Two numbers are in the ratio 3 : 7. If 6 is added to each of them, the new ratio becomes 5 : 9. What are the two original numbers?

Introduction / Context:
This is a ratio transformation problem where two numbers initially have a given ratio, and after adding the same constant to each number, a new ratio is obtained. We must find the original numbers from this information. It tests understanding of how adding a constant affects ratios and how to set up and solve a simple linear equation.

Given Data / Assumptions:
• Original ratio of the two numbers = 3 : 7. • When 6 is added to each number, new ratio = 5 : 9. • Both numbers are positive.

Concept / Approach:
Let the original numbers be 3x and 7x for some positive x. After adding 6 to each, the numbers become 3x + 6 and 7x + 6. The new ratio (3x + 6) : (7x + 6) is given as 5 : 9. We set up a proportion and solve for x. Once x is known, we substitute back to get the two numbers.

Step-by-Step Solution:
Step 1: Let the original numbers be 3x and 7x. Step 2: After adding 6 to each, they become 3x + 6 and 7x + 6. Step 3: Given new ratio (3x + 6) : (7x + 6) = 5 : 9. Step 4: Set up the equation: (3x + 6) / (7x + 6) = 5 / 9. Step 5: Cross-multiply: 9(3x + 6) = 5(7x + 6). Step 6: Expand both sides: 27x + 54 = 35x + 30. Step 7: Rearrange: 27x + 54 − 35x − 30 = 0 ⇒ −8x + 24 = 0. Step 8: Solve for x: −8x = −24 ⇒ x = 3. Step 9: Original numbers are 3x = 3 * 3 = 9 and 7x = 7 * 3 = 21.

Verification / Alternative check:
Check the new ratio: after adding 6, numbers become 9 + 6 = 15 and 21 + 6 = 27. The ratio 15 : 27 simplifies by dividing both terms by 3 to 5 : 9, which matches the given new ratio. The original ratio 9 : 21 simplifies to 3 : 7 after dividing both by 3, confirming both conditions are satisfied.

Why Other Options Are Wrong:
• 11 & 17, 7 & 17 and 13 & 23 do not satisfy both the original 3 : 7 ratio and the transformed ratio 5 : 9 when 6 is added to each number.

Common Pitfalls:
A common mistake is to add 6 directly to the ratio terms (3 and 7) instead of to the actual numbers 3x and 7x. Another error is in cross-multiplication or simplification when solving the linear equation. Always check that both the original and new ratios are satisfied by your final answer.

Final Answer:
The two original numbers are 9 and 21.