Sharing wages for work

When sev­eral peo­ple are paid together for one job, how should the money be split? Not equally, and not by the num­ber of days each per­son turned up, but by how much of the work each per­son actu­ally did. This les­son turns that fair-shar­ing rule into a method, using the 1 day's work idea from the pre­vi­ous lessons, and then works out daily wages.

The rule#

Money is shared in the ratio of the work done.

If every­one works for the same num­ber of days, the work each does is in the ratio of their 1 day's work. If they work for dif­fer­ent num­bers of days, first find the total work each per­son did, then share in that ratio.

Recall how to share an amount in a ratio. To share ₹1200 in the ratio $5 : 3$, add the parts ($5 + 3 = 8$) and give each per­son their parts out of $8$: $\displaystyle 1200 \times \frac{5}{8} = 750$ and $\displaystyle 1200 \times \frac{3}{8} = 450$, so the shares are ₹750 and ₹450.

Every­one works the whole time#

Exam­ple 1#

A can do a piece of work in $12$ days and B alone in $15$ days. With the help of C they fin­ish it in $5$ days. If they are paid ₹960 for the whole work, how should the money be divided among them?

First find C's 1 day's work.

$$\displaystyle \text{A's 1 day's work} = \frac{1}{12}, \qquad \text{B's} = \frac{1}{15}$$

$$\displaystyle (A + B)\text{'s 1 day's work} = \frac{1}{12} + \frac{1}{15} = \frac{9}{60} = \frac{3}{20}$$

$$\displaystyle (A + B + C)\text{'s 1 day's work} = \frac{1}{5}$$

$$\displaystyle \text{C's 1 day's work} = \frac{1}{5} - \frac{3}{20} = \frac{4 - 3}{20} = \frac{1}{20}$$

All three work for the same $5$ days, so the money is shared in the ratio of their 1 day's work:

$$\displaystyle \frac{1}{12} : \frac{1}{15} : \frac{1}{20} = \frac{5}{60} : \frac{4}{60} : \frac{3}{60} = 5 : 4 : 3$$

The parts add to $5 + 4 + 3 = 12$.

$$\displaystyle \text{A's share} = 960 \times \frac{5}{12} = 400$$

$$\displaystyle \text{B's share} = 960 \times \frac{4}{12} = 320$$

$$\displaystyle \text{C's share} = 960 \times \frac{3}{12} = 240$$

A gets ₹400, B gets ₹320 and C gets ₹240. Check: $400 + 320 + 240 = 960$.

To turn frac­tions into a whole-num­ber ratio, write them over a com­mon denom­i­na­tor (here $60$) and then keep only the numer­a­tors.

Peo­ple work for dif­fer­ent lengths of time#

Exam­ple 2#

A can do a piece of work in $10$ days and B in $15$ days. They work together for $5$ days, and the rest of the work is fin­ished by C in $2$ days. If they get ₹1920 for the whole work, how should they divide the money, and what are their daily wages?

$$\displaystyle (A + B)\text{'s 1 day's work} = \frac{1}{10} + \frac{1}{15} = \frac{3 + 2}{30} = \frac{5}{30} = \frac{1}{6}$$

$$\displaystyle (A + B)\text{'s 5 days' work} = \frac{5}{6}, \qquad \text{Remaining work} = 1 - \frac{5}{6} = \frac{1}{6}$$

So C's $2$ days' work is $\displaystyle \frac{1}{6}$, and C's 1 day's work is $\displaystyle \frac{1}{12}$.

Now find how much of the job each per­son did in total:

$$\displaystyle \text{A's 5 days' work} = \frac{5}{10} = \frac{1}{2}, \qquad \text{B's 5 days' work} = \frac{5}{15} = \frac{1}{3}, \qquad \text{C's 2 days' work} = \frac{2}{12} = \frac{1}{6}$$

These add to $\displaystyle \frac{3 + 2 + 1}{6} = 1$, the whole job, which is a good check. Share in the ratio

$$\displaystyle \frac{1}{2} : \frac{1}{3} : \frac{1}{6} = 3 : 2 : 1$$

The parts add to $6$.

$$\displaystyle \text{A's share} = 3 \times \frac{1920}{6} = 960, \quad \text{B's share} = 2 \times \frac{1920}{6} = 640, \quad \text{C's share} = 1 \times \frac{1920}{6} = 320$$

Daily wage $=$ share $\div$ num­ber of days worked:

Notice that C earns a higher daily wage than B. C worked fewer days, but C is the faster worker (C would fin­ish the whole job in $12$ days, B in $15$), so each of C's days is worth more.

A com­mon mis­take#

It is tempt­ing to share by the days each per­son worked. In Exam­ple 2 that would give the ratio $5 : 5 : 2$, which pays A and B the same even though A did more of the job. Always share by work done, never by days alone.

Prac­tice#

1. A can do a job in $6$ days and B in $8$ days. They do it together and are paid ₹2800. Find each per­son's share.
2. A and B are paid ₹1500 for a job they do together. A alone would take $10$ days and B alone $15$ days. Find each share.
3. A can do a job in $10$ days and B in $15$ days. With C's help they fin­ish it in $4$ days. They are paid ₹4500. Find each per­son's share.
4. A can do a job in $12$ days and B in $18$ days. They work together for $4$ days, and C fin­ishes the rest in $4$ days. They are paid ₹3600. Find each per­son's share and daily wage.
5. A, B and C can do a job in $20$, $30$ and $60$ days. They work together and are paid ₹6000. Find each share.
6. A and B can do a job in $16$ and $24$ days. They work together for $6$ days, and A fin­ishes the rest alone. They are paid ₹4000. How much does B get?

Answers#