The ratio of apples to pears is 23 to 2*x*, where *x* is a positive constant.  If the number of apples is reduced by 12, by how many must the number of pears by reduced to maintain the ratio?



Let's start by analyzing the problem step by step.

1. **Define the Variables:**
   - Let the number of apples be \( a \).
   - Let the number of pears be \( p \).
   - The ratio of apples to pears is given as \( \frac{a}{p} = \frac{23}{2x} \).

   Therefore, we have:
   \[
   a = \frac{23}{2x} \times p
   \]

2. **Account for the Reduction in Apples:**
   The problem states that the number of apples is reduced by 12. Let the new number of apples be \( a' = a - 12 \).

   To maintain the same ratio after this reduction:
   \[
   \frac{a - 12}{p'} = \frac{23}{2x}
   \]
   where \( p' \) is the new number of pears after the reduction.

3. **Express the New Ratio:**
   From the original ratio, we know:
   \[
   a = \frac{23}{2x} \times p
   \]
   Substituting \( a \) in the equation:
   \[
   \frac{\frac{23}{2x} \times p - 12}{p'} = \frac{23}{2x}
   \]

4. **Solve for \( p' \):**
   To find \( p' \), we multiply both sides by \( p' \) and by \( \frac{2x}{23} \):
   \[
   \frac{23}{2x} \times p - 12 = \frac{23}{2x} \times p'
   \]

   Simplifying further:
   \[
   \frac{23}{2x} \times p - \frac{23}{2x} \times p' = 12
   \]

   Factoring out \( \frac{23}{2x} \):
   \[
   \frac{23}{2x} \times (p - p') = 12
   \]

   Now, solve for \( p - p' \):
   \[
   p - p' = \frac{12 \times 2x}{23}
   \]

   The quantity \( p - p' \) represents the reduction in the number of pears required to maintain the ratio.

Thus, the number of pears must be reduced by **\(\frac{24x}{23}\)** to maintain the ratio.

Would you like to explore this further or have any other questions?

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Here are 8 related questions that might interest you:
1. If \( x = 3 \), what is the exact reduction in the number of pears?
2. What happens to the ratio if the number of apples is increased instead of decreased?
3. How would the ratio change if the number of pears increased by a certain amount?
4. What is the effect on the ratio if \( x \) is doubled?
5. Can you determine the original number of pears given the number of apples and \( x \)?
6. How does the ratio change if both apples and pears are reduced by the same number?
7. What if the ratio were reversed, with pears to apples as \( 2x \) to 23?
8. How many pears would be needed to achieve the same ratio if the number of apples were doubled?

**Tip:** When working with ratios, try expressing everything in terms of a single variable to simplify comparisons.

Explore how to maintain the ratio between apples and pears when the number of apples is reduced by 12. Learn the step-by-step solution involving ratio and proportion concepts. Suitable for Grades 9-12.

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