If the larger of two numbers is divided by the smaller, the quotient and the remainder are 2 each. If 5 times the smaller number is divided by the larger, the quotient and the remainder are still 2 each. find the two numbers.
let x be smaller, y be larger
y = 2x+2
5x = 2y+2
-2x + y = 2
5x - 2y = 2
x = 6
y = 14
To solve this problem, let's assume the smaller number as "x" and the larger number as "y".
From the given conditions, we can establish the following equations:
1) When the larger number is divided by the smaller, the quotient and remainder are both 2:
y = 2x + 2
2) When 5 times the smaller number is divided by the larger, the quotient and remainder are both 2:
5x = 2y + 2
Now, let's solve these equations simultaneously to find the values of x and y.
First, we'll substitute the value of y from equation 1 into equation 2:
5x = 2(2x + 2) + 2
Simplifying this equation:
5x = 4x + 4 + 2
5x = 4x + 6
Next, we'll move the 4x term to the left side and the 6 term to the right side:
5x - 4x = 6
x = 6
Now that we've found the value of x, we can substitute it back into equation 1 to determine the value of y:
y = 2x + 2
y = 2(6) + 2
y = 12 + 2
y = 14
Therefore, the two numbers are x = 6 and y = 14.