Question

Divide $4500$ into two parts such that $5\% $ of the first part is equal to the $10\% $ of the second part.

Answer 1

Hint: We know that if $x$ and $y$ are two parts of the number $z$ then we can write $x + y = z$. We will use this information and given data. Then, we will get two linear equations. We will solve those equations by a simple elimination method to find required parts.

Complete step-by-step answer:
In this problem, we need to find two parts of the number $4500$ such that $5\% $ of the first part is equal to $10\% $ of the second part. For this, let us assume that the first part of number $4500$ is $x$ and the second part is $y$. As we assume $x$ and $y$ are two parts of number $4500$, we can write $x + y = 4500 \cdots \cdots \left( 1 \right)$
Now $5\% $ of the first part $x$ is equal to $\dfrac{{5x}}{{100}}$ and $10\% $ of the second part $y$ is equal to $\dfrac{{10y}}{{100}}$. It is given that $5\% $ of the first part is equal to $10\% $ of the second part. So, we can write
$\dfrac{{5x}}{{100}} = \dfrac{{10y}}{{100}} \cdots \cdots \left( 2 \right)$. Let us simplify the equation $\left( 2 \right)$. So, we can write
$5x = 10y$
$ \Rightarrow x = \dfrac{{10y}}{5}$
$ \Rightarrow x = 2y \cdots \cdots \left( 3 \right)$
Let us solve the equations $\left( 1 \right)$ and $\left( 3 \right)$ to find the values of $x$ and $y$. Let us substitute $x = 2y$ from equation $\left( 3 \right)$ in equation $\left( 1 \right)$ and simplify the obtained equation. So, we get
$x + y = 4500$
$ \Rightarrow 2y + y = 4500$
$ \Rightarrow 3y = 4500$
$ \Rightarrow y = \dfrac{{4500}}{3}$
$ \Rightarrow y = 1500$
Substitute the value of $y$ in equation $\left( 3 \right)$, we get
$x = 2y$
$ \Rightarrow x = 2\left( {1500} \right)$
$ \Rightarrow x = 3000$
Hence, the required parts are $1500$ and $3000$. Hence, we can say that two parts of $4500$ are $1500$ and $3000$ such that $5\% $ of the first part is equal to $10\% $ of the second part.

Note: In this problem, we need to find values of two unknown variables. So, we need only two equations. We can solve two linear equations by using a simple elimination method. Also remember that $x\% $ of $y$ is equal to $\left( {\dfrac{x}{{100}} \times y} \right)$.