Let's analyze the problem step-by-step.
Problem Breakdown
- Given:
- A certain number of people working h hours per day can produce 30 units in 10 days.
- Find:
- How many days will 10 people working 10 hours per day take to produce 10 units of the product?
Step 1: Define variables
- Let:
- PPP = number of people initially (unknown)
- hhh = number of hours each person works per day (unknown)
- d1=10d_1=10d1=10 days (initial days)
- U1=30U_1=30U1=30 units (initial units produced)
- P2=10P_2=10P2=10 people (new number of people)
- h2=10h_2=10h2=10 hours per day (new hours per person)
- U2=10U_2=10U2=10 units (new units to produce)
- d2=?d_2=?d2=? days (unknown, what we want to find)
Step 2: Express total work done in terms of people, hours, and days
Assuming:
- Work done (units produced) is directly proportional to:
- Number of people (PPP)
- Number of hours worked per day (hhh)
- Number of days worked (ddd)
So,
Work∝P×h×d\text{Work}\propto P\times h\times dWork∝P×h×d
Step 3: Calculate total work in first scenario
Total work done in first scenario:
W1=P×h×d1=30unitsW_1=P\times h\times d_1=30\quad \text{units}W1=P×h×d1=30units
Step 4: Calculate total work in second scenario
Total work done in second scenario:
W2=P2×h2×d2=10unitsW_2=P_2\times h_2\times d_2=10\quad \text{units}W2=P2×h2×d2=10units
Step 5: Use proportionality to relate the two scenarios
Since the efficiency and product are the same, the work done per unit time per person is constant.
W1W2=P×h×d1P2×h2×d2\frac{W_1}{W_2}=\frac{P\times h\times d_1}{P_2\times h_2\times d_2}W2W1=P2×h2×d2P×h×d1
Rearranged to solve for d2d_2d2:
d2=P×h×d1×W2P2×h2×W1d_2=\frac{P\times h\times d_1\times W_2}{P_2\times h_2\times W_1}d2=P2×h2×W1P×h×d1×W2
Step 6: Find P×hP\times hP×h from given data
We don't know PPP or hhh individually, but we can find their product from the first scenario:
P×h=W1d1=3010=3units per dayP\times h=\frac{W_1}{d_1}=\frac{30}{10}=3\quad \text{units per day}P×h=d1W1=1030=3units per day
Step 7: Calculate d2d_2d2
Substitute values:
d2=(3)×1010×10=30100=0.3 daysd_2=\frac{(3)\times 10}{10\times 10}=\frac{30}{100}=0.3\text{ days}d2=10×10(3)×10=10030=0.3 days
Step 8: Interpret the result
- d2=0.3d_2=0.3d2=0.3 days = 0.3 × 24 hours = 7.2 hours.
Final answer:
10 people working 10 hours per day can make 10 units of the product in 0.3 days (or approximately 7.2 hours). If you want the answer in days, it's 0.3 days ; if you want it in hours, it's about 7.2 hours.
