In this Worksheet on Reducing Algebraic Fractions, we can see various questions on reducing an algebraic fraction to the lowest term, simplifying the fraction, that is useful to the students so that they can improve skills and knowledge in the particular concept. Practice questions from reducing algebraic fractions worksheet to score best marks in the exam. Below given are types of questions with solutions that are helpful to understand the concept better. Assess your preparation level by solving the Worksheet on Reducing Algebraic Fractions on your own.

Firstly, find the factors of the numerator, the denominator of the algebraic fractions, and cancel the like terms to get the lowest term of an algebraic fraction. Get Reducing Algebraic Fractions questions along with solutions, and practice them.

1. Reduce the following algebraic fractions to the lowest terms:

(a) [x² – 5x] / [x – 5]

(b) [a² – 49] / [a² – 7a]

(c) [a² + 6a + 5] / [a² + 8a + 15]

(d) [x + 2] / [x² – 4]

## Solution:

(a) [x² – 5x] / [x – 5]

The factors of numerator and denominator are

= x(x – 5) / (x – 5)

Cancel the common term (x – 5) in both denominator and numerator.

= x, which is the lowest form of the given algebraic fraction.

(b) [a² – 49] / [a² – 7a]

Factorize the numerator and denominator of the given algebraic fraction,

= [a² – 7²] / [a² – 7a]

= (a + 7) (a – 7) / a(a – 7)

Cancel the common term (a – 7) in both numerator and denominator.

= (a + 7) / a, which is the lowest form of the given algebraic fraction.

(c) [a² + 6a + 5] / [a² + 8a + 15]

By factorizing the numerator, denominator of the given algebraic fraction

= [a² + 5a + a + 5] / [a² + 5a + 3a + 15]

= [a(a + 5) + 1(a + 5)] / [a(a + 5) + 3(a + 5)]

= (a + 5) (a + 1) / (a + 5) (a + 3)

We observed that in the numerator and denominator of the algebraic fraction, (a + 5) is the common factor.

Now, when the numerator and denominator of the algebraic fraction is divided by this common factor or their H.C.F. the algebraic fraction becomes,

= [(a + 5) (a + 1)] / (a + 5) / [(a + 5) (a + 3)] / (a + 5)

= (a + 1) / (1 + 3), which is the lowest form of the given algebraic fraction.

2. Reduce the following rational expression to its lowest terms:

(a) ab / [a²x² – ax]

(b) 20x²y²z² / 25(y² – x²y²)

(c) [25p² – 16q²] / [5p² + 50p + 4pq+ 40q]

## Solution:

(a) ab / [a²x² – ax]

Factorize the numerator and denominator of the algebraic fraction

= ab / ax[ax – 1]

We can observe that in the denominator and numerator of the algebraic fraction, a is the common factor. Now cancel the common factor and write the remaining terms as it is.

= b / x [ax – 1], which is the lowest form of the given algebraic fraction.

(b) 20x²y²z² / 25(y² – x²y²)

Get the factors of the numerator and denominator of the algebraic fraction.

= [5 x 4 x x²y²z²] / [5 x 5 x y²(1 – x²)]

= [5 x 4 x x²y²z²] / [5 x 5 x y²(1² – x²)]

= [5 x 4 x x²y²z²] / [5 x 5 x y² (1 – x) (1 + x)]

We observed that in the numerator and denominator of the algebraic fraction, 5y² is the common factor. Now cancel the common factor in both and write the remaining factor as it is.

= [4 x x²z²] / [5 (1 – x) (1 + x)]

= 4x²z² / 5(1 – x) (1 + x), which is the lowest form of the given algebraic fraction.

(c) [25p² – 16q²] / [5p² + 50p + 4pq+ 40q]

Factorize the numerator, denominator of the algebraic fraction.

= [(5p)² – (4q)²] / [5p(p + 10) + 4q(P + 10)]

= (5p – 4q) (5p + 4q) / [(5p + 4q) (p + 10)]

We observed that in the numerator and denominator of the algebraic fraction, (5p + 4q) is the common factor. Now cancel the common factor in both and write the remaining factor as it is.

= (5p – 4q) / (p + 10), which is the lowest form of the given algebraic fraction.

3. Reduce the algebraic fractions to their lowest terms:

(a) [x(2a² – 3ax)] / [a(4a²x – 9x³)]

(b) [x³ + 9x² + 20x] / [x² + 2x – 15]

(c) [x⁴ + 20x³ + 150x² + 500x + 625] / [x² – x – 30]

(d) [3a² – 6ab] / [2a²b – 4ab²]

## Solution:

(a) [x(2a² – 3ax)] / [a(4a²x – 9x³)]

Get the factors of the numerator and denominator of the algebraic fraction.

= [x(2a² – 3ax)] / [ax(4a² – 9x²)]

= [x(2a² – 3ax)] / [ax((2a)² – (3x)²)]

= [ax(2a – 3x)] / [ax(2a – 3x) (2a + 3x)]

We observed that in the numerator and denominator of the algebraic fraction, (2a + 3x), a, x are the common factors. Now cancel the common factors in both and write the remaining factor as it is.

= 1 / (2a + 3x), which is the lowest form of the given algebraic fraction.

(b) [x³ + 9x² + 20x] / [x² + 2x – 15]

Factorize the algebraic expressions.

= [x(x² + 9x + 20)] / [x² + 5x – 3x – 15]

= [x(x² + 5x + 4x + 20)] / [x(x + 5) – 3(x + 5)]

= [x(x (x + 5) + 4(x + 5)] / (x – 3) (x + 5)

= [x(x + 5) ( x + 4)] / (x – 3) (x + 5)

We observed that in the numerator and denominator of the algebraic fraction, (x + 5) are the common factor. Now cancel the common factor in both and write the remaining factor as it is.

= x(x + 4) / (x – 3), which is the lowest form of the given algebraic fraction.

(c) [x⁴ + 20x³ + 150x² + 500x + 625] / [x² – x – 30]

Get the factors of the numerator and denominator of the given algebraic fraction.

= (x + 5)⁴ / [x² – 6x + 5x – 30]

= (x + 5)⁴ / [x(x – 6) + 5(x – 6)]

= (x + 5)⁴ / (x – 6) (x + 5)

We observed that in the numerator and denominator of the algebraic fraction, (x + 5) are the common factor. Now cancel the common factor in both and write the remaining factor as it is.

= (x + 5)³ / (x – 6), which is the lowest form of the given algebraic fraction.

4. Reduce the given algebraic fractions to the lowest term.

(a) [3a² – 6ab] / [2a²b – 4ab²]

(b) [x + 2] / [x² – 4]

## Solution:

(a) [3a² – 6ab] / [2a²b – 4ab²]

Get the factors of the numerator and denominator of the given algebraic fraction.

= [3a(a – 2b)] / [2ab(a – 2b)]

We observed that in the numerator and denominator of the algebraic fraction, a(a – 2b) are the common factor. Now cancel the common factor in both and write the remaining factor as it is.

= 3 / 2b, which is the lowest form of the given algebraic fraction.

(b) [x + 2] / [x² – 4]

Find the factors of the numerator and denominator of the fraction.

= (x + 2) / (x² – 2²)

= (x + 2) / (x + 2) (x – 2)

We can observe that in the denominator and numerator of the algebraic fraction, (x + 2) is the common factor. Now cancel the common factor and write the remaining terms as it is.

= 1 / (x – 2), which is the lowest form of the given algebraic fraction.