Question 1209283
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There are at least three ways to answer this question. 


Method 1


{{{1/a + 1/b = 5/12}}}


{{{b/(ab) + a/(ab) = 10/24}}}


{{{(b + a)/(ab) = 10/24}}}


{{{(a+b)/(24) = 10/24}}} Plug in ab = 24.


{{{a+b = 10}}} 


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Method 2


{{{1/a + 1/b = 5/12}}}


{{{ab*(1/a + 1/b) = ab*(5/12)}}} Multiply both sides by ab


{{{ab*(1/a) + ab*(1/b) = ab*(5/12)}}} Distribute


{{{b + a = ab*(5/12)}}}


{{{a+b = 24*(5/12)}}} Plug in ab = 24


{{{a+b = 10}}} 


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Method 3


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


Then,
{{{1/a + 1/b = 5/12}}}


{{{1/a + a/24 = 5/12}}} Replaced 1/b with a/24


{{{(1/a + a/24)*24a = (5/12)*24a}}} Multiply both sides by the LCD 24a to clear out the fractions.


{{{(1/a)*24a + (a/24)*24a = 10a}}}


{{{24 + a^2 = 10a}}}


{{{a^2-10a+24 = 0}}}


{{{(a-6)(a-4) = 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/a + 1/b = 1/4 + 1/6 = 3/12 + 2/12 = (3+2)/12 = 5/12}}}


In short {{{1/4 + 1/6 = 5/12}}} which helps us confirm we have the correct a,b values.


Then a+b = 4+6 = <font color=red>10</font>


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Answer: <font color=red>10</font>
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