# Work and Wages

**Wages** are given in proportion to the work done and in indirect (or inverse) proportion to the time taken by the individual.

When a person receives some money for a certain work, the received money is called wages of the person for that particular work.

Total wages = Wages of 1 day work × Total number of days

- Wages is directly proportional to the work done.
- More money will be received for more work and less money will be received for less work.
- Wages is indirectly proportional to the time taken by the individual.

## Formulas for Work and Wages

- If a person can do a piece of work in ‘n’ days, then in one day, the person will do ‘1/n’ work. Conversely, if the person does ‘1/n’ work in one day, the person will require ‘n’ days to finish the work.
- In questions where there is a comparison of work and efficiency, we use the formula

M1 D1 H1 E1 / W1 = M2 D2 H2 E2 / W2, where

M = Number of workers

D = Number of days

H = Number of working hours in a day

E = Efficiency of workers

W = Units of work - In case we have more than one type of workers, then the formula modifies to

∑(Mi Ei) D1 H1 / W1 = ∑(Mj Ej) D2 H2 / W2, where ‘i’ and ‘j’ may vary as per the number of workers. - If a person A is ‘n’ times more efficient than person B, then

Ratio of work done by A and B in one day (Ratio of efficiencies) = n : 1

Ratio of time taken by A and B = 1 : n

**Total work** = No. of Days x Efficiency

If a group of people are given a salary for a job they do together, their individual salaries are in the ratio of their individual efficiencies if they work for the same number of days. Otherwise, salaries are divided in the ratio of units of work done.

### Some Important Rule

**Rule** **1**: If P can do a piece of work in x days and Q can do the same work in y days, the ratio of their wages will be y : x. Then the wages earned by P and Q will be

P's wages = $\frac{Totalwages}{(x+y)}\times y$

Q's wages = $\frac{Totalwages}{(x+y)}\times x$

**Rule** **2:** If P, Q and R can do a piece of work in x, y and z days respectively, the ratio of their wages will be yz : xz : xy. Then, wages earned by P, Q and R respectively will be

P's wages = $\frac{Totalwages}{(xy+yz+zx)}\times yz$

Q's wages = $\frac{Totalwages}{(xy+yz+zx)}\times xz$

R's wages = $\frac{Totalwages}{(xy+yz+zx)}\times xy$

**Rule 3:** P can do a piece of work in x days. With the help of Q, P can do the same work in y days. If they get ₹ a for that work, then

Share of P = ₹$\frac{ay}{x}$

Share of Q = ₹$\frac{a(x-y)}{x}$

**Rule 4:** P, Q and R undertake to do a work for ₹ a. If together they do only x/y of the work and rest is done by R alone, then

Share of R = a $(1-\frac{x}{y})$