COMPOUND FRACTIONS

A compound fraction is a fraction in which the numerator, the denominator, or both, are themselves fractional expressions.

Examples : 

(1/a)/(b/2), (x + 1/y)/(y + 1/x)

Simplifying a Compound Fraction : Method I

Step 1 : 

Simplify the numerator and the denominator of the complex fraction so that each is a single fraction.

Step 2 : 

To divide the fraction in numerator by the fraction in denominator, invert the divisor and multiply.

Step 3 : 

Simplify if possible.

Example 1 : 

(2x/27y²)/(6x²/9)

Solution : 

=  (2x/27y²)/(6x²/9)

The numerator of the compound fraction is already a single fraction, and so is the denominator.

Invert the the fraction in denominator and multiply.

=  (2x/27y²) ⋅ (9/6x²)

=  (2x ⋅ 9) / (6x² ⋅ 27y²)

Cancel common factors. 

=  1 / (3x ⋅ 3y²)

=  1/9xy²

Example 2 : 

(x/y + 1)/(1 - y/x)

Solution : 

=  (x/y + 1)/(1 - y/x)

We combine the terms in the numerator into a single fraction and do the same in the denominator. Then we invert and multiply.

=  [(x + y)/y]/[(x - y)/x]

Invert the the fraction in denominator and multiply.

=  [(x + y)/y)] ⋅ [x/(x - y)]

=  [x(x + y)] / [y(x - y)]

Simplifying a Compound Fraction : Method II

Step 1 : 

Find the least common denominator (LCD) of the fractions in both numerator and denominator of the compound fraction.

Step 2 : 

Multiply the numerator and the denominator of the compound fraction by the LCD found in step 1. 

Step 3 : 

Simplify if possible.

Example 3 : 

[5x/(x + 2)]/[10/(x - 2)]

Solution : 

=  [5x/(x + 2)]/[10/(x - 2)]

The least common denominator of the fractions in both numerator and denominator is (x + 2)(x - 2). 

Multiply numerator and denominator by the LCD.

=  [5x/(x + 2)](x + 2)(x - 2) / [10/(x - 2)](x + 2)(x - 2)

Simplify.

=  [5x(x - 2)]/[10(x + 2)]

=  x(x - 2)/2(x + 2)

Example 4 : 

[x/y² + 1/y] / [y/x² + 1/x]

Solution : 

=  [x/y² + 1/y] / [y/x² + 1/x]

The least common denominator of the fractions in both numerator and denominator is x²y². 

Multiply numerator and denominator by the LCD.

=  [x/y² + 1/y](x²y²) / [y/x² + 1/x](x²y²)

Use the distributive property.

=  [x/y² ⋅ x²y² + 1/y ⋅ x²y²] / [y/x² ⋅ x²y² + 1/x ⋅ x²y²]

Simplify. 

=  [x³ + x²y]/[y³ + xy²]

Factor. 

=  [x²(x + y)]/[y²(y + x)]

=  x²/y²

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