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$X$ 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?

1. $30$
2. $35$
3. $40$
4. $45$

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Migrated from GO Mechanical 5 months ago by Arjun

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.

by Veteran (423k points)

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.

by (140 points)