Question

In a dairy farm, 40 cows eat 40 bags of husk in 40 days. In how many days one cow will eat one bag of husk?
(a). $1$
(b). $\dfrac{1}{40}$
(c). $40$
(d). $80$

Answer 1

- Hint: Here, we have cows and days are in indirect proportion and bags and days are in direct proportion. For cows and days in indirect proportion we have \[40:1::x:40\], for bags and days in direct proportion we have \[40:1::40:x\]. Now, we have to find the value of $x$.

Complete step-by-step solution -

Here, we are given that 40 cows eat 40 bags of husk in 40 days.
Now, we have to calculate the number of days in which one cow will eat one bag of husk.
We know there are two types of proportion. They are the direct proportion and indirect proportion.
Here, one is in direct proportion and the other is in indirect proportion.
We can say that two variables \[x\] and \[y\] are directly proportional to each other when the ratio \[x:y\] or $\dfrac{x}{y}$ is a constant. This means that \[x\] and \[y\] would either increase together or decrease together by an amount that would not change the ratio.
Similarly, we can say that two variables \[x\] and \[y\] are inversely proportional or indirectly proportional to each other when their product is a constant. This means that if \[x\] increases\[y\]decreases, vice-versa, by an amount such that $xy$ remains the same.
Here, if we consider cows and days we have more number of cows and less number of days which means they are in indirect proportion as one is more and the other is less.
Similarly we have less bags and less days which means they are in direct proportion as one is less then, other is also less.
Hence, for cows and days in inverse proportion we have:
\[40:1::x:40\]
Similarly, for bags and days in direct proportion we have:
\[40:1::40:x\]
Now, to make both the proportions similar change any one of the proportion, consider the indirect proportion, we can write it as:
\[1:40::40:x\]
Hence, if we combine the proportion we will get:
$\left. \begin{align}
  & Cows\text{ }1:40 \\
 & Bags\text{ }40:1 \\
\end{align} \right\}::40:x$
This can be written as:
$1\times 40\times x=40\times 1\times 40$
 Now, by cross multiplication we get:
$x=\dfrac{40\times 1\times 40}{1\times 40}$
Now, by cancellation we obtain:
$x=40$
 Therefore, we can say that one cow will eat one bag of husk in 40 days.
 Hence, the correct answer for this question is option (c)

Note: Here, this problem involves both direct and inverse proportion. Hence, if $d$ varies directly as $w$ and inversely as $m$, this involves three variables, then we have a formula, $\dfrac{{{m}_{1}}{{d}_{1}}}{{{w}_{2}}}=\dfrac{{{m}_{2}}{{d}_{2}}}{{{w}_{1}}}$. In this problem $d$ is days, $w$ is cows and $m$ is bags. Here, ${{d}_{2}}$ is the unknown quantity. So, you can find ${{d}_{2}}$ by cross multiplication.