Question

Divide $170$ into three parts such that the first part is $10$ more than the second and its ratio with the third part is $2:5$.
(A) $40,30,100$
(B) $20,30,100$
(C) $40,50,100$
(D) $50,30,100$

Answer 1

Hint: We know that if $x,y$ and $z$ are three parts of the number $w$ then we can write $x + y + z = w$. We will use this information and given data to find the required three parts of the given number $170$.

Complete step-by-step answer:
In the given problem, we need to find three parts of the number $170$ such that the first part is $10$ more than the second part and its ratio with the third part is $2:5$. For this, let us assume that the first, second and third parts of the number $170$ are $x,y,z$ respectively. As we assume that $x,y$ and $z$ are three parts of the number $170$, we can write
$x + y + z = 170 \cdots \cdots \left( 1 \right)$
It is given that the first part $x$ is $10$ more than the second part $y$. So, we can write
$x = y + 10$
$ \Rightarrow x - 10 = y$
$ \Rightarrow y = x - 10 \cdots \cdots \left( 2 \right)$
It is also given that the ratio of first part $x$ with the third part $z$ is $2:5$. So, we can write
$x:z = 2:5$
$ \Rightarrow \dfrac{x}{z} = \dfrac{2}{5}$
$ \Rightarrow z = \dfrac{{5x}}{2} \cdots \cdots \left( 3 \right)$
Note that in equation $\left( 2 \right)$, we have $y$ in terms of $x$ and in equation $\left( 3 \right)$, we have $z$ in terms of $x$. Let us convert the equation $\left( 1 \right)$ into the form of a single variable equation. That is, we will put $y = x - 10$ and $z = \dfrac{{5x}}{2}$ in equation $\left( 1 \right)$, we get
$x + \left( {x - 10} \right) + \dfrac{{5x}}{2} = 170$
$ \Rightarrow x + x - 10 + \dfrac{{5x}}{2} = 170$
Simplify the above equation by multiplying $2$ with each term, we get
$2x + 2x - 20 + 5x = 340$
$ \Rightarrow 9x - 20 = 340$
$ \Rightarrow 9x = 340 + 20$
$ \Rightarrow 9x = 360$
$ \Rightarrow x = \dfrac{{360}}{9}$
$ \Rightarrow x = 40$
Hence, we can say that the first part of the number $170$ is $x = 40$. Let us substitute $x = 40$ in equation $\left( 2 \right)$. So, we can write
$y = 40 - 10$
$ \Rightarrow y = 30$
Hence, we can say that the second part of the number $170$ is $y = 30$. Let us substitute $x = 40$ in equation $\left( 3 \right)$. So, we can write
$z = \dfrac{{5\left( {40} \right)}}{2}$
$ \Rightarrow z = 5 \times 20$
$ \Rightarrow z = 100$
Hence, we can say that the third part of the number $170$ is $z = 100$.
Hence, the required parts are $40,30$ and $100$. Hence, option A is correct.

So, the correct answer is “Option A”.

Note: In the given problem, we need to find values of three unknown variables $x,y,z$. So, we need three equations. To find values of $n$ unknowns, we need $n$ equations. A linear equation in one variable can be solved using basic algebraic operations.