Question

A and B each have a certain number of oranges. A says to B, “If you give me 10 of your oranges, I will have twice the number of oranges left with you.” B replies, “If you give me 10 of your oranges, I will have the same number of oranges as left with you.” Find the number of oranges with A and B separately.

Answer 1

Hint: First suppose person A has x oranges and person B has y oranges then form two equations by using the provided statements after this simplifies the equations to find the values of x and y.

Complete step by step answer:
Our task is to find the number of oranges with A and B separately. For this, we suppose that person A has x number of oranges, and person B has y number of oranges.
In the question two statements are given, so we will make two equations with the help of these statements.
We are considering the statement which A says to B “If you give me 10 of your oranges, I will have twice the number of oranges left with you.” From this statement equation \[x + 10 = 2\left( {y - 10} \right)\]is formed.
Simplifying this equation,
\[
  x + 10 = 2y - 20 \\
  x = 2y - 20 - 10 \\
  x = 2y - 30 \\
 \]
Marking this simplified equation as (1) to simplify our further calculation.
\[x = 2y - 30\] …. (1)
Now, we are considering the statement which B replies, “If you give me 10 of your oranges, I will have the same number of oranges as left with you.” From this statement the equation is formed.
Simplifying this equation,
\[
  y = x - 10 - 10 \\
  y = x - 20 \\
 \]
Marking this simplified equation as (2) to simplify our further calculation.
\[y = x - 20\] …. (2)
Now we have to solve equation (1) and (2) to get the value of x and y.
For this we are substituting x from equation (1) into equation (2) and then will solve for y,
\[
  y = 2y - 50 \\
  y - 2y = - 50 \\
   - y = - 50 \\
  y = 50 \\
 \]
Now we have the value of y, substituting this value of y into equation (1) to get the value of x,
\[
  x = 2\left( {50} \right) - 30 \\
  x = 100 - 30 \\
  x = 70 \\
 \]
Now we have the value of x and y.
Thus, person A has 70 oranges and person B has 50 oranges.

Note: In this type of equation we have two-equations which can be solved either by substitution method or by elimination method. In the elimination method, we eliminate any of the variables by using mathematical operations on both the equations simultaneously so that we can find the value of another variable.