Numerical aptitude · Time and motion
Time and work, made easy
💡 The idea
Turn days into work per day. If Ali paints a wall in 4 days, he paints ¼ of the wall each day.
🧭 The method: 3 steps
- Write each person’s work per day: 1 ÷ days.
- Working together? Add them. Something undoing the work? Take it away.
- Time for the whole job = 1 ÷ (work per day).
🎬 Every kind of question, in pictures
Pick a kind. Guess the answer first, then watch the picture work it out. Tap any step to see it in the picture.
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✏️ Worked examples
Example 1. Ali paints a wall in 4 days, Ben in 6 days. Together?
- Ali: ¼ a day. Ben: ⅙ a day.
- Together: ¼ + ⅙. Same bottom (12): 3/12 + 2/12 = 5/12 a day
- Whole wall (12/12) ÷ 5/12 = 12 ÷ 5 = 2.4 days
⚡ Shortcut: (4 × 6) ÷ (4 + 6) = 24 ÷ 10 = 2.4 days.
Example 2. Together they finish in 6 days. Ali alone takes 10. Ben alone?
- Together per day: 1/6. Ali: 1/10.
- Ben = 1/6 − 1/10 = 5/30 − 3/30 = 2/30 = 1/15
- Ben alone takes 15 days
⚡ Shortcut: (10 × 6) ÷ (10 − 6) = 60 ÷ 4 = 15.
Example 3. 6 workers build a wall in 10 days. How long for 4 workers?
- Total work = 6 × 10 = 60 worker-days
- 4 workers: 60 ÷ 4 = 15 days
⚠️ Where people go wrong
- Never add the days (4 + 6 = 10 days is wrong): together they must be faster than either alone.
- More workers → fewer days (multiply and divide the opposite way).
- “p% more efficient” means divide the time by (1 + p/100), not take p% off.
📌 Remember
- Work per day = 1 ÷ days
- Two together = (a × b) ÷ (a + b)
- Workers × days = total work