To solve this problem, we need to find the larger number when the sum of two numbers is given, and the relationship between the two numbers is provided. Let's break this down step-by-step:
- Given that the sum of two numbers is 6000. Let's denote the smaller number as \(x\) and the larger number as \(y\). \(y + x = 6000\)
- It is also given that \(\frac{y}{x} = 2\) with a remainder of 600. This indicates a division equation: \(y = 2x + 600\)
- We now have two equations:
- Equation 1: \(y + x = 6000\)
- Equation 2: \(y = 2x + 600\)
- We can substitute the value of \(y\) from Equation 2 into Equation 1: \((2x + 600) + x = 6000\)
- Simplify the equation: \(3x + 600 = 6000\)
- Subtract 600 from both sides: \(3x = 5400\)
- Divide both sides by 3 to solve for \(x\): \(x = 1800\)
- Now substitute the value of \(x\) back into the expression for \(y\): \(y = 2 \times 1800 + 600 = 4200\)
- Verify: The sum of the numbers is \(1800 + 4200 = 6000\), and dividing the larger by the smaller gives: \(\frac{4200}{1800} = 2\) with a remainder of 600, confirming our solution.
Thus, the larger number is 4200.