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Question

A piece of work can be done by $$2$$ men and $$7$$ boys in $$4$$ days. The same piece of work can be done by $$4$$ men and $$4$$ boys in $$3$$ days. How long it would take to do the same work by one man or one boy?


Solution

Let $$x$$ be the number of days in which one man can complete the work.
Let y be the number of days in which one boy can complete the work.
In one day a man can do $$\dfrac {1}{x}th$$ work and a boy does $$\dfrac {1}{y}th$$ work.
Now $$2$$ men and $$7$$ boys can complete the work in $$4$$ days. Hence they can complete $$\dfrac {1}{4}$$th of the work in one day. Thus we obtain
$$\dfrac {2}{x} + \dfrac {7}{y} = \dfrac {1}{4}$$ ..............($$1$$).
Again $$4$$ men and $$4$$ boys together complete the work in $$3$$ days. Hence they can complete $$\dfrac {1}{3}rd$$ of the work in a day. Thus we also get
$$\dfrac {4}{x} + \dfrac {4}{y} = \dfrac {1}{3}......... (2)$$
Multiply $$(1)$$ by $$2$$ and subtract $$(2)$$ from it, you get
$$\dfrac {4}{x} + \dfrac {14}{y} - \dfrac {4}{x} - \dfrac {4}{y} = \dfrac {2}{4} - \dfrac {1}{3}$$.
This reduces to $$\dfrac {10}{y} = \dfrac {1}{6}$$. Hence $$y = 60$$. From equation $$(2)$$ we get
$$\dfrac {4}{x} = \dfrac {1}{3} - \dfrac {4}{y} = \dfrac {1}{3} - \dfrac {4}{60} = \dfrac {1}{3} - \dfrac {1}{15} = \dfrac {4}{15}$$.
Hence $$x = 15$$.
Thus, one man can complete the work in $$15$$ days and one boy can do the work in $$60$$ days.

Maths

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