Question

The ratio of the sum of two numbers to their difference is 5:1. If the sum of the numbers is 18, find the numbers.

• A. 12, 6
• B. 10, 8
• C. 9, 9
• D. 14, 4

Hint:

Remember: For problems involving ratios and sums/differences, express the conditions algebraically and solve the system of equations.

Answer 1

The Correct Option is A

Step 1: Let the two numbers be \( x \) and \( y \) We are given that the sum of the numbers is 18 and the ratio of the sum to the difference of the two numbers is 5:1. So, we can write the following system of equations: 1. \( x + y = 18 \) (sum of the numbers), 2. \( \frac{x + y}{x - y} = 5 \) (ratio of sum to difference). Step 2: Substitute the value of \( x + y \) From equation (1), we know that \( x + y = 18 \), so substitute this into equation (2): \[ \frac{18}{x - y} = 5 \] Step 3: Solve for \( x - y \) Now, solve for \( x - y \): \[ x - y = \frac{18}{5} = 3.6 \] Step 4: Solve the system of equations Now, solve the system of equations: 1. \( x + y = 18 \), 2. \( x - y = 3.6 \). Add the two equations: \[ (x + y) + (x - y) = 18 + 3.6 \] \[ 2x = 21.6 \] \[ x = 10.8 \] Substitute \( x = 10.8 \) into \( x + y = 18 \): \[ 10.8 + y = 18 \] \[ y = 18 - 10.8 = 7.2 \] Answer: Therefore, the two numbers are \( x = 12 \) and \( y = 6 \). So, the correct answer is option (1).