Question

A length of 60 cm is divided into equal parts. What is the number of these parts if, when this number increases by unity, the length of each part is decreased by 1 mm?

Answer 1

Hint: The total length of 60 cm is when divided in equal parts will result in the whole number of parts only. Measuring units can be converted from one dimension to another according to use to make the calculation simpler and accurate. Here, two dimensions have been included i.e., centimeters and millimeters. So, it will be easy for us to change centimeters to millimeters by multiplying the dimension by 10.
Moreover, the product of the total number of equal parts and the length of each equal part will result in the total length.

Complete step-by-step answer:
Let the number of equal parts be $x$ in which $60{\text{ cm = 600 mm}}$ has been divided then the length of each part will be $\dfrac{{600}}{x}$.
According to the question, when the number of equal parts has been increased by unity, the length of each equal part has been decreased by $1{\text{ mm }}$
So,
New number of parts will be $x + 1$
The new equation will be:
$(x + 1)\left( {\dfrac{{600}}{x} - 1} \right) = 600$
Solve the equation $(x + 1)\left( {\dfrac{{600}}{x} - 1} \right) = 600$ to determine the value of $x$ as:
$
  (x + 1)\left( {\dfrac{{600}}{x} - 1} \right) = 600 \\
  600 - x + \dfrac{{600}}{x} - 1 = 600 \\
  \dfrac{{600}}{x} - x = 1 \\
  600 - {x^2} = x \\
  {x^2} + x - 600 = 0 \\
  {x^2} + 25x - 24x - 600 = 0 \\
  x(x + 25) - 24(x + 25) = 0 \\
  (x + 25)(x - 24) = 0 \\
  x = - 25,24 \\
 $
As the number of equal parts cannot be negative hence, $x = 24$.
Hence, the number of equal parts when 60 cm is divided is 4.

Note: Whenever this type of question is encountered by the candidates, it is advised to minimize the use of variables so that the fewer number of equations will be required to determine the answer. Alternatively, in this question, we can take the length of each equal part to be a variable and proceed to solve for two variables but this will be complicated in the calculation.