Question

If 3 cans of dog food will feed 2 dogs for 1 day, how many cans are required to feed 8 dogs for a week?

Answer 1

Hint: We first try to form the proportionality equation for the variables. We take an arbitrary constant. We use the given values of the variables to find the value of the constant. Finally, we put the constant’s value to find the equation.

Complete step by step solution:
We have been given the relation between three variables where we assume cans of dog food as r, number of dogs as p and number of days as t.
The inversely proportional number is actually directly proportional to the inverse of the given number.
The relation between r and p is a direct relation. The same goes for r and t.
It’s given r varies directly as p and t which gives $r\propto pt$.
To get rid of the proportionality we use the proportionality constant which gives $r=kpt$.
Here, the number k is the proportionality constant.
It’s given $r=3$ when $p=2,t=1$.
We put the values in the equation $r=kpt$ to find the value of k.
So, $3=k\times 2\times 1$. Simplifying we get
\[\begin{align}
  & 3=k\times 2\times 1 \\
 & \Rightarrow k=\dfrac{3}{2} \\
\end{align}\]
Therefore, the equation becomes with the value of k as $r=\dfrac{3}{2}pt$.
Now we simplify the equation to get the value of r for number of dogs being 8 and number of days being 7
\[\begin{align}
  & r=\dfrac{3}{2}pt \\
 & \Rightarrow r=\dfrac{3}{2}\times 8\times 7 \\
 & \Rightarrow r=3\times 4\times 7=84 \\
\end{align}\]

Therefore, the number of cans required to feed 8 dogs for a week is 84.

Note: In a direct proportion, the ratio between matching quantities stays the same if they are divided. They form equivalent fractions. In an indirect (or inverse) proportion, as one quantity increases, the other decreases. In an inverse proportion, the product of the matching quantities stays the same.