There are two boxes, red and white in colour. The ratio of the number of chocolates in the white box to the number of biscuits in the red box is ` 3 : 2` and the ratio of the number of biscuits in the white box to the number of chocolates in the red box is ` 3 : 4 `. If the ratio of the total number of chocolates and biscuits in the white box to the total number of chocolates and biscuits in the red box is ` 15 : 16`. Find the ratio of the total number of chocolates to the total number of biscuits in the two boxes.

Answer

To solve the problem, we need to find the ratio of the total number of chocolates to the total number of biscuits in the two boxes (red and white). Let's denote the variables as follows:

- Let \( C_W \) be the number of chocolates in the white box.
- Let \( B_W \) be the number of biscuits in the white box.
- Let \( C_R \) be the number of chocolates in the red box.
- Let \( B_R \) be the number of biscuits in the red box.

### Step 1: Set up the ratios based on the problem statement

From the problem, we have the following ratios:

1. The ratio of the number of chocolates in the white box to the number of biscuits in the red box is \( \frac{C_W}{B_R} = \frac{3}{2} \).
- We can express this as:
\[ C_W = 3x \quad \text{and} \quad B_R = 2x \]

2. The ratio of the number of biscuits in the white box to the number of chocolates in the red box is \( \frac{B_W}{C_R} = \frac{3}{4} \).
- We can express this as:
\[ B_W = 3y \quad \text{and} \quad C_R = 4y \]

### Step 2: Set up the equation for the total chocolates and biscuits

The total number of chocolates and biscuits in the white box is:
\[ C_W + B_W = 3x + 3y \]
The total number of chocolates and biscuits in the red box is:
\[ C_R + B_R = 4y + 2x \]
According to the problem, the ratio of the total number of chocolates and biscuits in the white box to the total number in the red box is \( \frac{C_W + B_W}{C_R + B_R} = \frac{15}{16} \).

### Step 3: Set up the equation

We can set up the equation:
\[ \frac{3x + 3y}{4y + 2x} = \frac{15}{16} \]

### Step 4: Cross-multiply to eliminate the fraction

Cross-multiplying gives us:
\[ 16(3x + 3y) = 15(4y + 2x) \]

### Step 5: Expand both sides

Expanding both sides:
\[ 48x + 48y = 60y + 30x \]

### Step 6: Rearranging the equation

Rearranging gives:
\[ 48x - 30x = 60y - 48y \]
\[ 18x = 12y \]

### Step 7: Simplifying the equation

Dividing both sides by 6:
\[ 3x = 2y \]
This implies:
\[ \frac{x}{y} = \frac{2}{3} \]

### Step 8: Substitute back to find the ratios

Now we can substitute \( y = \frac{3}{2}x \) back into our expressions for \( C_W, B_W, C_R, \) and \( B_R \):
- \( C_W = 3x \)
- \( B_W = 3y = 3 \cdot \frac{3}{2}x = \frac{9}{2}x \)
- \( C_R = 4y = 4 \cdot \frac{3}{2}x = 6x \)
- \( B_R = 2x \)

### Step 9: Calculate the total chocolates and biscuits

Now we can find the total chocolates and biscuits:
- Total chocolates \( = C_W + C_R = 3x + 6x = 9x \)
- Total biscuits \( = B_W + B_R = \frac{9}{2}x + 2x = \frac{9}{2}x + \frac{4}{2}x = \frac{13}{2}x \)

### Step 10: Find the final ratio

Now, we can find the ratio of total chocolates to total biscuits:
\[ \text{Ratio} = \frac{9x}{\frac{13}{2}x} = \frac{9 \cdot 2}{13} = \frac{18}{13} \]

### Final Answer

The ratio of the total number of chocolates to the total number of biscuits in the two boxes is \( \frac{18}{13} \).