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# How do you solve for $f$ in $w=fd$?

Last updated date: 10th Aug 2024
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Hint: We first try to describe the relation between the denominator and the numerator to find the simplified form. We use the G.C.D of the denominator and the numerator to divide both of them. We get the simplified form when the G.C.D is 1. We divide both sides of the equation $w=fd$ by $d$. That gives the value of $f$ in $w=fd$.

Complete step by step answer:
We need to solve for $f$ in $w=fd$. Here $w=fd$ has been expressed as the multiplication of $f$ and $d$. We divide both sides of the equation $w=fd$ with $d$.
So, $\dfrac{w}{d}=\dfrac{fd}{d}$. The right side of the equation becomes $\dfrac{fd}{d}=f$
This simplified part can be expressed as using their GCD.
Simplified form is achieved when the G.C.D of the denominator and the numerator is 1.
This means we can’t eliminate any more common root from them other than 1.
For any fraction $\dfrac{p}{q}$, we first find the G.C.D of the denominator and the numerator. If it’s 1 then it’s already in its simplified form and if the G.C.D of the denominator and the numerator is any other number x then we need to divide the denominator and the numerator with x and get the simplified fraction form as $\dfrac{{}^{p}/{}_{x}}{{}^{q}/{}_{x}}$.
For our given fraction $\dfrac{fd}{d}$, the G.C.D of the denominator and the numerator is $d$.
Now we divide both the denominator and the numerator with $d$ and get $\dfrac{{}^{fd}/{}_{d}}{{}^{d}/{}_{d}}=\dfrac{f}{1}=f$.
Therefore, the solution for $f$ in $w=fd$ is $\dfrac{w}{d}$.

Note:
The process is similar for both proper and improper fractions. In case of mixed fractions, we need to convert it into an improper fraction and then apply the case. Also, we can only apply the process on the proper fraction part of a mixed fraction.