SOLUTION: If 1/a + 1/b = 5/12 and ab = 24, what is the value of a+b?

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Question 1209283: If 1/a + 1/b = 5/12 and ab = 24, what is the value of a+b?
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

There are at least three ways to answer this question.

Method 1

1%2Fa+%2B+1%2Fb+=+5%2F12

b%2F%28ab%29+%2B+a%2F%28ab%29+=+10%2F24

%28b+%2B+a%29%2F%28ab%29+=+10%2F24

%28a%2Bb%29%2F%2824%29+=+10%2F24 Plug in ab = 24.

a%2Bb+=+10

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Method 2

1%2Fa+%2B+1%2Fb+=+5%2F12

ab%2A%281%2Fa+%2B+1%2Fb%29+=+ab%2A%285%2F12%29 Multiply both sides by ab

ab%2A%281%2Fa%29+%2B+ab%2A%281%2Fb%29+=+ab%2A%285%2F12%29 Distribute

b+%2B+a+=+ab%2A%285%2F12%29

a%2Bb+=+24%2A%285%2F12%29 Plug in ab = 24

a%2Bb+=+10

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Method 3

ab = 24 rearranges to b = 24/a which has reciprocal 1/b = a/24

Then,
1%2Fa+%2B+1%2Fb+=+5%2F12

1%2Fa+%2B+a%2F24+=+5%2F12 Replaced 1/b with a/24

%281%2Fa+%2B+a%2F24%29%2A24a+=+%285%2F12%29%2A24a Multiply both sides by the LCD 24a to clear out the fractions.

%281%2Fa%29%2A24a+%2B+%28a%2F24%29%2A24a+=+10a

24+%2B+a%5E2+=+10a

a%5E2-10a%2B24+=+0

%28a-6%29%28a-4%29+=+0

a-6=0 or a-4=0

a+=+6 or a+=+4

If a = 6, then b = 24/a = 24/6 = 4.
If a = 4, then b = 24/a = 24/4 = 6.
The order of a,b doesn't matter so we arrive at the same pairing.

The pair of values is a = 4 and b = 6 (or a = 6 and b = 4).
As a way to check, 1%2Fa+%2B+1%2Fb+=+1%2F4+%2B+1%2F6+=+3%2F12+%2B+2%2F12+=+%283%2B2%29%2F12+=+5%2F12

In short 1%2F4+%2B+1%2F6+=+5%2F12 which helps us confirm we have the correct a,b values.

Then a+b = 4+6 = 10

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Answer: 10