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25 Nov 2019, 06:11
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
85%
(hard)
Question Stats:
51% (02:41) correct
49%
(02:46)
wrong
based on 76
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A string is cut into two parts. The ratio of the length of the whole string to that of the smaller part is the square of the ratio of the lengths of the larger and smaller parts. Approximately, what fraction of the length of the bigger part is the length of the smaller part?
a) 0.32
b) 0.42
c) 0.62
d) 0.72
e) 0.88
a) 0.32
b) 0.42
c) 0.62
d) 0.72
e) 0.88
26 Nov 2019, 18:53
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CaptainLevi
Let the parts be in ratio x:y.
\((x+y):y=x^2:y^2.....(x+y)*y^2=y*x^2.........(x+y)y=x^2......\frac{xy+y^2}{x^2}=1.......\frac{y}{x}+(\frac{y}{x})^2=1\)
Let \(\frac{y}{x}=a \)..
\(\frac{y}{x}+(\frac{y}{x})^2=1....a+a^2=1........a(a+1)=1\)
Check which choice fits in..
a) 0.32....0.32(0.32+1)=0.3*1.3=0.39
b) 0.42...0.42(0.42+1)=0.4*1.4=0.56
c) 0.62...0.62(0.62+1)=0.6*1.6=0.96~1...YES
d) 0.72...0.72(0.72+1)=0.7*1.7=1.19
e) 0.88.. Will be >1.19
C
28 Nov 2019, 17:40
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CaptainLevi
We can let the ratio of the larger part to the smaller part = L/S, so the square of that is L^2/S^2, and we have:
L^2/S^2 = (L + S)/S
(L/S)^2 = L/S + 1
Letting x = L/S, we have:
x^2 = x + 1
Since (5/3)^2 = 25/9 ≈ 2.78 and 5/3 + 1 = 8/3 ≈ 2.67, we see that x ≈ 5/3. Since x = L/S, S/L = 1/x. So S/L ≈ 1/(5/3) = 3/5. Therefore, the smaller part is approximately 3/5 of the larger part. Since 3/5 = 0.6 is closest to 0.62, choice C would be the correct answer.
Alternate Solution:
We can let the ratio of the larger part to the smaller part = L/S, so the square of that is L^2/S^2, and we have:
L^2/S^2 = (L + S)/S
(L/S)^2 = L/S + 1
Letting x = L/S, we have:
x^2 = x + 1
x^2 - x - 1 = 0
Using the quadratic formula, we obtain:
x = (1 +/- √5)/2 ≈ 3.24/2 ≈ 32/20 = 8/5
Now, since x = L/S, we see that L/S = 8/5. Thus, the ratio of S/L is 5/8, or 0.625, which is closest to 0.62, choice C.
Answer: C
