Question

A piece of string is 40 centimeters long. It is cut into three pieces. The longest piece is 3 times as long as the middle-sized and the shortest piece is 23 centimeters shorter than the longest piece. Find the length of the shortest piece (in cm).
(a) 27
(b) 5
(c) 4
(d) 9

Answer 1

Hint: Let us assume that the length of the middle-sized is x cm (centimeters) then according to the statement of the problem then the length of the longest piece is 3 times that of the middle-sized string. So, from this statement, we can find the length of the longest string. And the sum of the length of the longest, middle-sized, and smallest string is equal to 40 cm. So, from this addition, we can find the relation between the smallest-sized string and x. Now, the value of x is calculated by using the statement the shortest piece is 23 centimeters shorter than the longest piece.

Complete step-by-step solution:
In the above problem, a string of length 40 cm is given. Let us assume that the length of the middle-sized string is x cm.
Now, it is given that the longest string is 3 times that of the middle-sized string so multiplying x with 3 will give the length of the longest string.
$3x$
It is also given that the shorter piece is 23 centimeters shorter than the longest piece. In the above, we have found the length of the longest string as $3x$ so subtracting 23cm from $3x$ we get,
$3x-23$
We know that the sum of all the three different lengths of the strings is equal to 40 cm so add all the three segments of the string and equate them to 40cm.
$\begin{align}
  & x+3x+3x-23=40 \\
 & \Rightarrow 7x-23=40 \\
\end{align}$
Adding 23 on both the sides of the above equation we get,
$\begin{align}
  & 7x-23+23=40+23 \\
 & \Rightarrow 7x=63 \\
\end{align}$
Dividing 7 on both the sides of the above equation we get,
$x=\dfrac{63}{7}=9$
We are asked to find the length of the shortest string so putting the above value of x in $3x-23$ we get,
$\begin{align}
  & 3\left( 9 \right)-23 \\
 & =27-23 \\
 & =4 \\
\end{align}$
Hence, the correct option is (c).

Note: You can check the value of the shortest string by putting the value of x in all the three segments of the string and add the lengths of three segments of the string and see if the addition is equal to 40 or not.
$\begin{align}
  & x+3x+3x-23 \\
 & =7x-23 \\
\end{align}$
Substituting $x=9$ in the above equation we get,
$\begin{align}
  & 7\left( 9 \right)-23 \\
 & =63-23 \\
 & =40 \\
\end{align}$
As you can see that we are getting the sum as 40 so the value of the shortest string which we have calculated above is correct.