Time and Work: Efficiency, the LCM Method and CSAT Shortcuts

Time and Work is the CSAT topic that tests how quickly people or machines finish a job when they work alone or together. At its heart it is one simple idea: total work equals efficiency multiplied by time. CSAT Paper II is only qualifying, you need 33 percent, but 1 to 3 time-and-work questions appear almost every year, so a five-minute method you can trust is worth building.

The one-day-work idea that starts everything

Every time-and-work question reduces to one question: how much of the job is done in a single day (or hour or minute). If A finishes a job in 18 days, then in one day A does 1/18 of the job. If B finishes the same job in 9 days, B does 1/9 in a day. Working together they complete 1/18 + 1/9 = 3/18 = 1/6 of the job per day, so the whole job takes 6 days. This unit work (the fraction done in one unit of time) is the only definition you need, and every formula below is built from it.

Efficiency is the work done per unit of time, so efficiency and time are inversely proportional for a fixed job: someone twice as efficient finishes in half the time. A common CSAT trap hides here. If B is twice as efficient as A, B takes half the time. But if B is two times more efficient than A, careful reading is needed: most CSAT-style setters intend it to mean 3 times as efficient (twice extra over the base), so always convert the phrase into a clean ratio before calculating.

The LCM shortcut for combined work

Adding fractions like 1/16 + 1/24 is slow under exam pressure. The LCM method replaces fractions with whole units of work. Take the LCM of the individual days, call it the total work, give each worker efficiency units, add them, and divide.

Worked example 1. Ajay takes 16 days and Vijay takes 24 days for a job. How long do they take together? Total work = LCM of 16 and 24 = 48 units. Ajay's efficiency = 48/16 = 3 units/day, Vijay's = 48/24 = 2 units/day. Combined = 5 units/day. Time = 48/5 = 9.6 days. No fractions until the last step, and only one division.

Worked example 2. P, Q and R finish a job in 8, 16 and 24 days respectively. Total work = LCM of 8, 16, 24 = 48 units. Efficiencies: 6, 3 and 2 units/day. Combined = 11 units/day. Time together = 48/11 = 4.36 days.

Worked example 3. Three workers A, B and C finish a job in 12, 15 and 20 days. LCM = 60. Efficiencies 5, 4, 3. Combined 12 units/day. Time = 60/12 = 5 days.

Finding the unknown worker: the subtraction trick

When a team's time and one member's time are known, the unknown member's share is a subtraction. Combined rate minus known rate gives the missing rate, then invert it.

Worked example 4. A man finishes a job in 12 days, but with his friend's help he finishes it in 4 days. In how many days can the friend do it alone? Let the friend take x days. 1/12 + 1/x = 1/4, so 1/x = 1/4 - 1/12 = 2/12 = 1/6. The friend takes 6 days alone. With the LCM lens: total work 12 units, man does 1 unit/day, together they do 3 units/day, so the friend does 2 units/day, hence 12/2 = 6 days.

Worked example 5. C and D together complete a job in 6 days. C alone takes 15 days. How long does D take alone? 1/D = 1/6 - 1/15 = (5 - 2)/30 = 3/30 = 1/10. D alone takes 10 days. The trap to avoid: subtracting the days themselves (15 - 6 = 9) is wrong, because days are not additive, rates are.

Alternate-day work and efficiency ratios

Alternate-day work means two workers take turns, and the job is finished by counting day-pairs. Worked example 6. A and B finish a job in 10 and 20 days respectively, working on alternate days. Total work = 20 units; A does 2 units/day, B does 1 unit/day. A 2-day cycle completes 3 units. After 6 cycles (12 days), 18 units are done. If A starts day 13, he finishes the remaining 2 units that day: 13 days total. If B starts, day 13 adds 1 unit (19 done) and day 14 A needs only half a day for the last unit: 13.5 days. Whoever is more efficient starting first saves time, a favourite CSAT twist.

Efficiency ratios appear directly in CSAT stems. Worked example 7. X can do a job in 36 days. Y is four times as efficient as X. Time for Y = 36/4 = 9 days. Worked example 8. John finishes in 40 days; Peter is 25 percent more efficient than John. Peter's efficiency = 1.25 times, so his time = 40/1.25 = 32 days. When efficiency is given as a percent, convert it to a multiplier (25 percent more = 1.25x) and divide.

The pair method: A+B, B+C, C+A questions

CSAT often gives the times of pairs and asks for the trio or an individual. Add all three pair-rates; that sum equals twice the combined rate of all three, so halve it.

Worked example 9. A and B together take 15 days, B and C 20 days, C and A 30 days. Combined pair-rate = 1/15 + 1/20 + 1/30 = 4/60 + 3/60 + 2/60 = 9/60 = 3/20 of the job per day. This is twice A+B+C's rate, so the trio's rate = 3/40 per day and the trio takes 40/3 = 13.33 days. To find A alone: A = (A+B+C) - (B+C) = 3/40 - 1/20 = 1/40, so A alone takes 40 days.

Traps CSAT actually sets

First, the garrison or food problem is inverse proportion, not direct: food for 1000 soldiers for 30 days. After 10 days, 20 days of food for 1000 remains; 2000 soldiers now share it, so it lasts 10 days, not 20 (CSAT 2013). Second, comparing men and women: 24 men and 12 women finish in 30 days; how long for 12 men and 24 women? You cannot answer, because nothing says a man's efficiency equals a woman's. Data is inadequate (CSAT 2022). Third, partial-completion stems: Ram and Shyam complete 60 percent in 4 days together, then Shyam finishes the remaining 40 percent in 8 more days. Shyam's rate = 0.4/8 = 0.05/day (20 days alone). Combined rate = 0.6/4 = 0.15/day. Ram's rate = 0.15 - 0.05 = 0.10/day, so Ram alone takes 10 days (CSAT 2016). Fourth, percent-of-work conversions: if X completes 20 percent in 8 days, X's full time is 8 x 100/20 = 40 days, not 8 days.

Fifth, the rotating schedule: A (8 days), B (16 days), C (12 days) work Monday, Tuesday, Wednesday in rotation. A 3-day cycle completes 1/8 + 1/16 + 1/12 = 13/48 of the job. After 9 days (3 cycles), 39/48 is done; Wednesday's turn (C) adds 1/12 to reach 43/48; Thursday (A) finishes the rest. So the work finishes on Thursday but takes 11 days, not 10 (CSAT 2023). Always simulate the leftover after whole cycles.

Speed tips for the exam hall

Use the LCM method for every combined-work question; it keeps numbers whole and kills fraction errors. Memorise the two conversions: one worker's time = total work/efficiency and efficiency = total work/time. When a stem gives a percent of work and the time for it, scale to 100 percent first (20 percent in 8 days becomes 40 days alone). For pair problems, always write the three pair-rates and halve their sum before inverting. And read efficiency language literally: 'twice as efficient' halves the time, but 'two times more efficient' in these papers is usually used to mean three times as efficient, so pause and confirm the ratio from context.

Key Terms

• Efficiency is the amount of work done per unit of time. It is inversely proportional to time: doubling efficiency halves the time for the same job.
• Unit work (one day's work) is the fraction of the total job completed in one unit of time. If A takes 18 days, A's unit work is 1/18 of the job.
• Combined rate is the sum of individual unit works when people work together. If A does 1/10 and B does 1/30 per day, the combined rate is 4/30 per day.
• LCM method is a shortcut that takes the LCM of individual days as total work in units, converts each worker's time into whole-number efficiency units, and avoids fractions until the final division.
• Alternate-day work is a schedule where two or more workers take turns on successive days. Solve it by computing one full rotation's work and then handling the leftover.
• Inverse proportion describes two quantities where one grows as the other shrinks, such as soldiers and days of food: doubling the eaters halves the days the same food lasts.
• Pair method is the technique for A+B, B+C, C+A questions: adding the three pair-rates gives twice the rate of all three working together.
• Percent of work is a partial completion stated as a share of the whole job, for example 20 percent done in 8 days. Scale it to 100 percent to find the full solo time.