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The ratio of boys to girls in Class A is 1 to 4, and that in Class B

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The ratio of boys to girls in Class A is 1 to 4, and that in Class B [#permalink]

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The ratio of boys to girls in Class A is 1 to 4, and that in Class B is 2 to 5. In addition, there are twice as many students in Class A as in Class B. If the two classes are combined to form one class, what would the resulting ratio of boys to girls?

A) 1 to 3
B) 5 to 12
C) 8 to 27
D) 6 to 25
E) 13 to 27
[Reveal] Spoiler: OA

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Re: The ratio of boys to girls in Class A is 1 to 4, and that in Class B [#permalink]

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We have three ratios:

1) Classroom A -> 1:4 -> potion of Boys = \(1/(1+4)\) = \(1/5\)
2) Classroom B -> 2:5 -> potion of Boys = \(2/(2+5)\) = 2/7
3) Classrooms RATIO -> 2:1 -> portion of Class A = 2/3 and portion of Class B = \(1/3\)

I stated the weighted portion of Boys in both classrooms:

Total Portion of Boys
\(2/3*(1/5)+1/3*(2/7)\)
\(= 2/3*(1/5+1/7)\)
\(= 2/3*(12)/35\)
\(= 8/35\)

Total Portion of Girls:

\(1-8/35=27/35\)

So the final ratio is
Portion boys / Portion of girls
8/27

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Re: The ratio of boys to girls in Class A is 1 to 4, and that in Class B [#permalink]

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amathews wrote:
The ratio of boys to girls in Class A is 1 to 4, and that in Class B is 2 to 5. In addition, there are twice as many students in Class A as in Class B. If the two classes are combined to form one class, what would the resulting ratio of boys to girls?

A) 1 to 3
B) 5 to 12
C) 8 to 27
D) 6 to 25
E) 13 to 27


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Re: The ratio of boys to girls in Class A is 1 to 4, and that in Class B [#permalink]

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amathews wrote:
The ratio of boys to girls in Class A is 1 to 4, and that in Class B is 2 to 5. In addition, there are twice as many students in Class A as in Class B. If the two classes are combined to form one class, what would the resulting ratio of boys to girls?

A) 1 to 3
B) 5 to 12
C) 8 to 27
D) 6 to 25
E) 13 to 27


Let's use weighted averages.

Weighted average of groups combined = (group A proportion)(group A average) + (group B proportion)(group B average) + (group C proportion)(group C average) + ...

So, in this case, Weighted average of combined classes = (Class A proportion)(Class A average) + (Class B proportion)(Class B average)

CLASS A:
Ratio of boys to girls in Class A is 1 to 4
So, for every 5 students, we have 1 boy and 4 girls.
In other words, 1/5 of the students are boys.
So, we can say the class A average is 1/5 boys

CLASS B:
Ratio of boys to girls in Class B is 2 to 5
So, for every 7 students, we have 2 boys and 5 girls.
In other words, 2/7 of the students are boys.
So, we can say the class B average is 2/7 boys

There are twice as many students in Class A as in Class B.
So, for every 3 students in the COMBINED group, there are 2 students from Class A, and 1 student from Class B
In others words, Class A students comprise 2/3 of the COMBINED group, and Class B students comprise 1/3 of the COMBINED group

Now plug these values into our formula to get:
Weighted average of combined classes = (2/3)(1/5) + (1/3)(2/7)
= 2/15 + 2/21
= 14/105 + 10/105
= 24/105
= 8/35

So, in the combined group, 8/35 of the students are boys, which means 27/35 of the students are girls.
In other words, among every 35 students in the combined group, 8 are boys and 27 are girls.
So, the ratio of boys to girls = 8 to 27

Answer:
[Reveal] Spoiler:
C


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Re: The ratio of boys to girls in Class A is 1 to 4, and that in Class B [#permalink]

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amathews wrote:
The ratio of boys to girls in Class A is 1 to 4, and that in Class B is 2 to 5. In addition, there are twice as many students in Class A as in Class B. If the two classes are combined to form one class, what would the resulting ratio of boys to girls?

A) 1 to 3
B) 5 to 12
C) 8 to 27
D) 6 to 25
E) 13 to 27


There are several ways to do it. Let me discuss two of them.

Method 1:

    • The ratio of the number of boys to girls in Class A is 1 : 4

      o Therefore, we can say that the total number of students in Class A = 5x

    • The ratio of the number of boys to girls in Class B is 2 : 5

      o Therefore, we can say that the total number of students in Class B = 7y

    • As per the given condition:

      o \(5x = 2*7y\)
      o \(\frac{x}{y} = \frac{14}{5}\)
      o Take x = 14 and y = 5

    • Thus, the combined Boys to Girls Ratio would be

      \(= \frac{(x + 2y)}{(4x + 5y)}\)
      \(= \frac{(14 + 10)}{(56+25)}\)
      \(= \frac{24}{81}\)
      \(= \frac{8}{27}\)

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Re: The ratio of boys to girls in Class A is 1 to 4, and that in Class B [#permalink]

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Method 2:

Use the simple concept of weighted average to get the answer.

    • The ratio of the number of boys to girls in Class A is 1 : 4
      o Number of boys will be \(\frac{1}{(1+4)} = \frac{1}{5}th\) of total
      o Number of girls will be \(\frac{4}{(1+4)} = \frac{4}{5}th\) of total

    • The ratio of the number of boys to girls in Class B is 2 : 5
      o Number of boys will be \(\frac{2}{(2+5)} = \frac{2}{7}th\) of total
      o Number of girls will be \(\frac{5}{(2+5)} = \frac{5}{7}th\) of total

    • Ratio of class A : Class B = 2 : 1

Thus weighted average = (Number of boys in Class A + Class B) / (Number of Girls in Class A + Class B)
= \((2 * \frac{1}{5} + 1 * \frac{2}{7})/(2*\frac{4}{5} + 1 * \frac{5}{7})\)
= \(\frac{24}{35}/\frac{81}{35}\)
= \(\frac{8}{27}\)

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Re: The ratio of boys to girls in Class A is 1 to 4, and that in Class B [#permalink]

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New post 19 Apr 2017, 14:24
Another way of solving this. I believe this way is faster.

Class A: ratio a:4a and total student 5a ------------- Class B: ratio 2b:5b and total student 7b

as per question stem, 5a=2*(7b)

Ratio of the new class is (a+2b):(4a+5b)
which same ratio as (5a+10b):(20a+25b)
substitution leads to ratio 24b:81b or 24:81
Hence 8:27

Answer is C
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Re: The ratio of boys to girls in Class A is 1 to 4, and that in Class B [#permalink]

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New post 19 Apr 2017, 14:25
Another way of solving this. I believe this way is faster.

Class A: ratio a:4a and total student 5a ------------- Class B: ratio 2b:5b and total student 7b

as per question stem, 5a=2*(7b)

Ratio of the new class is (a+2b):(4a+5b)
which same ratio as (5a+10b):(20a+25b)
substitution leads to ratio 24b:81b or 24:81
Hence 8:27

Answer is C
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Re: The ratio of boys to girls in Class A is 1 to 4, and that in Class B [#permalink]

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New post 24 May 2017, 07:21
Not sure if my approach is easier or harder but I used the LCM method. I find it helps simplify a lot of questions.

1:4 = 5
2:5 = 7

5 & 7 LCM = 35. Since A is twice as much as B. then A must be 70 and B = 35. (even if its 140 & 70, the ratio will be same).

A= 14 boys, 56 girls.
B= 10 boys, 25 girls.

24 boys : 81 girls divide by 3... = 8:27.
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Re: The ratio of boys to girls in Class A is 1 to 4, and that in Class B   [#permalink] 24 May 2017, 07:21
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