Castor.kim wrote:
If n, k are positive integers, (1/n) - [1/(n+k)] < 1/10 ?
1) n > 10
2) k < 10
Target question:
Is (1/n) - [1/(n+k)] < 1/10 ?
Looks like a nice candidate for rephrasing the target question.
Let's first combine the two expressions on the left side. To do this, we'll need a common denominator.
We get: (n+k)/[n(n+k)] - (n)/[n(n+k)] < 1/10
Simplify: k/[n(n+k)] < 1/10
Multiply both sides by 10: 10k/[n(n+k)] < 1
Multiply both sides by [n(n+k)]: 10k < n(n+k)
Aside: This last step was allowable because we can be certain that [n(n+k)] is positive.
Simplify: 10k < n^2 + nk
Subtract nk from both sides: 10k - nk < n^2
Factor: k (10 - n) < n^2
So, the inequality k (10 - n) < n^2 is equivalent to the inequality (1/n) - [1/(n+k)] < 1/10
This means we can now rephrase the target question .
Rephrased target question:
Is k (10 - n) < n^2?
Statement 1: n > 10
If n > 10, then (10 - n) must be negative.
Since k is positive, we can see that
k (10 - n) must have a
negative value.
Conversely,
n^2 must have a
positive value.
So,
k (10 - n) must be less than n^2
Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: k < 10
There are several pairs of numbers that meet this condition. Here are two:
Case a: k=1, n=10, in which case
k (10 - n) is less than n^2
Case b: k=1, n=1, in which case
k (10 - n) is not less than n^2
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Answer =
A
Cheers,
Brent
