 # Multiplication and division of fraction

Multiplication of Fractions

The product of the two fractions a/2 and c/d is defined in previous Chapter to be (ac)/(bd);

that is,  a/b*c/d=(ac)/(bd)

Thus the product of two fractions is a fraction whose numerator is the product of the numerators and whose denominator is the product of the denominators. In general,

Note    Always reduce the resulting fraction to lowest terms.

EXAMPLE  Find the product of (27a^3b^2)/(8x^2y) and (16x^3y)/(81a^2b^3)

Solution

(27a^3b^2)/(8x^2y)*(16x^3y)/(81a^2b^3)==(27*16a^3b^2x^3y)/(8*81x^2ya^2b^3)==(2ax)/(3b)

Note     It is easier to reduce (27*16)/(8*81) then 432/648, which is the result of the products.

That is. numbers should not be multiplied together until the fraction has been simpliﬁed.

To multiply fractions whose numerators or denominators are polynomials, ﬁrst factor polynomials completely Consider the fractions as just one fraction, and divide the numerators and denominators by their greatest common factor to get an equivalent fraction in lowest terms.

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EXAMPLE  Simplify (x^2-3x)/(2x^2+11x+5)*(6x^2+x-1)/(3x^2-10x+3)

Solution

EXAMPLE  Simplify (12x^2-13x+3)/(3x^2-5x-2)*(2x^2-x-6)/(9-6x-8x^2)

Solution

Division of Fractions

From the deﬁnition of division of fractions, discussed before, we have

a/b ÷ c/d = a/b*d/c

The above result shows how to transform division of fractions into multiplication of fractions.

The fractions c/d and d/c are called multiplicative inverses or reciprocals

Note   The reciprocal of the expression a+b is 1/(a+b, not 1/a+1/b

Note   The reciprocal of 1/a+1/b is 1/(1/a+1/b), or simplified, ab/(b+a)

1/(1/a+1/b)==1/(1/a+1/b)*ab/ab==(ab)/((ab)/1(1/a+1/b))==(ab)/((ab/a+ab/b))==ab/(b+a)

EXAMPLE  Simplify (3a^3)/(5b^2) ÷ (9a^2)/(20b)

Solution   (3a^3)/(5b^2) ÷ (9a^2)/(20b)== ((3a^3)/(5b^2)) *((20b)/(9a^2))== (4a)/(3b)

Note  Note The difference between

a/b÷ (c/d*e/f) == a/b*d/c*e/f == ade/bcf

and

a/b÷(c/d*e/f)== a/b ÷ ce/df== a/b*df/ce== adf/bce

EXAMPLE  Simplify (9a^2b^4)/(49x^2y^3) ÷ (a^2b)/(14x^2y)*(21y)/(ab^2)

Solution

(9a^2b^4)/(49x^2y^3) ÷ (a^2b)/(14x^2y)*(21y)/(ab^2)==(9a^2b^4)/(49x^2y^3)*(14x^2y)/(a^2b)*(21y)/(ab^2)==(54b)/(ay)

EXAMPLE  Simplify (a^3b^2)/(x^2y^3) ÷ ((a^2b^5)/(x^5y)*(x^3y^2)/(ab^3))

Solution

(a^3b^2)/(x^2y^3) ÷((a^2b^5)/(x^5y)*(x^3y^2)/(ab^3))==((a^3b^2)/(x^2y^3)) ÷ ((a^2b^5*x^3y^2)/(x^5y*ab^3))==(a^3b^2)/(x^2y^3)*(x^5y*ab^3)/(a^2b^5*x^3y^2)==a^2/y^4

EXAMPLE  Simplify (8x^2+2x-3)/(4x^2-17x-15) ÷ (12x^2-20x+7)/(6x^2-37x+35)

Solution    As in multiplication of fractions, we factor the numerators and denominators

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