Understanding the Machine Production Problem

This problem involves calculating the time required to complete a production target given initial conditions, followed by changes in the number of machines, working hours, and efficiency.

Initial Production Rate Calculation

First, let's determine the rate at which the machines produce units.

• Number of machines (M1): 12
• Hours per day (H1): 8
• Number of days (D1): 10
• Total units produced (U1): 960

The total work done can be represented as the product of machines, hours per day, and days. We can assume the production rate (units per machine-hour) is constant initially.

Total machine-hours = M1 $\times$ H1 $\times$ D1 = 12 $\times$ 8 $\times$ 10 = 960 machine-hours.

The production rate is calculated as:

Rate = Total Units / Total Machine-hours = 960 units / 960 machine-hours = 1 unit per machine-hour.

Production in the First 4 Days

The machines work under the initial conditions for the first 4 days.

• Number of machines: 12
• Hours per day: 8
• Number of days: 4

Machine-hours in the first 4 days = 12 machines $\times$ 8 hours/day $\times$ 4 days = 384 machine-hours.

Units produced in the first 4 days = Rate $\times$ Machine-hours = 1 unit/machine-hour $\times$ 384 machine-hours = 384 units.

Remaining Production Calculation

Now, we need to find out how many units are left to be produced.

Total units required = 960 units.

Units produced in the first 4 days = 384 units.

Remaining units = 960 units - 384 units = 576 units.

Calculating Production Under New Conditions

After 4 days, the conditions change:

• Number of machines break down: 3
• Remaining machines (M2): 12 - 3 = 9
• New hours per day (H2): 10
• Efficiency reduced by 15%, so new efficiency (E2) = 100% - 15% = 85% = 0.85.

The effective work done per day by the remaining machines needs to be calculated.

Effective machine-hours per day = M2 $\times$ H2 $\times$ E2

Effective machine-hours per day = 9 machines $\times$ 10 hours/day $\times$ 0.85

Effective machine-hours per day = 90 $\times$ 0.85 = 76.5 effective machine-hours per day.

Calculating Additional Days Needed

We need to produce the remaining 576 units with the new conditions.

The production rate remains 1 unit per (effective) machine-hour.

Total effective machine-hours required = Remaining units / Rate

Total effective machine-hours required = 576 units / (1 unit/machine-hour) = 576 effective machine-hours.

Additional days needed (D2) = Total effective machine-hours required / Effective machine-hours per day

D2 = 576 / 76.5

To simplify the division, we can write 76.5 as $\frac{153}{2}$:

D2 = $\frac{576}{\frac{153}{2}} = \frac{576 \times 2}{153} = \frac{1152}{153}$

Now, simplify the fraction. Both numbers are divisible by 3:

$\frac{1152 \div 3}{153 \div 3} = \frac{384}{51}$

Both numbers are again divisible by 3:

$\frac{384 \div 3}{51 \div 3} = \frac{128}{17}$

Convert the improper fraction to a mixed number:

128 $\div$ 17 = 7 with a remainder of 9 (since $17 \times 7 = 119$, and $128 - 119 = 9$).

So, D2 = $7\frac{9}{17}$ days.

Conclusion

The number of additional days needed to complete the production is $7\frac{9}{17}$ days.