March 30, 2013

Simplify a fraction multiplication before multiplying

I'm having difficulty in solving this question which involves calculating fractions - this question relates to finding an arc length.

140 divided by 360, multiplied by 2, multiplied by 22 divided by 7, multiplied by 12:


140

360
× 2 × 22

7
× 12


Solution:

You can either put everything in the calculator, multiplying the top numbers, then dividing by 360 and 7.

Or, you can simplify before you multiply. This process is actually quite handy!

For example, the first fraction 140/360 can be simplified into 14/36, and then further into 7/18 before you multiply.

We get

7

18
× 2 × 22

7
× 12


Now, the 7 in the numerator and the 7 in the denominator cancel out.

Why? Every time we have the same number in the numerator and the denominator, and the only other operation involved is multiplication (like in our example), that number cancels out. It becomes the same situation as if you multiply by 7 and divide by 7: the result is 1. As a shortcut, we can cancel out those numbers and write 1's in their places.

Now we get

1

18
× 2 × 22

1
× 12


Then, 22 and 18 have a common factor 2... so that 2 cancels out. You can think of it as being...

1

2 × 9
× 2 × 2 × 11

1
× 12


... or you can think of it as if  the fraction 22/18 was in there, which simplifies to 11/9.

1

9
× 2 × 11

1
× 12


One last simplification: 12 in the top and 9 in the bottom have a common factor 3... so, divide both 12 and 9 by that 3 and get:

1

3
× 2 × 11

1
× 4


Now it is easy to multiply mentally (regular fraction multiplication):

1

3
× 2 × 11

1
× 4 = 88

3
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