`
Answer: a
Frequently Asked Questions
Q.`1`. Do we change anything when dividing fractions?
Answer: Yes, when dividing fractions, we change the operation to multiplication and multiply the first fraction by the reciprocal of the second fraction. This is the key step in the process. However, it's essential to remember that we don't change anything in the first fraction; we just multiply it by the reciprocal of the second fraction.
Q.`2`. Why does division of fractions involve multiplying by the reciprocal?
Answer: The concept of dividing fractions by multiplying by the reciprocal comes from the idea that division is the same as multiplying by the reciprocal. When you divide by a fraction, you are essentially multiplying by its reciprocal. This rule simplifies the process of dividing fractions and is consistent with the rules of arithmetic.
Q.`3`. Can you divide a fraction by zero?
Answer: No, you cannot divide a fraction by zero. When we divide a fraction by zero, we get an undefined value.
Q.`4`. Can you divide zero by a fraction?
Answer: Yes, you can divide zero by a fraction. When we divide zero by a fraction, we get zero as the answer.
Q.`5`. How do you check if the answer is correct when dividing fractions?
Answer: To check if the answer to a division of fractions is correct, you can multiply the result by the second fraction. If this product is equal to the first fraction, then your answer is correct. This is a useful way to verify your work and ensure the accuracy of the division.