Question

The sum of numbers is $3500$. If $6\%$ of one number equal to $8\%$ of the other number. Find the number?

Answer 1

Hint: From the question, we have to find the unknown numbers. First, we will frame the given to the mathematical expression by using some variables and convert the given percentage into fraction form then after some simple calculation, we get the required unknown numbers.

Complete step-by-step answer:
By the given, the sum of the numbers is $3500$.
Now, we frame the above in the mathematical expression as mentioned above in the formula. We get,
 ${\text{x}} + {\text{y}} = 3500$
Then, by the given, we have to frame the mathematical expression for $6\%$ of one number equal to $8\%$ of the other number. By using the formula mentioned above. We get,
Let $6\%$ of one number be $6{\text{x}}$ and $8\%$ of other numbers be $8{\text{y}}$.
Then, $6\%$ of one number equal to $8\%$ of another number be $6{\text{x}} = 8{\text{y}}$.
For the solution, now we have to change $6{\text{x}} = 8{\text{y}}$ into two ways:
$1.$${\text{x}} = \dfrac{{8{\text{y}}}}{6}$. On dividing the numerator and denominator by $2$, we get ${\text{x}} = \dfrac{{{\text{4y}}}}{3}$ .
$2.$ ${\text{y}} = \dfrac{{6{\text{x}}}}{8}$ . On dividing the numerator and denominator by $2$, we get ${\text{y}} = \dfrac{{3{\text{x}}}}{4}$.
Here, I took the option $1$ for finding the required unknown numbers.
Now, we have to substitute ${\text{x}} = \dfrac{{{\text{4y}}}}{3}$in the${\text{x}} + {\text{y}} = 3500$. Then, we get
${\text{x}} + {\text{y}} = 3500$
$ \Rightarrow \dfrac{{4{\text{y}}}}{3} + {\text{y}} = 3500$
Here, we are going to solve the above equation by taking Least Common Multiple (LCM) on the left hand side (LHS).
The Least Common Multiple (LCM) of $1$ and $3$ is $3$. Then, we get
 $ \Rightarrow \dfrac{{4{\text{y}} \times 1}}{{3 \times 1}} + \dfrac{{{\text{y}} \times 3}}{{1 \times 3}} = 3500$
$ \Rightarrow \dfrac{{4{\text{y}}}}{3} + \dfrac{{3{\text{y}}}}{3} = 3500$
Now, the denominators are the same. So, we have to add the numerators.
$ \Rightarrow \dfrac{{4{\text{y}} + 3{\text{y}}}}{3} = 3500$
$ \Rightarrow \dfrac{{7{\text{y}}}}{3} = 3500$
On cross multiplying $3$ into the right hand side (RHS). Then, we get
$ \Rightarrow 7{\text{y}} = 3500 \times 3$
On cross multiplying $7$ into the left hand side (LHS). Then, we get
$ \Rightarrow {\text{y}} = \dfrac{{3500 \times 3}}{7}$
Now, dividing the left hand side (LHS) numerator and denominator by $7$, we get
$ \Rightarrow {\text{y}} = 500 \times 3$
$ \Rightarrow {\text{y}} = 1500$.
Therefore, we got one of the required unknown numbers: ${\text{y}} = 1500$.
 Now, we are going to find another required number${\text{x}}$ by substituting the value of ${\text{y}}$ in ${\text{x}} + {\text{y}} = 3500$. Then, we get
$ \Rightarrow {\text{x}} + 1500 = 3500$
$ \Rightarrow {\text{x}} = 3500 - 1500$
$ \Rightarrow {\text{x}} = 2000$.
Therefore, another number is ${\text{x}} = 2000$.

Hence, the required unknown numbers are ${\text{x}} = 2000$ and ${\text{y}} = 1500$.

Note: If ${\text{x}}\%$ can be expressed in the fractional form as $\dfrac{{\text{x}}}{{100}}$.
If ${\text{x}}\% $ of ${\text{y}}$ can be expressed in the fractional form as $\left( {\dfrac{{\text{x}}}{{100}}} \right) \times {\text{y}}$ and then it also can be written as $\dfrac{{{\text{xy}}}}{{100}}$.