at ABC academy, the ratio of the number of girls to the number of boys is 5 to 3. if 20 boys and 20 girls were to leave, the ratio of the number of boys to the number of girls would be 2 to 5. How many children are at ABC academy?
A 56
B 60
C 72
D 84
E 96
ratio question
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Let B = current # of boyshutch27 wrote:at ABC academy, the ratio of the number of girls to the number of boys is 5 to 3. if 20 boys and 20 girls were to leave, the ratio of the number of boys to the number of girls would be 2 to 5. How many children are at ABC academy?
A 56
B 60
C 72
D 84
E 96
Let G = current # of girls
The ratio of the number of girls to the number of boys is 5 to 3
So, G/B = 5/3
Cross multiply to get 5B = 3G
If 20 boys and 20 girls were to leave, the ratio of the number of boys to the number of girls would be 2 to 5.
(B-20)/(G-20) = 2/5
Cross multiply to get: 2(G-20) = 5(B-20)
Expand: 2G - 40 = 5B - 100
We now have two equations with 2 variables.
2G - 40 = 5B - 100
5B = 3G
NOTE: If 5B = 3G, then we'll take the red equation and replace 5B with 3G to get: 2G - 40 = 3G - 100
Solve to get G = 60
If G = 60, then B = 36 (since we already know that 5B = 3G)
So the student population = B + G = 36 + 60 = 96
Answer: E
Cheers,
Brent
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In most cases, I think one should use as few variables as possible. However, with ratio questions, I find it easier to organize my thoughts if I use two variables.hutch27 wrote:Brent, I don't understand that approach. Why start with two variables B and G? Is that the only way to tackle this problem?
Here's the 1 variable solution:
The ratio of the number of girls to the number of boys is 5 to 3
Let B = current # of boys
So 5B/3 = current # of girls
If 20 boys and 20 girls were to leave. . .
Future # of boys = B - 20
Future # of girls = 5B/3 - 20
. . . the ratio of the number of boys to the number of girls would be 2 to 5.
(B - 20)/(5B/3 - 20) = 2/5
Cross multiply to get: 5(B - 20) = 2(5B/3 - 20)
Expand: 5B - 100 = 10B/3 - 40
Eliminate fractions by multiplying both sides by 3: 15B - 300 = 10B - 120
5B = 180
B = 36
So, # of boys = 36
# of girls = 5B/3
= 5(36)/3
= 60
So the current student population = 36 + 60 = 96
Answer: E
Cheers,
Brent
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Another approach
let total number of students be T
let total number of students in future = T-40
number of boys = 3T/8
number of boys in future = 2(T-40)/7
Number of Boys decreased by 20
Hence
3T/8 - 20 = 2(T-40)/7
21T - (20*8*7) = 16T - (80*8)
5T = (20*8*7) - (80*8)
5T = (80*14) - (80*8)
T = (80*6)/5 = 16*6 = 96
let total number of students be T
let total number of students in future = T-40
number of boys = 3T/8
number of boys in future = 2(T-40)/7
Number of Boys decreased by 20
Hence
3T/8 - 20 = 2(T-40)/7
21T - (20*8*7) = 16T - (80*8)
5T = (20*8*7) - (80*8)
5T = (80*14) - (80*8)
T = (80*6)/5 = 16*6 = 96
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An alternate -- and perhaps the fastest -- approach is to plug in the answer choices, which represent the total number of children.hutch27 wrote:at ABC academy, the ratio of the number of girls to the number of boys is 5 to 3. if 20 boys and 20 girls were to leave, the ratio of the number of boys to the number of girls would be 2 to 5. How many children are at ABC academy?
A 56
B 60
C 72
D 84
E 96
G:B = 5:3, implying that there are 5 girls for every 3 boys.
Since = 5+3 = 8, the total number of children must be a multiple of 8.
Eliminate B and D.
After 40 children leave, G:B = 5:2, implying that there are 5 girls for every 2 boys.
Since 5+2 = 7, the resulting number of children must be a multiple of 7.
Thus, when 40 is subtracted from the correct answer choice, the result must be a multiple of 7:
A: 56-40 = 16.
C: 72-40 = 32.
E: 96-40 = 56.
Only the result in red is a multiple of 7.
The correct answer is E.
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
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