4. Efficiency: more daily work means less time
Today’s goal: connect efficiency with daily work, then completion time. Compare the same job under the same conditions.
A does 6 units/day; B does 3. How efficient is A compared with B?
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Correct answer: A
Efficiency here compares daily work. 6 ÷ 3 = 2.
Option B: A does more work per day, not less.
Option C: 6 and 3 are different rates.
Option D: 3 is the difference, not the ratio.
Hint: Compare daily rates by division.
See “twice as efficient”
Choose a 12-unit job. B does 1 unit/day; A does 2 units/day.
| Worker | Units/day | Days for 12 units |
|---|---|---|
| B | 1 | 12 |
| A | 2 | 6 |
Twice the daily work means half the time for the same total work. It does not mean two days fewer.
Use rates to find time
- B’s rate = 1 unit/day; A’s = 2 units/day.
- B’s 12 days mean total work = 1 × 12 = 12 units.
- A needs 12 ÷ 2 = 6 days.
Complete one
A is twice as efficient as B. B takes 18 days for the same job. A takes...?
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Correct answer: D
Twice the daily work means half the time: 18 ÷ 2 = 9.
Option A: More daily work means less time for the same job.
Option B: Equal times would require equal rates.
Option C: 6 days would require three times B’s rate.
Hint: The same work is finished with twice the daily progress.
Try a different ratio
Efficiency ratio A : B is 3 : 2. B takes 15 days. How long does A take for the same job?
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Correct answer: B
Let rates be 3 and 2. Total work = 15 × 2 = 30; A takes 30 ÷ 3 = 10 days.
Option A: The faster worker should take less time.
Option C: Their efficiencies are unequal.
Option D: 5 is the difference between their times, not A’s time.
Hint: Use daily rates of 3 and 2 units.
Connect it with working together
A is twice as efficient as B. B takes 12 days alone. How long together?
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Correct answer: C
B does 1/day, A does 2/day. Total = 12 units. Together 3/day; time = 12 ÷ 3 = 4 days.
Option A: 6 is A’s time alone; B also helps.
Option B: Do not add solo times.
Option D: Use combined daily work, not a time difference.
Hint: Give B one unit per day and A two.
Remember: efficiency compares daily work; time for the same job goes the other way.
If you can explain that, stop here. Try lesson 5 in a later sitting, ideally tomorrow, before rereading. Doing it immediately checks immediate understanding, not next-day recall.
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