If the sum of the reciprocals of two consecutive

Let us assume that two consecutive odd integers are a and b where b = a+2. Then, sum of their reciprocal is (a+b)/ab. We have been given that sum of two consecutive odd integers is 12/35.

As GCD of two consecutive integers is always 1, sum of these two is 12 and multiplication of these two integers is 35.

Obviously, these two odd integers are 5 and 7.

Thus, larger integer is 7. So, answer is C.

Vincen wrote:If the sum of the reciprocals of two consecutive odd integers is 12/35 then the greater of the two integers is

A) 3
B) 5
C) 7
D) 9
E) 11

We can let the first odd integer = x and the next odd integer = x + 2; thus, the reciprocals are 1/x and 1/(x + 2). Thus:

1/x + 1/(x+2) = 12/35

Multiplying by 35x(x+2), we have:

35(x + 2) + 35x = 12x(x + 2)

35x + 70 + 35x = 12x^2 + 24x

12x^2 - 46x - 70 = 0

6x^2 - 23x - 35 = 0

(6x + 7)(x - 5) = 0

Thus, x = -7/6 or x = 5.

Since x is an integer, x must be 5 and the greater integer is x + 2 = 7.

Alternate solution:

We are given that the sum of the reciprocals of two consecutive odd integers is 12/35. We see that the denominator is 35. It’s not difficult to conjecture that the integers have to be 5 and 7 since 5 x 7 = 35. Finally, we can check the sum of 1/5 and 1/7 to see if they sum to 12/35:

1/5 + 1/7 = 7/35 + 5/35 = 12/35

Since they do add up to 12/35, the larger integer is 7.

Answer: C