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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