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. 10.8, 7.2
• 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

Correct Answer is Option 1: 10.8, 7.2

Step 1: Define Variables

Let the two numbers be represented by x and y. The problem provides two key pieces of information:

• The sum of the numbers is 18: x + y = 18
• The ratio of their sum to their difference is 5:1. Assuming x>y, the difference is x - y. This yields: (x + y) / (x - y) = 5

Step 2: Formulate Equations

The problem can be represented by the following two equations:

1. x + y = 18
2. (x + y) / (x - y) = 5

Substitute x + y = 18 into the second equation:

18 / (x - y) = 5

Step 3: Calculate the Difference

Cross-multiplication simplifies the equation:

18 = 5(x - y)

Divide both sides by 5:

x - y = 18 / 5

x - y = 3.6

We now have:

• x + y = 18
• x - y = 3.6

Step 4: Solve the System of Equations

Add the two equations to eliminate y:

(x + y) + (x - y) = 18 + 3.6

2x = 21.6

Divide by 2:

x = 21.6 / 2

x = 10.8

Substitute x = 10.8 into the first equation to find y:

10.8 + y = 18

y = 18 - 10.8

y = 7.2

Step 5: Verify the Solution

Confirm that x = 10.8 and y = 7.2 satisfy the problem conditions:

• Sum: 10.8 + 7.2 = 18 (Verified)
• Difference: 10.8 - 7.2 = 3.6
• Ratio: 18 / 3.6 = 5 (Verified, matching the 5:1 ratio)