BTGModeratorVI wrote: ↑Tue Mar 10, 2020 7:24 am
Is mn < 10?
(1) m < 5 and n < 2
(2) 1 < m < 3 and n^2 < 25
Answer:
C
Source: Manhattan Prep
Target question: Is mn < 10?
Statement 1: m < 5 and n < 2
Let's TEST some values.
There are several values of m and n that satisfy statement 1. Here are two:
Case a: m = 0 and n = 0. In this case mn = (0)(0) = 0. So, the answer to the target question is
YES, mn IS less than 10
Case b: m = -5 and n = -5. In this case mn = (-5)(-5) = 25. So, the answer to the target question is
NO, mn is NOT less than 10
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: 1 < m < 3 and n² < 25
Let's TEST some values again.
There are several values of m and n that satisfy statement 2. Here are two:
Case a: m = 1 and n = 0. In this case mn = (1)(0) = 0. So, the answer to the target question is
YES, mn IS less than 10
Case b: m = 2.9 and n = 4. In this case mn = (2.9)(4) = 11.6. So, the answer to the target question is
NO, mn is NOT less than 10
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
IMPORTANT: When we combine the statements, we must adhere to the most RESTRICTIVE case for m and n each
For example, statement 1 tells us that m < 5, and statement 2 tells us that 1 < m < 3
Since both statements are TRUE, we can conclude that
1 < m < 3 since those possible values of m also satisfy the condition that m < 5
Likewise, for the value of n, statement 1 tells us that n < 2, and statement 2 tells us that n² < 25 (which is the SAME as saying
-5 < n < 5)
Since both statements are TRUE, we can conclude that
-5 < n < 2
So, we know the following about m and n:
1 < m < 3
-5 < n < 2
So, one possible case is that m = 2 and n = 0. In this case mn = (2)(0) = 0. So, the answer to the target question is
YES, mn IS less than 10
IMPORTANT: Since we've already shown that mn can be LESS THAN 10 (yielding a YES answer to the target question)
Is it possible for mn to be GREATER THAN or equal to 10 (which would yielding a NO answer to the target question)?
To find out, let's maximize the values of m and n.
So, m COULD equal 2.9999999... and n COULD equal 1.9999999....., in which case mn is LESS THAN 10
This tells us that, if
1 < m < 3 and
-5 < n < 2, then it's IMPOSSIBLE for mn to be GREATER THAN or equal to 10
So, the answer to the target question is
YES, mn IS less than 10
Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
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