Time and Work Problems: Formulas, Shortcuts, Examples

If one person can finish a job in n days, the fraction of the job done in one day is 1/n. This single idea — one day's work equals one divided by the total days — turns every time-and-work problem into a simple fraction addition. You never add the days; you add the daily fractions.

Time and work is the mathematics of planning: how many workers a project needs, how long a job will take with extra help, or when a machine plus a backup crew will finish. Contractors, factory managers and project planners use exactly this logic every day.

Key Concepts

• One day's work: if a person finishes a job in n days, their daily share is 1/n of the job.
• Combined work: when people work together, their daily shares add up: 1/a + 1/b.
• Efficiency: work done per unit time; a person who takes fewer days is more efficient. Efficiency and time are inversely related.
• Work left: 1 minus the fraction already done.
• Man-days: total effort = number of workers × number of days (assumes all workers are equally efficient).

Formulas

• Daily work of one person = 1 ÷ (days taken alone).
• Two people together: days needed = 1 ÷ (1/a + 1/b) = (a × b) ÷ (a + b), where a and b are their solo days. Assumes both work the whole time at constant speed.
• Work done in t days = t × (daily share).
• Wages share: when paid for a joint job, divide the wage in the ratio of daily work shares (1/a : 1/b), not in the ratio of days.

Worked Example

A can build a wall in 12 days and B can build the same wall in 15 days. How long will they take working together?

1. A's one-day work = 1/12; B's one-day work = 1/15.
2. Together in one day: 1/12 + 1/15 = 5/60 + 4/60 = 9/60 = 3/20 of the wall.
3. Days to finish the whole wall = 1 ÷ (3/20) = 20/3 = 6⅔ days, i.e. 6 days and 16 hours.

They take 6⅔ days. Check with the shortcut: (12 × 15) ÷ (12 + 15) = 180 ÷ 27 = 6⅔. Both methods agree.

Common Misconceptions

• "Add the days: 12 + 15 = 27 days." Working together is faster, not slower. You add the daily fractions, never the days.
• "Wages are split by days taken." Wrong direction. Wages follow the ratio of daily work (efficiency): if A does twice B's daily share, A earns twice the pay, even though A takes half the days.
• "If C joins halfway, just average everything." No — compute the work done before C joins, subtract it, then divide the remaining work by the new combined daily share.

Real-World Applications

• Construction managers decide crew sizes and deadlines from daily output rates.
• Factories schedule machines with different speeds on the same production line.
• Software teams estimate sprint completion from each member's velocity.
• Households split chores and costs fairly using efficiency ratios.

Time and work is fraction addition in disguise: convert each person's time into a daily share, add the shares, and flip the total to get the days. Never add days, and split any payment by daily work, not by time taken.