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The composite index of the production of a special blend coffee in the year 2024 based on the year 2021 is 115. Find values of a and b.

We are given the following details in the table, and our goal is to find the values of \( a \) (percentage for milk) and \( b \) (percentage for sugar). Additionally, the composite index of the production is 115, which is calculated as a weighted average of the individual price indices, using the respective percentages as weights.

### Formula for the Composite Index:
\[
\text{Composite Index} = \frac{(\text{Index for Coffee Beans} \times \text{Percentage for Coffee Beans}) + (\text{Index for Milk} \times a) + (\text{Index for Sugar} \times b) + (\text{Index for Flavour Additives} \times \text{Percentage for Flavour Additives})}{100}
\]
We know:
- Composite Index = 115
- Price Index for Coffee Beans = 125 and Percentage = 25%
- Price Index for Milk = 120 and Percentage = \( a \)
- Price Index for Sugar = 105 and Percentage = \( b \)
- Price Index for Flavour Additives = 110 and Percentage = 40%

Also, since the sum of percentages must equal 100%, we have:
\[
25 + a + b + 40 = 100
\]
This simplifies to:
\[
a + b = 35
\]

Now let's substitute these values into the composite index formula:
\[
115 = \frac{(125 \times 25) + (120 \times a) + (105 \times b) + (110 \times 40)}{100}
\]

### Solving step by step:

1. Calculate constants:
\[
125 \times 25 = 3125
\]
\[
110 \times 40 = 4400
\]

Substitute these into the equation:
\[
115 = \frac{3125 + 120a + 105b + 4400}{100}
\]
\[
115 = \frac{7525 + 120a + 105b}{100}
\]
Multiply both sides by 100:
\[
11500 = 7525 + 120a + 105b
\]
Simplify:
\[
120a + 105b = 3975
\]

### System of Equations:
We now have two equations:
1. \( 120a + 105b = 3975 \)
2. \( a + b = 35 \)

#### Solving the system:
From equation (2), solve for \( b \):
\[
b = 35 - a
\]

Substitute into equation (1):
\[
120a + 105(35 - a) = 3975
\]
Expand:
\[
120a + 3675 - 105a = 3975
\]
Simplify:
\[
15a = 300
\]
Solve for \( a \):
\[
a = 20
\]

Now substitute \( a = 20 \) into \( b = 35 - a \):
\[
b = 35 - 20 = 15
\]

### Final Answer:
- \( a = 20 \)
- \( b = 15 \)

Would you like more details or have any other questions?

### Related Questions:
1. How would the composite index change if the price of sugar increases by 10%?
2. What happens to the composite index if the percentage of coffee beans is reduced to 20%?
3. Can you derive a general formula for the composite index when more ingredients are involved?
4. How does the composite index help in predicting production costs?
5. If the price of milk fluctuates significantly, how would that impact the composite index?

### Tip:
Always check if the sum of percentages equals 100% when calculating weighted averages for indices.

Discover how to find the values of a and b when calculating the composite index for coffee production based on weighted averages and price indices. This problem involves solving a system of equations using algebra and is suitable for high school students (Grades 10-12).

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