P and Q can together complete a piece of work in 8 days.
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P and Q can together complete a piece of work in 8 days. If P worked at twice his original efficiency and Q worked at one-third of his original efficiency, the work will be completed in 9 days. In how many days can P alone complete the whole work?
Show answer & explanation
Let P's work rate be 1/p and Q's work rate be 1/q (fraction of work per day). From the first condition: 1/p + 1/q = 1/8. From the second condition with modified rates: 2/p + 1/(3q) = 1/9. Solving these simultaneous equations yields p = 24 days for P alone.
Step-by-step Derivation:
Step 1: Set up equations based on work rates.
Let P complete work alone in p days and Q in q days.
Work rate: P = 1/p, Q = 1/q
Equation 1 (original efficiency):
1/p + 1/q = 1/8
Equation 2 (modified efficiency):
2/p + 1/(3q) = 1/9
Step 2: Solve the system of equations.
From Equation 1: 1/q = 1/8 - 1/p
Substitute into Equation 2:
2/p + 1/3 × (1/8 - 1/p) = 1/9
2/p + 1/24 - 1/(3p) = 1/9
Step 3: Combine like terms.
2/p - 1/(3p) = 1/9 - 1/24
(6 - 1)/(3p) = (8 - 3)/72
5/(3p) = 5/72
Step 4: Solve for p.
3p = 72
p = 24 days
Step 5: Verify.
1/p + 1/q = 1/24 + 1/q = 1/8
→ 1/q = 1/8 - 1/24 = 3/24 - 1/24 = 2/24 = 1/12
→ q = 12 days
Check Equation 2: 2/24 + 1/(3×12) = 1/12 + 1/36 = 3/36 + 1/36 = 4/36 = 1/9 ✓