Two numbers are in the ratio 3 : 5. If 9 is subtracted from each, the new numbers are in the ratio 12 : 23. What is the smaller number?

Problem restatementOriginal numbers are 3k and 5k. After subtracting 9 from each, the ratio becomes 12 : 23. Find the smaller original number.

Given data

• Original: 3k, 5k.
• After subtraction: (3k − 9) : (5k − 9) = 12 : 23.

Concept/ApproachCross-multiply to solve for k, then compute 3k.

Step-by-step calculation 23(3k − 9) = 12(5k − 9) 69k − 207 = 60k − 108 9k = 99 ⇒ k = 11 Smaller number = 3k = 33

Verification/AlternativeCheck new ratio: (33 − 9) : (55 − 9) = 24 : 46 = 12 : 23.

Common pitfalls

• Using (3k − 9)/(5k − 9) = 23/12 inadvertently (reversing the ratio).

Final Answer33