Question

The sum of two numbers is 1000 and the difference between their squares is 256000. Find the numbers.

Answer 1

Hint- Here, we will proceed by assuming the two numbers as two different variables say x and y. Then, we will form two equations in these two variables and solve them with the help of the substitution method.

Complete step-by-step solution -
Let us suppose the two numbers be x and y
Given, Sum of these two numbers is 1000
i.e., \[ x + y = 1000 \\
   \Rightarrow y = 1000 - x{\text{ }} \to {\text{(1)}} \\
 \]
Here, let us assume that the square of number x is greater than the square of number y
i.e., ${x^2} > {y^2}$
It is also given that the difference between their squares is 256000.
i.e., ${x^2} - {y^2} = 256000$
By substituting the value of y from equation (1) in the above equation, we get
$
   \Rightarrow {x^2} - {\left( {1000 - x} \right)^2} = 256000 \\
   \Rightarrow {x^2} - \left[ {{{\left( {1000} \right)}^2} + {x^2} - 2000x} \right] = 256000 \\
   \Rightarrow {x^2} - 1000000 - {x^2} + 2000x = 256000 \\
   \Rightarrow 2000x = 256000 + 1000000 \\
   \Rightarrow 2000x = 1256000 \\
   \Rightarrow x = \dfrac{{1256000}}{{2000}} = 628 \\
 $
By putting x = 628 in equation (1), we get
\[ \Rightarrow y = 1000 - 628 = 372\]
Therefore, the two required numbers are 628 and 372.

Note- In this particular problem, we have assumed that the square of number x is greater than the square of number y. Here, if we would have assumed that the square of number y is greater than the square of number x then we would have got x = 372 and y = 628 as the result. Here, in order to solve the two equations in two variables we can also proceed by elimination method.