Hi commoditempore,Your chain of logic is spot on. You've correctly shown that next year's manufacturing count must be
strictly greater than
3x. So far, perfect.
The one thing to fix is what the phrase
"at least three times" actually claims. "At least three times" means
"three times or more." It is a
lower bound - it does
not say "exactly three times."
So when you prove that manufacturing next year is
strictly greater than
3x, you have
not contradicted answer A - you've actually confirmed it. Anything that is more than
3x is automatically also
at least3x. Being strictly greater than
3x is one of the cases that "at least
3x" covers.
Think of it this way:
- "More than
3x" - the value is somewhere above
3x.
- "At least
3x" - the value is
3xor anything above it.
Every value in the first list also sits in the second list. So proving the stronger "more than" guarantees the weaker "at least."
And remember the question asks which statement
must be true. A safely-true, slightly weaker claim is exactly what you want - it can never fail.
Quick gut-check to lock it in:Suppose I tell you I have
more than $10 in my wallet, and it turns out I have
$15.
- Is the statement "I have
at least $10" false? No - it's true.
- Would "I have
exactly $10" be true? No.
"At least" is satisfied by any amount from the floor upward. Same move here: manufacturing landing above
3x still makes "at least three times" true - so
A holds.
Answer: Acommoditempore
In 2016 in service 3x and in Manufacturing x. Now in next year employees in Manufacturing will be atleast equal to employees in service also employees in service will definitely larger than in 2016 i.e >3x so how employees in manufacturing in next year atleast 3x it has to be strictly greater than 3x