Mindyard | Two farmers, X and Y , hire a field. X keeps 18 oxen there for 5 months and 20 goats for 3 months. Y keeps 24 goats for 7 months and 30 sheep for 6 months. If 2 oxen eat as much as 5 goats, and 3 goats eat as much as 4 sheep, what fraction of the total rent should X pay? 90196, 95196, 10196, 111196

Two farmers, X and Y , hire a field. X keeps 18 oxen there for 5 months and 20 goats for 3 months. Y keeps 24 goats for 7 months and 30 sheep for 6 months. If 2 oxen eat as much as 5 goats, and 3 goats eat as much as 4 sheep, what fraction of the total rent should X pay?

Options

$\frac{90}{196}$

$\frac{95}{196}$

$\frac{10}{196}$

$\frac{111}{196}$

Correct Answer :

Option B

$\frac{95}{196}$

Solution :

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Clean Answer:
` \frac{95}{196} `

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Ensure every ` ... ` is either in its own `$ ...
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$\frac{95}{196}$

The correct option is $\frac{95}{196}$ .
To find the fraction of the total rent that Farmer X should pay, we must calculate the effective total consumption of fodder by both farmers in terms of a common unit. Let us take the consumption of 1 goat as our base unit.
Step 1: Determine the relative consumption rates of the animals.

We are given that 2 oxen eat as much as 5 goats:

$2 Oxen = 5 Goats$

This means 1 Ox eats as much as:

$1 Ox = \frac{5}{2} Goats = 2.5 Goats$

We are also given that 3 goats eat as much as 4 sheep:

$4 Sheep = 3 Goats$

This means 1 Sheep eats as much as:

$1 Sheep = \frac{3}{4} Goats = 0.75 Goats$
Step 2: Calculate the total consumption for Farmer X in goat-months.

Farmer X keeps 18 oxen for 5 months and 20 goats for 3 months.

Consumption by Oxen = 18 × 5 = 90 ox-months.

Converting ox-months to goat-months:

$90 \times 2.5 = 225 goat-months$

Consumption by Goats = 20 × 3 = 60 goat-months.

Total consumption by X = 225 + 60 = 285 goat-months.
Step 3: Calculate the total consumption for Farmer Y in goat-months.

Farmer Y keeps 24 goats for 7 months and 30 sheep for 6 months.

Consumption by Goats = 24 × 7 = 168 goat-months.

Consumption by Sheep = 30 × 6 = 180 sheep-months.

Converting sheep-months to goat-months:

$180 \times 0.75 = 135 goat-months$

Total consumption by Y = 168 + 135 = 303 goat-months.
Step 4: Find the total combined consumption and X's share of rent.

Total consumption of X and Y combined:

$285 + 303 = 588 goat-months$

Fraction of total rent X should pay:

$Fraction for X = \frac{285}{588}$

Simplifying the fraction by dividing both numerator and denominator by 3:

$\frac{285 \div 3}{588 \div 3} = \frac{95}{196}$
Thus, Farmer X should pay $\frac{95}{196}$ of the total rent.}}

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Two farmers, X and Y , hire a field. X keeps 18 oxen there for 5 months and 20 goats for 3 months. Y keeps 24 goats for 7 months and 30 sheep for 6 months. If 2 oxen eat as much as 5 goats, and 3 goats eat as much as 4 sheep, what fraction of the total rent should X pay?

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