Time and work

If $5$ workers paint a house in $12$ days, how long would $20$ workers take? These are time and work problems, and they are surprisingly common , in homes, on construction sites, on assembly lines, even in cricket fielding strategies.

Concept

The key idea: rate.

If a person finishes a job in $t$ days, their rate is $\dfrac{1}{t}$ of the job per day.

• A worker who finishes in $6$ days has rate $\tfrac{1}{6}$ per day.

When several workers work together, their rates add:

$\text{combined rate} = r_1 + r_2 + r_3 + \dots$

The time to finish together is the reciprocal:

$t_{\text{together}} = \dfrac{1}{r_1 + r_2 + \dots}.$

Example. A can do a job in $6$ days, B in $9$ days. Together, how many days?

• Rates: $\tfrac{1}{6} + \tfrac{1}{9} = \tfrac{3+2}{18} = \tfrac{5}{18}$ per day.
• Time: $\tfrac{18}{5} = 3.6$ days.

One-task scaling (inverse proportion). If the workers all work at the same rate, then "workers $\times$ days $=$ constant" for a fixed job.

• $5$ workers $\times$ $12$ days $= 60$ worker-days.
• For $20$ workers: $20 \times t = 60$ $\Rightarrow t = 3$ days.

The unit worker-days (or "person-hours") is the total amount of work , counted in workers $\times$ time.

Pipes and tanks. Same logic. If a tap fills a tank in $t_1$ hours and another in $t_2$ hours, together their rates add:

$\text{time together} = \dfrac{1}{1/t_1 + 1/t_2}.$

If one is an inlet (fills) and one an outlet (empties), one rate is positive and the other negative. The net rate tells you whether the tank fills or empties, and how fast.

Pitfalls.

• Adding times directly (wrong!). $6$ days + $9$ days $\ne$ $15$ days, nor $7.5$. Add rates.
• Forgetting that rates are reciprocals of times.

Worked examples

Example 1. $A$ can do a piece of work in $4$ days, $B$ in $6$ days. Together?

• Rate sum: $\tfrac{1}{4} + \tfrac{1}{6} = \tfrac{3+2}{12} = \tfrac{5}{12}$.
• Time: $\tfrac{12}{5} = 2.4$ days.

Example 2. $20$ workers can build a wall in $30$ days. How long for $40$ workers?

• Total work: $20 \cdot 30 = 600$ worker-days.
• Time for $40$: $600/40 = 15$ days.

Example 3. Pipe A fills a tank in $6$ h, pipe B in $4$ h. Both open, how long?

• Rates: $\tfrac{1}{6} + \tfrac{1}{4} = \tfrac{2+3}{12} = \tfrac{5}{12}$.
• Time: $\tfrac{12}{5} = 2.4$ h.

Example 4. Pipe A fills in $10$ h, leak empties in $15$ h. Both, how long to fill?

• Rates: $\tfrac{1}{10} - \tfrac{1}{15} = \tfrac{3-2}{30} = \tfrac{1}{30}$.
• Time: $30$ h.

Try it yourself

1. $A$ does a job in $8$ days, $B$ in $12$. Together?
2. $30$ men finish in $24$ days. How long for $40$ men?
3. Pipe X fills in $5$ h, Y in $6$ h. Together?
4. A leak can empty a tank in $9$ h. With a fill pipe of $6$ h, time to fill?
5. $A$ and $B$ together do a job in $4$ days; $A$ alone in $6$ days. $B$ alone?
6. $12$ painters can paint a hall in $15$ days. Painters needed to finish in $9$ days?
7. Three pipes fill a tank in $8, 12, 24$ hours. Together?
8. $A$, $B$, $C$ can do a job in $10, 15, 30$ days respectively. Together they finish in how many days?

Activity

Group homework. Try splitting up a longish task at home , washing dishes, cleaning a room, or doing a worksheet , among $2$ or $3$ people. Measure how long each person would take alone, then estimate (using the rate formula) the group time. Then actually do the task together. The estimate is usually surprisingly close to reality.