To solve fractions with fraction denominators (also known as complex fractions), you simplify the expression by eliminating the fraction in the denominator. The most common method involves multiplying both the numerator and the denominator of the complex fraction by the reciprocal of the denominator. This is equivalent to dividing by the denominator.
Here's a breakdown of the process:
1. Identify the Complex Fraction: Recognize the fraction that has another fraction in its denominator (or numerator).
2. Find the Reciprocal of the Denominator: Invert the fraction in the denominator. For example, the reciprocal of 1/2 is 2/1 (or simply 2).
3. Multiply Both Numerator and Denominator by the Reciprocal: Multiply both the top and bottom of the main fraction by the reciprocal you found in step 2. This is equivalent to multiplying by 1, so the value of the overall expression doesn't change.
4. Simplify: Perform the multiplication in both the numerator and the denominator. The fraction in the original denominator should now cancel out, leaving a simpler expression. Further simplification may be needed.
Example 1:
Simplify: (3 / (1/2))
- Denominator: 1/2
- Reciprocal of the Denominator: 2/1 (or 2)
- Multiply Numerator and Denominator: (3 2) / ((1/2) 2)
- Simplify: 6 / 1 = 6
Example 2:
Simplify: ((1/4) / (3/5))
- Denominator: 3/5
- Reciprocal of the Denominator: 5/3
- Multiply Numerator and Denominator: ((1/4) (5/3)) / ((3/5) (5/3))
- Simplify: (5/12) / 1 = 5/12
Alternative Method: Combining Fractions in Numerator and Denominator First
Sometimes, the numerator or denominator (or both) might contain sums or differences of fractions. In such cases, it's often helpful to:
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Simplify the Numerator and Denominator Separately: Find a common denominator and combine any fractions in the numerator and any fractions in the denominator into single fractions.
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Divide the Simplified Numerator by the Simplified Denominator: Follow the rule: dividing by a fraction is the same as multiplying by its reciprocal.
Example 3:
Simplify: ((1/2 + 1/4) / (2/3))
- Simplify Numerator: 1/2 + 1/4 = 2/4 + 1/4 = 3/4
- Rewrite: (3/4) / (2/3)
- Multiply by the Reciprocal: (3/4) * (3/2)
- Simplify: 9/8
In summary, solving fractions with fraction denominators involves strategically eliminating the fraction in the denominator by multiplying both the numerator and denominator by its reciprocal. Simplifying the numerator and denominator before inverting can also be a useful approach.
