450 toys are divided into three types A, B & C. All three types of toys are sold at the profit of` 9%, 10% and 12% `respectively. Overall profit from type A and B is `9 3/7` % . Overall profit from all three types is 10%. Find the number of toys kept in all three categories A, B & C.

Answer

To solve the problem step by step, we will use the concept of alligation to find the number of toys in each category A, B, and C.

### Step 1: Understand the Given Information
- Total number of toys = 450
- Profit percentages:
- Type A = 9%
- Type B = 10%
- Type C = 12%
- Overall profit from types A and B = \(9 \frac{3}{7}\)% = \(9.42857\)% (converting mixed fraction to decimal)
- Overall profit from all three types = 10%

### Step 2: Calculate the Profit from Types A and B
Let the number of toys in type A be \(x\) and in type B be \(y\). The profit from type A and B can be calculated as follows:

\[ \text{Total Profit from A and B} = \frac{x \times 9 + y \times 10}{x + y} \]

According to the problem, this total profit is equal to \(9 \frac{3}{7}\)%:

\[ \frac{x \times 9 + y \times 10}{x + y} = 9.42857 \]

### Step 3: Set Up the Equation
Cross-multiplying gives:

\[ x \times 9 + y \times 10 = 9.42857(x + y) \]

This simplifies to:

\[ 9x + 10y = 9.42857x + 9.42857y \]

Rearranging gives:

\[ (9 - 9.42857)x + (10 - 9.42857)y = 0 \]

This simplifies to:

\[ -0.42857x + 0.57143y = 0 \]

### Step 4: Solve for the Ratio of A and B
Dividing through by 0.42857 gives:

\[ y = \frac{0.42857}{0.57143}x \]

Calculating this ratio gives:

\[ y = \frac{3}{4}x \]

Thus, the ratio of toys in A and B is \(x:y = 4:3\).

### Step 5: Express Total Toys in Terms of A and B
Let \(x = 4k\) and \(y = 3k\). The total number of toys in A and B is:

\[ x + y = 4k + 3k = 7k \]

Since the total number of toys is 450, we have:

\[ 7k + z = 450 \]

Where \(z\) is the number of toys in type C.

### Step 6: Calculate the Number of Toys in Type C
Now we need to find \(z\):

\[ z = 450 - 7k \]

### Step 7: Calculate Overall Profit Including Type C
The overall profit from all three types is given as 10%. Therefore, we can set up the equation:

\[ \frac{4k \times 9 + 3k \times 10 + z \times 12}{450} = 10 \]

Substituting \(z = 450 - 7k\):

\[ \frac{4k \times 9 + 3k \times 10 + (450 - 7k) \times 12}{450} = 10 \]

### Step 8: Solve for k
This simplifies to:

\[ 4k \times 9 + 3k \times 10 + 450 \times 12 - 7k \times 12 = 4500 \]

Calculating gives:

\[ 36k + 30k + 5400 - 84k = 4500 \]

Combining like terms:

\[ -18k + 5400 = 4500 \]

Solving for \(k\):

\[ -18k = 4500 - 5400 \]
\[ -18k = -900 \]
\[ k = 50 \]

### Step 9: Calculate the Number of Toys in Each Category
Now substituting \(k\) back to find \(x\), \(y\), and \(z\):

- \(x = 4k = 4 \times 50 = 200\)
- \(y = 3k = 3 \times 50 = 150\)
- \(z = 450 - 7k = 450 - 350 = 100\)

### Final Answer
- Number of toys in Type A = 200
- Number of toys in Type B = 150
- Number of toys in Type C = 100