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Zorluk: ZorRatios, Rates, and Proportions

In a school's robotics club, the ratio of experienced members to novice members was 3:53:5. After 44 experienced members graduated and 1212 new novice members joined the club, the ratio of experienced members to novice members became 1:31:3. What was the total number of members in the robotics club before the graduations and new members joined?

  1. A
    18
  2. B
    30
  3. C
    32
  4. 48Cevap
  5. E
    56

Cevap

48
The correct answer is 4848. Let the initial number of experienced members be 3x3x and novice members be 5x5x. After 44 experienced members graduate and 1212 novices join, the counts become 3x−43x - 4 and 5x+125x + 12 respectively. Setting up the ratio 3x−45x+12=13\frac{3x-4}{5x+12} = \frac{1}{3} and cross-multiplying yields 9x−12=5x+129x - 12 = 5x + 12. Solving this equation gives 4x=244x = 24, which means x=6x = 6. The initial total number of members was 3x+5x=8x=8(6)=483x + 5x = 8x = 8(6) = 48.

Adım Adım Çözüm

1
Define variables for the initial number of experienced and novice members using the given ratio.
Let the number of experienced members be 3x3x and the number of novice members be 5x5x, where xx is a constant. The initial total number of members is 3x+5x=8x3x + 5x = 8x.
Using a multiplier xx allows us to represent the parts of the ratio as actual quantities.
2
Set up an equation representing the ratio after the changes.
The new number of experienced members is 3x−43x - 4, and the new number of novice members is 5x+125x + 12. The new ratio is 3x−45x+12=13\frac{3x - 4}{5x + 12} = \frac{1}{3}.
Graduating members decreases the count by 44, and new members joining increases the count by 1212.
3
Solve the equation for xx.
Cross-multiplying gives 3(3x−4)=1(5x+12)3(3x - 4) = 1(5x + 12), which simplifies to 9x−12=5x+129x - 12 = 5x + 12. Subtracting 5x5x and adding 1212 to both sides gives 4x=244x = 24, so x=6x = 6.
Solving for the multiplier xx allows us to find the actual counts of members.
4
Calculate the initial total number of members.
The initial total number of members is 8x=8(6)=488x = 8(6) = 48.
The question asks for the total number of members before the changes occurred.

Anahtar Kavram

Setting up and solving equations involving ratios and proportions from word problems.
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Ratios, Rates, and Proportions

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