SOLUTION: Simplify: (a)27/20 x 8/63 divide by 6/35

Question 1015821: Simplify:
(a)27/20 x 8/63 divide by 6/35
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
dividing by a number is the same as multiplying by its reciprocal.

why is this true?

here's an example of why.

assume (5/6) / (3/5)

this is equivalent to (5/6) * (5/3)

here's the progression as to why that's true.

start with (5/6) / (3/5)

multiply both numerator and denominator by (5/3).

you can do this because (5/3) / (5/3) is equal to 1 and multiplying anything by 1 does not change its value.

you will get (5/6) * (5/3) divided by (3/5) * (5/3).

(3/5) * (5/3) in the denominator is equal to 1 because multiplying any number by its reciprocal is equal to 1.

you are left with (5/6) * (5/3) divided by 1 which is the same as (5/6) * (5/3).

you're done.

(5/6) / (3/5) is equivalent to (5/6) * (5/3).

you apply this concept as one of the steps to simplify your expression.

start with:

(27/20) * (8/63) / (6/35).

since dividing by (6/35) is the same as multiplying by (35/6), your expression becomes:

(27/20) * (8/63) * (35/6)

this can also be written as:

(27 * 8 * 35) / (20 * 63 * 6)

you can factor both numerator and denominator to get:

(3 * 9 * 8 * 5 * 7) / (4 * 5 * 9 * 7 * 6)

in the numerator, 27 became 3 * 9 and 35 became 5 * 7.
in the denominator, 20 became 4 * 5 and 63 became 9 * 7.

you've got a 5 in the numerator and denominator that cancel out.
you've got a 7 in the numerator and denominator that cancel out.
you've got a 9 in the numerator and denominator that cancel out.

you are left with:

(3 * 8) / (4 * 6)

this can be further factored to get:

(3 * 4 * 2) / (4 * 3 * 2)

in the numerator, 8 becomes 4 * 5..
in the denominator, 6 becomes 3 * 2.

you've got a 3 and a 4 and a 2 in the numerator and the same in the denominator that can cancel out.

you are left with:

1/1 = 1.

everything cancels out and you are left 1.

you can confirm by using your calculator and evaluating the original expression as is.

you will get (27/20) * (8/63) / (6/35) = 1.

factoring manually can be done in stages or it can be done all at once.
factoring in stages sometimes has benefits because you can simplify a piece at a time.

an alternative would be to factor each number down to its prime factors and then doing the cancellation.

when typing, don't forget that (a/b) * (c/d) * (e/f) can be written as (a * c * e) / (b * d * f)

it's sometimes easier to see what factors in the numerator can cancel out with what factors in the denominator when you type it that way.

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