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A string is cut into two parts. The ratio of the length of the whole

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A string is cut into two parts. The ratio of the length of the whole  [#permalink]

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New post 25 Nov 2019, 06:11
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Question Stats:

33% (01:36) correct 67% (02:17) wrong based on 18 sessions

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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
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Re: A string is cut into two parts. The ratio of the length of the whole  [#permalink]

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New post 26 Nov 2019, 18:53
CaptainLevi wrote:
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


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
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Re: A string is cut into two parts. The ratio of the length of the whole  [#permalink]

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New post 28 Nov 2019, 17:40
CaptainLevi wrote:
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



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
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Re: A string is cut into two parts. The ratio of the length of the whole   [#permalink] 28 Nov 2019, 17:40
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