Multiplication of Fractions

When multiplying fractions, we multiply the numbers in the numerators to get the answer for the numerator and separately we multiply the numbers in the denominators to get the answer for the denominator. Then we simplify the answer to its simplest form by eliminating common factors from the numerator and the denominator.


Or, we first eliminate any common factors from the numerator of one fraction with the denominators of the other fraction, before carrying out the multiplication.

For example


\frac { 2 }{ 5 } \times \frac { 15 }{ 17 } =\frac { 2\times 15 }{ 5\times 17 } =\frac { 30 }{ 85 }
But between 30 and 85, 5 is common. So we divide both numbers by 5 to give \frac { 6 }{ 17 } which is now in its simplest form.
Or, before carrying out the multiplication as we did above, we can see that 5 in \frac { 2 }{ 5 } and the 5 in 15=5\times 3 maybe eliminated, and we therefore have
\frac { 2 }{ 5 } \times \frac { 15 }{ 17 } =\frac { 2 }{ 1 } \times \frac { 3 }{ 17 } =\frac { 6 }{ 17 }
So, always watch out if there are common factors we can eliminate before multiplying out, it saves time.


More examples:


\frac { 5 }{ 8 } \times \frac { 2 }{ 15 }
\frac { 5 }{ 8 } \times \frac { 2 }{ 15 } =\frac { 5 }{ 2\times 4 } \times \frac { 2 }{ 5\times 3 }
=\frac { 1 }{ 4 } \times \frac { 1 }{ 3 } (eliminating 2 and 5)
=\frac { 1 }{ 12 }



\frac { 6 }{ 13 } \times \frac { 5 }{ 18 } =\frac { 6 }{ 13 } \times \frac { 5 }{ 6\times 3 }
=\frac { 5 }{ 13\times 3 }
=\frac { 5 }{ 39 }


\frac { 3 }{ 10 } \times \frac { 5 }{ 9 } \times \frac { 12 }{ 13 } =\frac { 3\div 3 }{ 10\div 5\div 2 } \times \frac { 5\div 5 }{ 9\div 3\div 3 } \times \frac { 12\div 4\div 2 }{ 13 } =\frac { 1\times 1\times 2 }{ 1\times 1\times 13 }
=\frac { 2 }{ 13 }


1\frac { 1 }{ 2 } \times 2\frac { 2 }{ 5 } \times 3\frac { 2 }{ 3 } =\frac { 11 }{ 3 }

Therefore, we have
\frac { 3 }{ 2 } \times \frac { 12 }{ 5 } \times \frac { 11 }{ 3 } =\frac { 3\div 3 }{ 2\div 2 } \times \frac { 12\div 6 }{ 5 } \times \frac { 11 }{ 3\div 3 }
=\frac { 6 }{ 5 } \times \frac { 11 }{ 1 }
=\frac { 66 }{ 5 }
=13\frac { 1 }{ 5 }


1\frac { 1 }{ 2 } \times 2\frac { 1 }{ 3 } \times 3\frac { 1 }{ 4 } \times 4\frac { 1 }{ 5 }

Again, we express the mixed fraction forms as improper fractions.

1\frac { 1 }{ 2 } =\frac { 3 }{ 2 } ,2\frac { 1 }{ 3 } =\frac { 7 }{ 3 } ,3\frac { 1 }{ 4 } =\frac { 13 }{ 4 } ,4\frac { 1 }{ 5 } =\frac { 21 }{ 5 }

Therefore, we have

\frac { 3 }{ 2 } \times \frac { 7 }{ 3 } \times \frac { 13 }{ 4 } \times \frac { 21 }{ 5 } =\frac { 3\div 1 }{ 2 } \times \frac { 7 }{ 3\div 1 } \times \frac { 13 }{ 4 } \times \frac { 21 }{ 5 }

=\frac { 1\times 7\times 13\times 21 }{ 2\times 4\times 5 }

=\frac { 1911 }{ 40 } =47\frac { 31 }{ 40 }