6 men or 10 boys can complete a piece of work in 15 days. If 7 men and $x$ boys complete the same piece of work in 9 days, then $x$ is equal to. (a) 4 (b) 5 (c) 6 (d) 7
ANSWER
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Hint: For solving this question we will first make some assumptions in terms of a few variables and then try to frame some equations as per the given data. Then, we will solve these equations to get the correct answer.
Complete step-by-step solution - Given: 6 men or 10 boys can finish a piece of work in 15 days and 7 men and $x$ boys can finish the same piece of work in 9 days. Let, there are $N$ units of work which is to be done. And 1 man can do $m$ units of work per day. And 1 boy can do $b$ units of work per day. Then, In 1 day 6 men can do $6m$ units of work per day and in 1 day 10 boys can do $10b$ units of work per day. And, it is given that 6 men or 10 boys can finish $N$ units of work in 15 days. Then, $\begin{align} & 6m\times 15=N \\ & \Rightarrow m=\dfrac{N}{90}\text{ }{}^{units}/{}_{day}\text{ }.........\left( 1 \right) \\ & 10b\times 15=N \\ & \Rightarrow b=\dfrac{N}{150}\text{ }{}^{units}/{}_{day}\text{ }............\left( 2 \right) \\ \end{align}$ Thus, in 1 day 1 man can do $\dfrac{N}{90}$ units of work and in 1 day 1 boy can do $\dfrac{N}{150}$ units of work. Now, it is given that 7 men and $x$ boys complete the $N$ units of work in 9 days. Then, 7 men can do $\dfrac{7N}{90}$ units of work per day and $x$ boys can do $\dfrac{Nx}{150}$ units of work per day. Then, $9\times \left( \dfrac{7N}{90}+\dfrac{Nx}{150} \right)=N$ $\begin{align} & \Rightarrow \dfrac{7N}{90}+\dfrac{Nx}{150}=\dfrac{N}{9} \\ & \Rightarrow \dfrac{Nx}{150}=\dfrac{N}{9}-\dfrac{7N}{90} \\ & \Rightarrow \dfrac{Nx}{150}=\dfrac{3N}{90} \\ & \Rightarrow x=\dfrac{3}{90}\times 150 \\ & \Rightarrow x=5 \\ \end{align}$ Thus, the value of $x$ is 5 boys. Hence, (b) is the correct option.
Note: Although the question is very easy to solve but the student must be careful while solving and avoid calculation mistakes. Moreover, one should not directly add the time taken by men and boys to complete the work alone directly to get the answer; it would be the wrong approach.Here we should have keep in mind that “OR”, “AND” have different means for this type of problem . ”OR” shows that men and boys work separately while “AND” shows that boys and men work together.