Dividing a fraction by another fraction is achieved by multiplying the first fraction by the reciprocal of the second fraction. This is the core concept. Here’s a detailed breakdown:
Understanding Reciprocals
Before diving into division, let's understand reciprocals:
- The reciprocal of a fraction is obtained by swapping its numerator and denominator. For example, the reciprocal of 2/3 is 3/2.
Steps for Dividing Fractions
The steps to divide fractions are straightforward:
- Find the Reciprocal: Determine the reciprocal of the second fraction (the divisor).
- Multiply: Replace the division symbol with a multiplication symbol and multiply the first fraction by the reciprocal of the second fraction. This involves:
- Multiplying the numerators of the two fractions to find the new numerator.
- Multiplying the denominators of the two fractions to find the new denominator.
- Simplify (If Necessary): Reduce the resulting fraction to its simplest form if possible. This is done by finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it.
Example
Let’s illustrate with an example:
Problem
Divide 2/3 by 1/4.
Solution
| Step | Explanation | Calculation |
|---|---|---|
| 1. Find the Reciprocal | The reciprocal of 1/4 is 4/1 | |
| 2. Multiply | Multiply the first fraction (2/3) by the reciprocal of the second (4/1) | (2/3) * (4/1) |
| 3. Calculate the new numerator | Multiply the numerators: 2 * 4 | 2 * 4 = 8 |
| 4. Calculate the new denominator | Multiply the denominators: 3 * 1 | 3 * 1 = 3 |
| 5. Result | The result of the division | 8/3 |
Therefore, 2/3 divided by 1/4 equals 8/3.
Key Takeaway
Dividing fractions is equivalent to multiplying the first fraction by the reciprocal of the second fraction.
Additional Considerations
- When dealing with mixed numbers, convert them to improper fractions before dividing.
- If a fraction involves variables, apply the same principles. For example, if you have x/y divided by a/b, the answer is (x/y) * (b/a) = xb/ya.
