10 men can make 20 toys in 12 days working 12h a day . Then in how many days can 24 persons make 32 toys working 16h a day ?

Answer

To solve the problem, we will use the concept of work done, which can be expressed using the formula:

\[ \text{Work} = \text{Men} \times \text{Days} \times \text{Hours} \]

Let's denote the variables for the two scenarios as follows:

- For the first scenario (10 men making 20 toys):
- \( M_1 = 10 \) (number of men)
- \( D_1 = 12 \) (number of days)
- \( H_1 = 12 \) (number of hours per day)
- \( W_1 = 20 \) (number of toys)

- For the second scenario (24 men making 32 toys):
- \( M_2 = 24 \) (number of men)
- \( D_2 = ? \) (number of days, which we need to find)
- \( H_2 = 16 \) (number of hours per day)
- \( W_2 = 32 \) (number of toys)

### Step 1: Calculate the total work done in the first scenario

Using the formula for work, we can calculate the total work done in the first scenario:

\[ \text{Work}_1 = M_1 \times D_1 \times H_1 = 10 \times 12 \times 12 \]

Calculating this gives:

\[ \text{Work}_1 = 10 \times 12 \times 12 = 1440 \text{ man-hours} \]

### Step 2: Calculate the work done per toy in the first scenario

Now we can find out how much work is required to make one toy:

\[ \text{Work per toy} = \frac{\text{Work}_1}{W_1} = \frac{1440}{20} = 72 \text{ man-hours per toy} \]

### Step 3: Calculate the total work needed for the second scenario

Now, we need to find out the total work required to make 32 toys in the second scenario:

\[ \text{Work}_2 = W_2 \times \text{Work per toy} = 32 \times 72 \]

Calculating this gives:

\[ \text{Work}_2 = 32 \times 72 = 2304 \text{ man-hours} \]

### Step 4: Set up the equation for the second scenario

Now we can set up the equation for the second scenario using the work formula:

\[ \text{Work}_2 = M_2 \times D_2 \times H_2 \]

Substituting the known values:

\[ 2304 = 24 \times D_2 \times 16 \]

### Step 5: Solve for \( D_2 \)

Now we can solve for \( D_2 \):

\[ 2304 = 384 \times D_2 \]

Dividing both sides by 384:

\[ D_2 = \frac{2304}{384} = 6 \]

### Conclusion

Thus, it will take 6 days for 24 persons to make 32 toys working 16 hours a day.

---