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# The ratio of caterpillars to butterflies in a certain lepidopterarium

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The ratio of caterpillars to butterflies in a certain lepidopterarium  [#permalink]

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21 Apr 2017, 07:54
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The ratio of caterpillars to butterflies in a certain lepidopterarium was 24 to 1. After some of the caterpillars turned into butterflies, butterflies made up 18% of the insects in the lepidopterarium. If the lepidopterarium houses only caterpillars and butterflies, and if no other changes took place, the number of caterpillars that turned into butterflies must have been at least:

A. 6
B. 7
C. 12
D. 14
E. 15

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Re: The ratio of caterpillars to butterflies in a certain lepidopterarium  [#permalink]

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21 Apr 2017, 08:59
1
Let the number if Caterpillars and Butterflies be, 24x and x respectively.

Let 'a' be the number of caterpillars changed into butterflies.

Total number of Caterpillars now = 24x-a
Total number of Butterflies now = x+a

(x+a)*100/25x = 18

4x+4a = 18x

4a=14x
a = 7x/2

Since a and x have to be integers, taking the minimum value for x = 2, a comes as 7.

Therefore the number of caterpillars that turned into butterflies must have been at least 7.

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Re: The ratio of caterpillars to butterflies in a certain lepidopterarium  [#permalink]

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06 Mar 2019, 03:53
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Q: The ratio of caterpillars to butterflies in a certain lepidopterarium was 24 to 1. After some of the caterpillars turned into butterflies, butterflies made up 18% of the insects in the lepidopterarium. If the lepidopterarium houses only caterpillars and butterflies, and if no other changes took place, the number of caterpillars that turned into butterflies must have been at least:

Solution:
Old ratio Caterpillars:butterflies 24:1 = 48:2 (multiple by 2)
New ratio Caterpillars:butterflies 82:18= 41:9 (divide by 2)

Caterpillars decreased by 48-41=7 = B
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The ratio of caterpillars to butterflies in a certain lepidopterarium  [#permalink]

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06 Mar 2019, 06:01
If 18% of the bugs were butterflies, then 9 out of 50 bugs were butterflies. That's completely reduced, so 50 is the smallest possible total number of bugs. If initially 1 out of 25 bugs were butterflies, with 50 bugs in total, we would have started with 2 butterflies, so the smallest possible increase is 7.

But there's a logical problem with the question. If the smallest possible number of new butterflies is 7, then it's true that "the number of caterpillars that turned into butterflies must have been at least" 7. But it also must have been at least 6, since 7 is larger than 6. So the question has two right answers, A and B.
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The ratio of caterpillars to butterflies in a certain lepidopterarium  [#permalink]

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06 Mar 2019, 09:23

But the Question stem states... "After some of the caterpillars turned into butterflies, butterflies made up 18% of the insects in the lepidopterarium."

Im assuming when you said "If 18% of the bugs were butterflies" you meant this was the initial number of butterflies
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Re: The ratio of caterpillars to butterflies in a certain lepidopterarium  [#permalink]

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06 Mar 2019, 09:47
AvidDreamer09 wrote:

But the Question stem states... "After some of the caterpillars turned into butterflies, butterflies made up 18% of the insects in the lepidopterarium."

Im assuming when you said "If 18% of the bugs were butterflies" you meant this was the initial number of butterflies

It doesn't matter when 18% of the bugs were butterflies - if you can take 18% of a group of people or a group of bugs or any group of things, then the number of things in total must be a multiple of 50, because 9/50 can't be reduced any further. But of course the 18% I mentioned is the 18% at the end, after the caterpillars became butterflies, as I hope I made clear by later discussing what the situation was "initially".
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Re: The ratio of caterpillars to butterflies in a certain lepidopterarium  [#permalink]

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08 Mar 2019, 07:57
Bunuel wrote:
The ratio of caterpillars to butterflies in a certain lepidopterarium was 24 to 1. After some of the caterpillars turned into butterflies, butterflies made up 18% of the insects in the lepidopterarium. If the lepidopterarium houses only caterpillars and butterflies, and if no other changes took place, the number of caterpillars that turned into butterflies must have been at least:

A. 6
B. 7
C. 12
D. 14
E. 15

We can create an initial ratio of C : B = 24x : x. If we let n = the number of caterpillars that turn into butterflies, we have:

(x + n)/(25x) = 18/100

(x + n)/(25x) = 9/50

50(x + n) = 225x

50x + 50n = 225x

50n = 175x

2n = 7x

n = 7x/2

Since n is a positive integer, x is divisible by 2 (since 7 isn’t). Therefore, the minimum possible value of n is 7(2)/2 = 7.

Alternate Solution:

To find the minimum possible value of the caterpillars that turn into butterflies, let’s first minimize the number of butterflies after some of the caterpillars turn. Since 18% = 18/100 = 9/50, the minimum number of butterflies after the transformation is 9 and the minimum number of insects is 50 (because both quantities must be integers and any smaller number for either quantity will result in a fractional value for the other quantity).

Let x denote the number of butterflies before the transformation. Since the ratio of caterpillars to butterflies was 24:1, the number of caterpillars is 24x. Then, the total number of insects is x + 24x = 25x. Since we know the minimum possible number of insects is 50, we have 25x = 50 and thus, x = 2.

Since there were x = 2 butterflies before the transformation and 9 butterflies afterwards, the minimum possible number of caterpillars that turn into butterflies is 9 - 2 = 7.

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Re: The ratio of caterpillars to butterflies in a certain lepidopterarium   [#permalink] 08 Mar 2019, 07:57
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