time and work

Time and work deal with the time taken by an individual or a group of individuals to complete a piece of work and the efficiency of the work done by each of them. Below are a few such important time and work formulas for your reference:

• If T is the time taken to complete a certain amount of work W then the Work Done (W) = Time (T) × Rate of Work (R). The number of units of works done per unit time is called the rate of work (R).

• Rate of work and Time are inversely proportional to each other, i.e. R = 1/T

• If a person does a piece of work in x days, it is clear that he can do \(\frac{1}{x}\) of the work in 1 day. 

• If A can do a work in x days and B can do the same work in y days, then the ratio of work done by A and B at the same time is  \(\frac{1}{x} : \frac{1}{y}\), that is y ∶ x 

• If A is x times as good a workman as B, then A will take \(\frac{1}{x}\) of the time that B takes to do certain work.

• If A is p% efficient than B, it means if B completes a work in x hours, A completes it in \([\frac{100}{100 +p}]x \) hours.

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Example 1: A can do a piece of work in 4 days and B can do it in 10 days. In what time can they do it working together?

Solution:

The amount of work done by A in 1 day is 1/4

The amount of work done by B in 1 day is 1/10

The amount of work done by A and B together in 1 day is \(\frac{1}{4} + \frac{1}{10} = \frac{7}{20}\)

∴ the time required by A and B working together to finish the work is \(\frac{20}{7} = 2\frac{6}{7}\)days
 

Example 2: A and B working together can finish a work in 6 days. A alone can do it in 10 days. In how many days can B alone do it?

Solution:

(A + B)'s 1-day work = \(\frac{1}{6}\)

A's 1-day work = \(\frac{1}{10}\)

∴ B's 1-day work is \(\frac{1}{6}\) − \(\frac{1}{10}\) = \(\frac{1}{15}\)

B alone can the work in 15 days.

Example 3: If A is 40% efficient than B, and B completes a job in 7 days, then A completes the job in how many days?

Solution:

A complete the same job in \([\frac{100}{100 +40}]\times 7 = \frac{100}{140}\times 7 = 5\)

A completes the same job in 5 days.

Example 4: A alone can do a piece of work in 3 days, B alone can do it in 6 days and C alone can do it in 9 days. If the total wages for the work are $781, then how should the money be divided among them?

Solution:

A's 1-day work is \(\frac{1}{3}\)

B's 1-day work is \(\frac{1}{6}\)

C's 1-day work is \(\frac{1}{9}\)

A, B, and C share the money in the ratio \(\frac{1}{3}\) ∶ \(\frac{1}{6}\) ∶ \(\frac{1}{9}\), i.e. in the ratio 6 ∶ 3 ∶ 2  [ \(\frac{1}{3} \times 18 : \frac{1}{6} \times 18 : \frac{1}{9} \times 18\) ]

A's share is \(\frac{6}{(6+3+2)} \times 781 = \frac{6}{11} \times 781 = 426\)

B's share is \(\frac{3}{(6+3+2)} \times 781 = \frac{3}{11} \times 781 = 213\)

C's share is \(\frac{2}{(6+3+2)} \times 781 = \frac{2}{11} \times 781 = 142\)

Note: While doing problems on wages, the wages obtained are always divided in the ratio of the work done by each person in 1 day.