Two numbers are in the ratio 3 : 5. If 6 is added to both numbers, the new ratio becomes 2 : 3. What are the two original numbers?

Introduction / Context:
This question deals with two numbers whose ratio changes when a constant is added to both. It tests understanding of ratios, how adding the same quantity to both terms affects the ratio, and how to set up and solve equations based on these conditions.

Given Data / Assumptions:

• Initially, the ratio of the two numbers is 3 : 5.• When 6 is added to each number, the new ratio becomes 2 : 3.• Both numbers are assumed to be positive integers.• We must find the values of the two original numbers.

Concept / Approach:
When two numbers are in the ratio 3 : 5, we can write them as 3k and 5k for some positive constant k. After adding 6 to each, they become 3k + 6 and 5k + 6. We are told that their new ratio is 2 : 3, so (3k + 6) / (5k + 6) = 2 / 3. This leads to a linear equation in k that can be solved using cross multiplication. Once k is determined, we substitute back to find the original numbers.

Step-by-Step Solution:
Step 1: Let the original numbers be 3k and 5k.Step 2: After adding 6 to each, the new numbers are 3k + 6 and 5k + 6.Step 3: The new ratio is given as 2 : 3, so (3k + 6) / (5k + 6) = 2 / 3.Step 4: Cross multiply to eliminate fractions: 3 * (3k + 6) = 2 * (5k + 6).Step 5: Expand both sides: 9k + 18 = 10k + 12.Step 6: Bring all terms involving k to one side: 9k − 10k = 12 − 18.Step 7: This simplifies to −k = −6.Step 8: So k = 6.Step 9: Therefore, the original numbers are 3k = 3 * 6 = 18 and 5k = 5 * 6 = 30.

Verification / Alternative check:
Check the conditions using the found numbers. Initially, the ratio of 18 to 30 is 18 : 30, which simplifies to 3 : 5 when both numbers are divided by 6. This matches the given initial ratio. After adding 6 to each, the numbers become 24 and 36. The new ratio is 24 : 36, which simplifies to 2 : 3 when both are divided by 12. This matches the given new ratio. Since both conditions are satisfied, the solution is verified as correct.

Why Other Options Are Wrong:
• 21 and 35: Their initial ratio is 21 : 35 = 3 : 5, but adding 6 gives 27 and 41, which do not have a ratio of 2 : 3.• 30 and 50: The initial ratio 30 : 50 = 3 : 5, but adding 6 yields 36 and 56, not in the ratio 2 : 3.• 24 and 40: The initial ratio is 3 : 5, but adding 6 gives 30 and 46, which again do not simplify to 2 : 3.

Common Pitfalls:
Some students try to guess the numbers by trial without systematically using algebra, which can be time consuming and error prone. Another common mistake is to misinterpret the new ratio and set up the equation incorrectly, for example by mixing up the order of the numbers. Algebraic errors during cross multiplication or simplification can also lead to wrong values for k. Careful equation setup and simplification step by step ensure accurate results.

Final Answer:
The two original numbers are 18 and 30.