GATE2019 EC: GA-3

Question

It would take one machine $4$ hours to complete a production order and another machine $2$ hours to complete the same order. If both machines work simultaneously at their respective constant rates, the time taken to complete the same order is ________ hours.

• A. $2/3$
• B. $3/4$
• C. $4/3$
• D. $7/3$ 

Answer 1

Machine $A$ takes $4$ hours and machine $B$ takes $2$ hours to complete the work.
$\text{Efficiency}=\dfrac{\text{Work}}{\text{Time}}$

Let the amount of work be $X.$

Efficiency of $A=\frac{X}{4}$
Efficiency of $B=\frac{X}{2}$

If both machines work simultaneously at their respective constant rates, then

Effective Efficiency $=$ Efficiency of $A$ $+$ Efficiency of $B=\frac{X}{4}+\frac{X}{2} = \frac{3X}{4}$

Effective Time$=\dfrac{X}{\frac{3X}{4}}=\dfrac{4}{3}$ hours.

So, the correct answer is $(C).$

Comments

$\:\:\:\:\:\:\:M_{1}\:\:\:\:\:M_{2}$

$T:\:\:4\:\:\:\:\:\:\:\:\:2\implies \text{Work}  = LCM(4,2) = 4$

$E:\:\:1\:\:\:\:\:\:\:\:\:2$

$\text{Rate of work} \times \text{Time} = \text{Work Done}$

$\text{Time} = \dfrac{\text{Work Done}}{\text{Rate of work}}$

$\text{Time} = \dfrac{4}{3}\:\text{hours}$

Answer 2

$Let\ W=4\ unit$

$M1\rightarrow4H-\ \ 1unit/H$

$M2\rightarrow2H-\ \ \underline{2unit/H}$

                      $3unit/H$                   

$1H\rightarrow 3unit$

$?\rightarrow 4unit$

$?=\dfrac{4}{3}H$

Answer 3

Let the production order be 8 items(any multiple of L.c.m(2,4) like 4,8,12,16... would work).

Since 1 st machine completes the order in 4 hours, it is producing at a rate of 2 units per hour.

Similarly 2 nd machine produces at a rate of 4 units per hour.

If both machines are working simultaneously, together they produce 6 units per hour.

$\therefore$ 8 units are produced in $8/6$ hours.

option c is the answer