Question

the sum of two integers is 10 and the sumof their reciprocal is 5/12. then the larger of this integers is?

Answer 1

Answer:

Let the two integers be m and n. It is given that the sum of the two integers is 10. This gives the equation

m+ n = 10………………….. ………………(1)

It is also given that the sum of their reciprocal is 5/12. The reciprocal of m is 1/m and the reciprocal of n is 1/n. Therefore we get the second equation

1/m + 1/n = 5/12

Or, (m+n)/mn = 5/12

Substituting for m+n = 10 from (1),

10/mn = 5/12

Divide both sides by 5 to get

2/mn = 1/12

Taking reciprocals on both sides,

mn/2 = 12

Cross-multiplying,

mn = 24

This gives m = 24/n

Substituting this value of m in (1),

24/n + n = 10

Taking LCM,

(24 + n^2)/n = 10

Cross-multiplying,

24 + n^2 = 10n

Or, n^2 -10n + 24 = 0

Factorising,

n^2 -6n -4n + 24 = 0

Or, n(n-6) - 4(n-6) =0

Or, (n-6) (n-4) = 0

This gives two solutions for n: 6 and 4

The corresponding solutions for m are obtained from (1) as 10-n and they are

4 and 6.

Thus there are two pairs of solutions for (m,n),

(4,6) and (6,4)

It follows from above that

The largest integer is 6.