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Complicated-Pls explain

by [email protected] » Tue Jun 10, 2014 7:55 pm
109) School A is 40% girls and School B is 60% girls. The ratio of the number of girls at School A to the number of girls at School B is 4:3. If 20 boys transferred from School A to School B and no other changes took place at the two schools, the new ratio of the number of boys at School A to the number of boys at School B would be 5:3. What would the difference between the number of boys at School A and at School B be after the transfer?


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by [email protected] » Tue Jun 10, 2014 8:37 pm
Hi Shibsriz,

This is a layered algebra question that would take a number of "math steps" to solve. The math involved isn't too hard, but you have to do a lot of work to get the job done....

"School A is 40% girls (thus 60% boys) and School B is 60% girls (thus 40% boys)"

We can write these ratios as:

School A
B:G
3:2

School B
B:G
2:3

Since ratios are all about multiples, I'm going to add a "variable" into each ratio (since the schools are different, I need to use a different variable for each).

School A
B:G
3x:2x

School B
B:G
2y:3y

"The ratio of girls at School A to girls at School B is 4:3"

We can write this ratio as...

2x/3y = 4/3

And cross-multiply....

6x = 12y

x = 2y

"If 20 boys transferred from School A to School B...the new ratio of boys at School A to boys at School B would be 5:3"

We can write this ratio as....

(3x - 20)/(2y + 20) = 5/3

And cross-multiply...

9x - 60 = 10y + 100

9x = 10y + 160

We now have 2 variables and 2 equations, so we can solve using "system" math...

x = 2y
9x = 10y + 160

By substituting in, we get...

9(2y) = 10y + 160

18y = 10y + 160

8y = 160

y = 20

Plugging back in, we get...

x = 40

The question asks for the DIFFERENCE in the number of boys at School A and at School B AFTER the transfer:

Number at School A after the transfer = 3x - 20 = 120-20 = 100
Number at School B after the transfer = 2y + 20 = 40 + 20 = 60

The difference = 100 - 60 = 40

Final Answer: B

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by Brent@GMATPrepNow » Tue Jun 10, 2014 9:49 pm
[email protected] wrote:109) School A is 40% girls and School B is 60% girls. The ratio of the number of girls at School A to the number of girls at School B is 4:3. If 20 boys transferred from School A to School B and no other changes took place at the two schools, the new ratio of the number of boys at School A to the number of boys at School B would be 5:3. What would the difference between the number of boys at School A and at School B be after the transfer?

A) 20
B) 40
C) 60
D) 80
E) 100
Here's another approach.

Let A = TOTAL population of School A
Let B = TOTAL population of School B

School A is 40% girls and School B is 60% girls
So, number of GIRLS at School A = 0.4A
And number of GIRLS at School B = 0.6B

The ratio of the number of girls at School A to the number of girls at School B is 4:3
We can write: 0.4A/0.6B = 4/3
Cross multiply to get: (3)(0.4A) = (4)(0.6B)
Simplify to get: 1.2A = 2.4B
Divide both sides by 1.2 to get A = 2B

Let's re-examine the following: School A is 40% girls and School B is 60% girls
So, current number of BOYS at School A = 0.6A
And current number of BOYS at School B = 0.4B

If 20 boys transferred from School A to School B....
So, NEW number of BOYS at School A = 0.6A - 20
And NEW number of BOYS at School B = 0.4B + 20

...., the new ratio of boys at School A to boys at School B would be 5:3
So, we can write: (0.6A - 20)/(0.4B + 20) = 5:3
Cross multiply to get: (3)(0.6A - 20) = (5)(0.4B + 20)
Simplify to get: 1.8A - 60 = 2B + 100
Add 60 to both sides to get: 1.8A = 2B + 160
Subtract 2B from both sides to get: 1.8A - 2B = 160

We now have a system of equations to solve:

Divide both sides by 1.2 to get A = 2B
1.8A - 2B = 160

Take 1.8A - 2B = 100 and replace A with 2B to get: 1.8(2B) - 2B = 160
Expand: 3.6B - 2B = 160
Simplify: 1.6B = 160
Solve: B = 100
So, the (current) TOTAL population at School B is 160
Since we know that A = 2B, we can conclude that the (current) TOTAL population at School A = (2)(100) = 200

Now that we know the values of A and B, we can complete our calculations.
Current number of BOYS at School A = 0.6A = 0.6(200) = 120
And current number of BOYS at School B = 0.4B = 0.4(100) = 40

So, NEW number of BOYS (after the transfer) at School A = 100
And NEW number of BOYS (after the transfer) at School B = 60

What would the difference between the number of boys at School A and at School B be after the transfer?
100 - 60 = [spoiler]40 = B[/spoiler]

Cheers,
Brent
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by GMATGuruNY » Wed Jun 11, 2014 1:56 am
[email protected] wrote:109) School A is 40% girls and School B is 60% girls. The ratio of the number of girls at School A to the number of girls at School B is 4:3. If 20 boys transferred from School A to School B and no other changes took place at the two schools, the new ratio of the number of boys at School A to the number of boys at School B would be 5:3. What would the difference between the number of boys at School A and at School B be after the transfer?


20


40


60


80


100
An alternate approach is to TEST CASES.
Since 20 boys are transferred, the number of boys at each school is almost certainly a MULTIPLE OF 10, implying that the number of girls at each school also is a MULTIPLE OF 10.

The ratio of the number of girls at School A to the number of girls at School B is 4:3.
(girls at A) : (girls at B) = 4:3 = 40:30 = 80:60 = 120:90 = 160:120...

Case 1: girls at A = 40, girls at B = 30
School A is 40% girls.
Since girls:boys = 40:60, girls = 40, boys = 60.
School B is 60% girls.
Since girls:boys = 60:40 = 30:20, girls = 30, boys = 20.

If 20 boys transferred from School A to School B and no other changes took place at the two schools, the new ratio of the number of boys at School A to the number of boys at School B would be 5:3.
School A:
new boys = 60-20 = 40.
School B:
new boys = 20+20 = 40.
Resulting ratio = 40:40 = 1:1.
Doesn't work.

Case 2: girls at A = 80, girls at B = 60
School A is 40% girls.
Since girls:boys = 40:60 = 80:120, girls = 80, boys = 120.
School B is 60% girls.
Since girls:boys = 60:40, girls = 60, boys = 40.

If 20 boys transferred from School A to School B and no other changes took place at the two schools, the new ratio of the number of boys at School A to the number of boys at School B would be 5:3.
School A:
new boys = 120-20 = 100.
School B:
new boys = 40+20 = 60.
Resulting ratio = 100:60 = 5:3.
Success!

Thus:
new boys at A - new boys at B = 100-60 = 40.

The correct answer is B.
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by mjpinvestor » Wed Jun 11, 2014 5:55 am
GMATGuruNY,

I started down this path and when I reached 1:1 (case 1) I stopped and went to Algebra. Needless to say, it took forever. I am curious, how do you know when to continue with TEST cases. With 2 minutes per question, missing the first test case causes panic and the thought that perhaps it wasn't the right approach. What if the test case for this question was the 4th case and not the 2nd, that would take well over 2 minutes also. Your advice is appreciated.
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by GMATGuruNY » Wed Jun 11, 2014 8:55 am
mjpinvestor wrote:GMATGuruNY,

I started down this path and when I reached 1:1 (case 1) I stopped and went to Algebra. Needless to say, it took forever. I am curious, how do you know when to continue with TEST cases. With 2 minutes per question, missing the first test case causes panic and the thought that perhaps it wasn't the right approach. What if the test case for this question was the 4th case and not the 2nd, that would take well over 2 minutes also. Your advice is appreciated.
Only 20 boys are transferred -- a relatively small number -- implying that the total number of boys attending the two schools is not too large.
Thus, we won't have to try many cases to find the one that satisfies all of the constraints in the problem.
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by GMATGuruNY » Wed Jun 11, 2014 9:30 am
[email protected] wrote:109) School A is 40% girls and School B is 60% girls. The ratio of the number of girls at School A to the number of girls at School B is 4:3. If 20 boys transferred from School A to School B and no other changes took place at the two schools, the new ratio of the number of boys at School A to the number of boys at School B would be 5:3. What would the difference between the number of boys at School A and at School B be after the transfer?

20
40
60
80
100
One more approach.

Let:
AG = girls in school A.
AB = boys in school A.
BG = girls in school B.
BB = boys in schools B.

Since school A is 40% girls, AG/AB = 40/60 = 2/3.
Since school B is 60% girls, BG/BB = 60/40 = 3/2.
It is given that AG/BG = 4/3.

Ratios can be MULTIPLIED:
AB/BB = AB/AG * AG/BG * BG/BB.
In the equation above, all of the values in red CANCEL OUT.
Thus:
AB/BB = 3/2 * 4/3 * 3/2 = 3/1.

Since AB/BB = 3:1 -- and the transfer of 20 boys implies that the number of boys in each school is a MULTIPLE OF 10 -- we get the following options:
AB=30, BB=10.
AB=60, BB=20.
AB=90, BB=30.
AB=120, BB=40.

After 20 boys are transferred from A to B in these four options, we get:
AB=10, BB=30.
AB=40, BB=40.
AB=70, BB=50.
AB=100, BB=60.

Only the option in red yields the required ratio of 5 to 3:
100:60 = 10:6 = 5:3.
Thus, after the transfer of 20 boys, AB - BB = 100-60 = 40.

The correct answer is B.
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by Mo2men » Wed Dec 23, 2015 2:11 pm
GMATGuruNY wrote:
[email protected] wrote:109) School A is 40% girls and School B is 60% girls. The ratio of the number of girls at School A to the number of girls at School B is 4:3. If 20 boys transferred from School A to School B and no other changes took place at the two schools, the new ratio of the number of boys at School A to the number of boys at School B would be 5:3. What would the difference between the number of boys at School A and at School B be after the transfer?


20


40


60


80


100
An alternate approach is to TEST CASES.
Since 20 boys are transferred, the number of boys at each school is almost certainly a MULTIPLE OF 10, implying that the number of girls at each school also is a MULTIPLE OF 10.

Hi,

Great Explanation.

I do not understand how we can conclude that number of boys will be multiple of 10? The number of boys could be 85 or 62 or any number. How can we conclude that for girls also?


Thanks for you help
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