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Difficulty: HardPercents and Percent Change

In a certain school, the number of students in the science club was 25%25\% of the number of students in the math club. During a membership drive, the number of students in the science club increased by 30%30\%, and the number of students in the math club increased by 10%10\%. After the drive, the total number of students in both clubs combined was 228228. If no student is a member of both clubs, what was the total number of students in both clubs combined before the membership drive?

Answer: 200 students

Answer

The total number of students in both clubs combined before the membership drive was 200.
To find the initial total number of students, we define the initial math club members as MM. Since the science club has 25%25\% of the size of the math club, its initial size is 0.25M0.25M. After a 30%30\% increase, the science club size is 1.30(0.25M)=0.325M1.30(0.25M) = 0.325M. After a 10%10\% increase, the math club size is 1.10M1.10M. The new combined total is 0.325M+1.10M=2280.325M + 1.10M = 228. Simplifying gives 1.425M=2281.425M = 228, which yields M=160M = 160. The initial science club size is 0.25(160)=400.25(160) = 40. The total number of students before the drive was 160+40=200160 + 40 = 200.

Step-by-Step Solution

1
Define variables for the initial number of students in each club.
Let MM be the initial number of students in the math club, and S=0.25MS = 0.25M be the initial number of students in the science club.
This establishes a relationship between the sizes of the two clubs based on the given percentage.
2
Express the new number of students in each club after the percent increases.
Science club: 1.30×S=1.30(0.25M)=0.325M1.30 \times S = 1.30(0.25M) = 0.325M. Math club: 1.10M1.10M.
An increase of 30%30\% is represented by multiplying by 1.301.30, and an increase of 10%10\% is represented by multiplying by 1.101.10.
3
Set up and solve an equation using the new combined total.
0.325M+1.10M=228  ⟹  1.425M=228  ⟹  M=1600.325M + 1.10M = 228 \implies 1.425M = 228 \implies M = 160.
The sum of the new club sizes equals the new total of 228228 students.
4
Calculate the initial science club members and the initial total combined members.
Initial science members: S=0.25×160=40S = 0.25 \times 160 = 40. Total initial members: M+S=160+40=200M + S = 160 + 40 = 200.
This answers the final question asking for the combined total before the membership drive.

Key Concept

Applying percent changes to individual components to determine initial values in a combined total system.

Alternative Method

Instead of setting up equations using MM, you can choose an arbitrary starting value for the math club that is easily divisible, such as 100100. If the math club starts with 100100 students, the science club starts with 2525 students. After the increases, the math club has 110110 students and the science club has 25×1.3=32.525 \times 1.3 = 32.5 students, for a total of 142.5142.5 students. Since the actual final total is 228228, we scale our initial values by a factor of 228142.5=1.6\frac{228}{142.5} = 1.6. The initial combined total is therefore 125×1.6=200125 \times 1.6 = 200.
Estimated Time:2m 0s
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Topics

Percents and Percent Change

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