Why it happens: Ram alone does 1 unit in an hour and Shyam does 2/3 of a unit in an hour, so together they do 1 + 2/3 = 5/3 units every hour. Working out how much is done in one hour is what makes the two rates addable — you cannot add “1 hour” and “1.5 hours” directly, but you can add the shares of work done in the same hour.
Check it yourself: In 3/5 hour Ram finishes 1 × 3/5 = 3/5 of the job and Shyam finishes 2/3 × 3/5 = 2/5 of it. Together 3/5 + 2/5 = 1, the whole job. ✓
Q2.
Is the quantity of work and time taken to complete it directly or inversely proportional?
Answer
Directly proportional — for a fixed working rate, twice the work takes twice the time.
Working together, they finish 5/3 units in 1 hour.
So 5/3 : 1 :: 1 : x, and by the rule of three
x = (1 × 1) ÷ 5/3 = 3/5 hour
Why it happens: This is the place where the two ideas of the chapter are easiest to confuse, so it is worth being exact about what is being held fixed.
Here the pair of workers is fixed, so their rate is fixed at 5/3 units per hour. Then work = rate × time, and work ÷ time stays at 5/3. Work and time rise together — direct.
In Example 3 the road was fixed, and the number of workers changed. More workers, fewer days — inverse, with workers × days constant.
The same word “work” appears in both, so the label cannot be read off the vocabulary. Ask instead which quantity is being kept constant: a fixed rate gives a direct proportion, a fixed total job gives an inverse one.
Tip: A quick reality check settles it. If they had cut vegetables for 2 hours instead of 1, would they have cut more or less? More — twice as much. Quantities that grow together are directly proportional.