17. If b and c are positive numbers and 1/b=b/c=c/8 , then b + c =
(A) 4
(B) 6
(C) 7
(D) 8
(E) 9
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Hi RiyaR,
This question is a real "pattern-matcher's" question. You don't have to do a lot of math to solve it, but you do have to think about how fractions can be equal to one another....
We're told that B and C are positive.
We're also told that 1/B = B/C = C/8
Let's start with that last fraction....C/8. We're told that the other two fractions are also equal to C/8 so they both need to have denominators that are either FACTORS of 8 or MULTIPLES of 8. Since the answer choices are relatively small, we're probably going to be dealing with FACTORS of 8....
So, what divides evenly into 8? 1, 2, 4, 8.....
B/C = C/8
If C = 8, then B = 8, but then 1/B would not equal the other two fractions, so C CANNOT = 8
If C = 4, then B = 2.....and we have 1/2 = 2/4 = 4/8. This is exactly what we're looking for.
The question asks us for B+C.....2+4 = 6
Final Answer: B
GMAT assassins aren't born, they're made,
Rich
This question is a real "pattern-matcher's" question. You don't have to do a lot of math to solve it, but you do have to think about how fractions can be equal to one another....
We're told that B and C are positive.
We're also told that 1/B = B/C = C/8
Let's start with that last fraction....C/8. We're told that the other two fractions are also equal to C/8 so they both need to have denominators that are either FACTORS of 8 or MULTIPLES of 8. Since the answer choices are relatively small, we're probably going to be dealing with FACTORS of 8....
So, what divides evenly into 8? 1, 2, 4, 8.....
B/C = C/8
If C = 8, then B = 8, but then 1/B would not equal the other two fractions, so C CANNOT = 8
If C = 4, then B = 2.....and we have 1/2 = 2/4 = 4/8. This is exactly what we're looking for.
The question asks us for B+C.....2+4 = 6
Final Answer: B
GMAT assassins aren't born, they're made,
Rich
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Brent@GMATPrepNow
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Here's an algebraic solution.RiyaR wrote:If b and c are positive numbers and 1/b = b/c = c/8 , then b + c =
(A) 4
(B) 6
(C) 7
(D) 8
(E) 9
Begin with 1/b = b/c = c/8
The lowest common denominator of b, c and 8 is 8bc.
So, multiply all three sides by 8bc to get: 8c = 8b² = bc²
Let's focus on 8c = 8b²
Divide both sides by 8 to get: c = b²
Now, let's focus on 8c = bc²
Divide both sides by c to get: 8 = bc
So, we have the following system:
c = b²
8 = bc
Take the red equation, and replace c with b² to get: 8 = b(b² )
Simplify: 8 = b³
Solve: b = 2
Now that we know that b = 2, we can use the fact that 1/b = c/8 to get 1/2 = c/8
This means that c = 4
So, b + c = 2 + 4 = [spoiler]6 = B[/spoiler]
Cheers,
Brent
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Given that the answer choices are integers, b and c are almost certainly integers themselves.RiyaR wrote:17. If b and c are positive numbers and 1/b=b/c=c/8 , then b + c =
(A) 4
(B) 6
(C) 7
(D) 8
(E) 9
1/b = c/8.
bc = 8.
8 has only two factor pairs:
1 and 8
2 and 4.
Since 1/b = b/c, c = b².
Implication:
c is a PERFECT SQUARE.
In the factor pairs for 8, only 1 and 4 are perfect squares.
If c=1 and b=8, then c ≠b².
Thus:
c=4 and b=2, implying that b+c = 2+4 = 6.
The correct answer is B.
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I'd proceed more or less the same way GMATGuru did: by assuming that the answers are friendly, familiar integers.
Since 1/b = c/8, it's probably the case that 1/b is in reduced terms and c/8 is not.
c/8 has a few reducible possibilities: 2/8, 4/8, and 8/8.
If c/8 = 2/8, then 1/b = 1/4. But then b/c = 4/2, which is not the same as 1/4 or 2/8.
If c/8 = 4/8, then 1/b = 1/2. Then b/c = 2/4, which IS the same as 1/2 and 4/8. Success!
Hence b = 2, c = 4, and b + c = 6.
Since 1/b = c/8, it's probably the case that 1/b is in reduced terms and c/8 is not.
c/8 has a few reducible possibilities: 2/8, 4/8, and 8/8.
If c/8 = 2/8, then 1/b = 1/4. But then b/c = 4/2, which is not the same as 1/4 or 2/8.
If c/8 = 4/8, then 1/b = 1/2. Then b/c = 2/4, which IS the same as 1/2 and 4/8. Success!
Hence b = 2, c = 4, and b + c = 6.
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We see that the equation can be re-expressed as three separate equations: 1/b = b/c, b/c = c/8, and 1/b = c/8.
Using the first equation, we have:
1/b = b/c
b^2 = c
Using the third equation, we have:
1/b = c/8
bc = 8
Since c = b^2, we have:
b(b^2) = 8
b^3 = 8
b = 2
So c = 2^2 = 4, and therefore, b + c = 2 + 4 = 6.
Answer: B
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