GATE2017 ME-2: GA-7

Question

$X$ bullocks and $Y$ tractors take $8$ days to plough a field. If we have half the number of bullocks and double the number of tractors, it takes $5$ days to plough the same field. How many days will it take $X$ bullocks alone to plough the field?

• A. $30$
• B. $35$
• C. $40$
• D. $45$

Comments

(X + Y )*8 = (0.5 X + 2 Y)*5

11X = 4Y

Y = (11/4) X

Y put in the one of the eq  (x +11/4x)*5=1 (total work is 1 )

x =1/30 this is one day work so total work take 30 day 

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Good method

Answer 1

Rate of work of $X+Y$ (in fraction of field ploughed per day) $ = \frac{1}{8}$

Rate of work of $\frac{X}{2}+2Y$ (in fraction of field ploughed per day) $ = \frac{1}{5}$

We can write

• $(X + Y) \propto \frac{1}{8} \quad \to (1) $
• $\left(\frac{X}{2}+2Y\right)\propto \frac{1}{5} \quad \to (2)$

From $(1), (2)$

$8X + 8Y = \frac{5X}{2} + 10Y$

$\implies 11X = 4Y$

Putting in $(1)$ we get

$\left(X + \frac{11}{4} X\right) \propto \frac{1}{8}$

$\implies \frac{15X}{4} \propto \frac{1}{8}$

$\implies X \propto \frac{1}{30}.$

i.e., Rate of work of $X$ bullock is $\frac{1}{30}$ ploughs per day and so it will take $30$ days for $X$ bullocks alone to plough the field.   

Correct Option: A.

Answer 2

$X\times 8\ B-D + Y\times 8\ T-D=\dfrac{X}{2}\times 5\ B-D + 2Y\times 5\ T-D$

$11\times X\ B=4\times Y\ T$

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$X\ B+Y\ T=X\ B+\dfrac{11\times X\ }{4}\ B=\dfrac{15X}{4}\ B$

$\dfrac{15X}{4}\ B\rightarrow8\ days$

$X\rightarrow\ ?$

$?=30\ days$

Comments

I especially lookout for your answers before going to best answers ;)

Answer 3

Let efficiency of a bullock and a tractor be B unit/day and T unit/day respectively.

Work done by X bullocks and Y tractors in 8 days to plough a field : 

Work done by X bullocks = X * B unit/day * 8 days
Work done by Y tractors = Y * T unit/day * 8 days
Total work done = (X * B * 8 + Y * T * 8) units 

Work done by X / 2 bullocks and 2 * Y tractors in 5 days to plough a field :
Work done by X bullocks = (X / 2) * B unit/day * 5 days
Work done by Y tractors = 2 * Y * T unit/day * 5 days
Total work done = (X * B * 2.5 + Y * T * 10) units

Both bullocks and tractors are used to plouge same field.
Therefore,
(X * B * 8 + Y * T * 8) units = (X * B * 2.5 + Y * T * 10) units

On solving, Y * T = 2.75 * X * B  

Let X bullocks take m days to plough a field
Total work done = Work done by X bullocks 
                = X * B unit/day * m days
                = (X * B * m) units

Here X bullocks plough the same field
Therefore,
(X * B * m) units = (X * B * 8 + Y * T * 8) units
X * B * m = X * B * 8 + (2.75 * X * B ) * 8

On solving, m = 30

Note: In question, Number of bullocks are halved in second statement.