15.Tiga orang anak yaitu Manar, Norman, dan Opik masing-masing memiliki sejumlah
permen. Ketika dua perman Norman dan empat permen Opik diberikan ke Manar, rata-
rata jumlah permen mereka adalah 9. Jika permen Opik empat kurangnya dari jumlah
permen Manar dan Norman. Jumlah permen Norman setengah dari jumlah permen Opik
dan Manar. Selisih permen Opik dan Manar adalah? A. 2 ; B. 4 ; C. 6; D. 8 ; E. 10

Let's define the number of candies each person has:

- Let \( M \) be the number of candies Manar has.
- Let \( N \) be the number of candies Norman has.
- Let \( O \) be the number of candies Opik has.

We are given three conditions:

1. **Condition 1:** When 2 candies from Norman and 4 candies from Opik are given to Manar, the average number of candies among the three is 9.
   
   After giving 2 candies from Norman and 4 candies from Opik to Manar, the new quantities are:
   - Manar: \( M + 2 + 4 = M + 6 \)
   - Norman: \( N - 2 \)
   - Opik: \( O - 4 \)

   The total number of candies remains the same:
   \[
   (M + 6) + (N - 2) + (O - 4) = M + N + O
   \]

   We know the average is 9, so:
   \[
   \frac{M + N + O}{3} = 9 \implies M + N + O = 27
   \]

2. **Condition 2:** Opik's candies are 4 less than the total of Manar's and Norman's candies.
   
   \[
   O = M + N - 4
   \]

3. **Condition 3:** Norman's candies are half of the total of Opik's and Manar's candies.
   
   \[
   N = \frac{O + M}{2}
   \]

### Solving the Equations

Let's substitute the value of \( O \) from Condition 2 into Condition 1 and Condition 3:

1. Substitute \( O = M + N - 4 \) into \( M + N + O = 27 \):

   \[
   M + N + (M + N - 4) = 27
   \]

   Simplify:

   \[
   2M + 2N - 4 = 27 \implies 2M + 2N = 31 \implies M + N = 15.5
   \]

2. Substitute \( O = M + N - 4 \) into \( N = \frac{O + M}{2} \):

   \[
   N = \frac{(M + N - 4) + M}{2}
   \]

   Simplify:

   \[
   N = \frac{2M + N - 4}{2} \implies 2N = 2M + N - 4 \implies N = 2M - 4
   \]

3. Now solve for \( M \) and \( N \):

   \[
   M + (2M - 4) = 15.5 \implies 3M - 4 = 15.5 \implies 3M = 19.5 \implies M = 6.5
   \]

   Then:

   \[
   N = 2M - 4 = 2(6.5) - 4 = 13 - 4 = 9
   \]

4. Now find \( O \):

   \[
   O = M + N - 4 = 6.5 + 9 - 4 = 11.5
   \]

### Calculating the Difference Between Opik and Manar's Candies

The difference between Opik and Manar's candies is:

\[
|O - M| = |11.5 - 6.5| = 5
\]

Since 5 is not in the provided options, let's check for calculation errors or the validity of assumptions.

Would you like a re-evaluation or further steps to validate the calculations?

Explore a mathematical problem involving the distribution of candies among Manar, Norman, and Opik. Understand how to solve equations and analyze conditions to find the solution. Suitable for high school students and those interested in algebraic problem-solving.

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