Question

The $22\dfrac{1}{2}$ m of silk costs Rs. $214.20$. What is the cost of $6\dfrac{3}{4}$ m?

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 answer:
We have been given the relation between two variables where we assume length of the silk as $r$ and cost as $t$. The inversely proportional number is actually directly proportional to the inverse of the given number. The relation between r and t is inverse relation. It’s given that $r$ varies directly as $t$ which gives $r\propto t$.

To get rid of the proportionality we use the proportionality constant which gives $r=kt$.We convert the mixed fractions into improper fractions. So,
$22\dfrac{1}{2}=\dfrac{45}{2}$ and $6\dfrac{3}{4}=\dfrac{27}{4}$.
Here, the number $k$ is the proportionality constant.
It’s given $r=\dfrac{45}{2}$ when $t=214.2$.
We put the values in the equation $r=kt$ to find the value of $k$. So,
$\dfrac{45}{2}=k\times 214.2$.
The simplified value is $k=\dfrac{45}{2\times 214.2}=\dfrac{25}{238}$
Therefore, the equation becomes with the value of k as $r=\dfrac{25}{238}t$.
Now we simplify the equation to get the value of t for length of the silk being $\dfrac{27}{4}$
\[\dfrac{27}{4}=\dfrac{25}{238}t \\
\therefore t=\dfrac{27\times 238}{4\times 25}=64.26 \\ \]
Therefore, the cost of $6\dfrac{3}{4}$m silk is Rs. \[64.26\].

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.