Question

A pole height $3$meters is struck by a speeding car and breaks into two pieces such that the first piece is $\dfrac{1}{2}$of the second. Find the length of both pieces?

Answer 1

Hint: Before going to solve this problem, we will be discussing the concept of Ratio.
Comparison/simplification of two quantities by the method of division is known as ratio. The result of this simplification gives the number of times a quantity is equal to another, or you can say the ratio is used to express a quantity as a fraction of the other two numbers of a ratio. It can be compared when they have the same unit. ‘$:$’ is the sign used to denote a ratio, which is represented as $a:b$.
We will solve the problem by assuming the length of one portion of the pole as $x$, and the length of the second portion is given by the relation ‘first piece is $\dfrac{1}{2}$of the second’. After that we will find the ratio of the both lengths and find the values of the lengths using the value length of the pole.

Complete step by step answer:
Given that, length of pole is $l=3$m
Let the length of the first piece is ${{l}_{1}}=x$
Let the length of the second piece is ${{l}_{2}}$.
Given that the first piece is $\dfrac{1}{2}$of the second, hence
$\begin{align}
  & {{l}_{1}}=\dfrac{1}{2}{{l}_{2}} \\
 &\Rightarrow {{l}_{2}}=2{{l}_{1}} \\
 & \Rightarrow {{l}_{2}}=2x
\end{align}$
Ratios of the lengths is
$\begin{align}
  & {{l}_{1}}:{{l}_{2}}=x:2x \\
 & \Rightarrow {{l}_{1}}:{{l}_{2}}=1:2 \\
\end{align}$
Sum of the terms in the ratio $=1+2=3$
Hence the length of the first part is
$\begin{align}
  & {{l}_{1}}=\dfrac{1}{3}\times 3 \\
 & =1m
\end{align}$
The length of the second part is
$\begin{align}
  & {{l}_{2}}=\dfrac{2}{3}\times 3 \\
 & =2m
\end{align}$

Hence the lengths of the first and second piece are 1m and 2m.

Note: We can also solve the above problem by not going to the concept of ratios. We will add the both lengths and equate them to $3m$, then
$\begin{align}
  & {{l}_{1}}+{{l}_{2}}=3 \\
 &\Rightarrow x+2x=3 \\
 &\Rightarrow 3x=3 \\
 &\Rightarrow x=1
\end{align}$
Hence the length of the first piece is $x=1m$ and length of second piece is $2x=2m$
From both methods, we got the same answer.