if a number is added to the numerator of 11/12 and twice as much is added to the denominator, the result is 8/11. What is the number?

Let x be the unknown number we want to find. The old numerator is 11 and it updates to 11+x after we add on this unknown number. The old denominator is 12 which becomes 12+2x after adding twice the unknown number onto it.

We go from 11/12 to (11+x)/(12+2x). We are told that this new fraction is equal to 8/11. Let's equate the two expressions, cross multiply, and solve for x

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(11+x)/(12+2x) = 8/11

11(11+x) = 8(12+2x) ... cross multiply

11(11)+11(x) = 8(12)+8(2x) ... distribute

121 + 11x = 96 + 16x

121 + 11x - 96 = 96 + 16x - 96 ... subtract 96 from both sides

25 + 11x = 16x

25 + 11x-11x = 16x-11x ... subtract 11x from both sides

25 = 5x

5x = 25

5x/5 = 25/5 ... divide both sides by 5

x = 5

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Since x = 5, we can say

11+x = 11+5 = 16

12+2x = 12+2*5 = 12+10 = 22

So, (11+x)/(12+2x) = 16/22 = 8/11

which confirms we have the right answer.

note: to reduce 16/22 to 8/11, you divide both parts by the GCF 2.