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Difficulty: MediumTranslating and Solving Algebraic Word Problems

The ratio of the number of students in a school's science club to the number of students in its art club is 3:53:5. There are 88 more students in the art club than in the science club. If xx students from the art club leave to join the science club, the ratio of the number of science club members to the number of art club members becomes 3:13:1. What is the value of xx?

  1. A
    4
  2. B
    6
  3. 12Answer
  4. D
    24
  5. E
    48

Answer

12
The correct answer is 12. Let the initial number of students in the science and art clubs be 3k3k and 5k5k, respectively. Since there are 8 more students in the art club, we set up the equation 5k=3k+85k = 3k + 8, which simplifies to 2k=82k = 8, giving k=4k = 4. This means there are initially 12 science club members and 20 art club members. When xx students transfer from the art club to the science club, the new sizes are 12+x12 + x and 20−x20 - x. Setting up the new ratio gives 12+x20−x=3\frac{12 + x}{20 - x} = 3. Solving this equation yields 12+x=60−3x12 + x = 60 - 3x, which simplifies to 4x=484x = 48, resulting in x=12x = 12.

Step-by-Step Solution

1
Represent the initial number of students in each club using the given ratio.
Let the number of science club members be 3k3k and the number of art club members be 5k5k, where kk is a positive constant.
The ratio of science club members to art club members is 3:53:5.
2
Set up and solve a linear equation to find the value of kk and the starting number of members in each club.
5k=3k+8⇒2k=8⇒k=45k = 3k + 8 \Rightarrow 2k = 8 \Rightarrow k = 4. Thus, the science club has 3(4)=123(4) = 12 members and the art club has 5(4)=205(4) = 20 members.
There are 88 more students in the art club than in the science club, so the difference between the two groups is 88.
3
Write the new member counts after xx students transfer and set up the ratio equation.
The new science club size is 12+x12 + x and the new art club size is 20−x20 - x. The new ratio equation is 12+x20−x=31\frac{12 + x}{20 - x} = \frac{3}{1}.
The xx students leave the art club (subtracting xx) and join the science club (adding xx), resulting in a new ratio of 3:13:1.
4
Solve the rational equation for xx.
12+x=3(20−x)⇒12+x=60−3x⇒4x=48⇒x=1212 + x = 3(20 - x) \Rightarrow 12 + x = 60 - 3x \Rightarrow 4x = 48 \Rightarrow x = 12.
Cross-multiplying and isolating xx yields the number of students who transferred.

Key Concept

Translating verbal statements involving ratios and changes in quantities into solvable linear algebraic equations.
Estimated Time:1m 30s
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Topics

Translating and Solving Algebraic Word Problems

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