P and Q can do a piece of work in 20 days and 30 days, respectively. They work together for 6 days, and then Q is replaced by R. If the rest of the work is completed by P and R in 8 more days, in how many days can R alone complete 60% of the work?

Correct — D. Track the fraction of the work done in each stage.

P and Q per day: 1⁄20 + 1⁄30 = 5⁄60 = 1⁄12
Six days together: 6 × 1⁄12 = 1⁄2, leaving 1⁄2
P's work in the last 8 days: 8 × 1⁄20 = 2⁄5
R's work in those 8 days: 1⁄2 − 2⁄5 = 1⁄10

R's rate: 1⁄10 ÷ 8 = 1⁄80, so R alone needs 80 days for the whole job
60% of the job: 0.6 × 80 = 48 days → option (d)

Why the others are wrong

• (a)
  28 days — 28 days is far too fast for R. R did only 1⁄10 of the work in 8 days, so 6⁄10 takes six times as long: 6 × 8 = 48 days.
• (b)
  34 days — 34 days at R's rate of 1⁄80 a day covers only 34⁄80 = 42.5% of the work, short of the 60% asked.
• (c)
  64 days — 64 days is 80% of R's 80 days, not 60%. At 1⁄80 of the work a day, 60% takes 0.6 × 80 = 48 days.

Time and work runs on rates: a worker who finishes in n days does 1⁄n of the job each day, and rates add when people work together.

When the team changes midway, split the job into stages. Each stage's work is rate × days, and the stages add up to the whole job.

Here R appears only in the second stage, so subtract everything else first. What is left is R's work alone.

Key facts

• A worker who completes a job in n days does 1⁄n of it per day.
• Work done = rate × time, and the rates of people working together add.
• If a job takes D days, p% of it takes (p ÷ 100) × D days at the same rate.

Common traps

• Crediting all of the remaining 1⁄2 to R and forgetting that P keeps working, which makes R far too fast.
• Answering for the whole job (80 days) or for the wrong share (64 days is 80%).

No directly related past PYQ was found.