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How do you write a direct proportion equation that relates $a$ and $b$ if $a$ is directly proportional to $b$ and $a=14,b=42$?

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Last updated date: 22nd Jul 2024
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Answer
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Hint: We first take an arbitrary constant for the proportional relation $a\propto b$. We put the given values of $a$ and $b$ to find the values of $k$. This solution gives the proportion equation of $a$ and $b$.

Complete step-by-step solution:
It’s given that $a$ is directly proportional to $b$. The mathematical way to represent the given statement is $a\propto b$.
We will bring a ratio constant to find the relation between $a$ and $b$.
The relation of $a\propto b$ gives $ak=b$ where $k$ is the proportional constant.
We have to find the value of $k$ from the given values of $a$ and $b$.
The values of $a$ and $b$ for a particular case is $a=14,b=42$.
We place the values in the equation of $ak=b$.
Therefore, $14\times k=42$ gives $14k=42$
We divide both sides of the equation $14k=42$ with 14 to get $k=\dfrac{42}{14}$.
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 d then we need to divide the denominator and the numerator with d and get the simplified fraction form as $\dfrac{{}^{p}/{}_{d}}{{}^{q}/{}_{d}}$.
For our given fraction $\dfrac{42}{14}$, the G.C.D of the denominator and the numerator is 14.
$\begin{align}
  & 2\left| \!{\underline {\,
  14,42 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7,21 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1,3 \,}} \right. \\
\end{align}$
Now we divide both the denominator and the numerator with 14 and get $\dfrac{{}^{42}/{}_{14}}{{}^{14}/{}_{14}}=\dfrac{3}{1}=3$.
Therefore, the simplified form of $k$ is 3. The equation between $a$ and $b$ is $3a=b$.

Note: We can also use the relation $a=kb$ where we will get the value of $k$ as $\dfrac{1}{3}$. The value of two numbers $a$ and $b$ is dependent on each other. If we take the value of $a$ as $a=5$, the value of $b$ becomes $b=5\times 3=15$.