Numbers with shifted ratio: Two numbers are in the ratio 4 : 7. If 5 is subtracted from each number, the ratio becomes 1 : 2. Find the greater number.

Introduction / Context:
Ratio problems with additive shifts are solved by expressing numbers via a common multiplier and then applying the shifted condition. Solving a simple linear equation yields the multiplier and thus the original numbers.

Given Data / Assumptions:

• Original ratio = 4 : 7.
• Subtract 5 from each number ⇒ new ratio = 1 : 2.
• Find the larger original number.

Concept / Approach:
Let the numbers be 4x and 7x. After subtracting 5 from each, the ratio becomes (4x − 5) : (7x − 5) = 1 : 2. Cross-multiply and solve for x.

Step-by-Step Solution:
2(4x − 5) = 1(7x − 5).8x − 10 = 7x − 5 ⇒ x = 5.Numbers are 4x = 20 and 7x = 35; greater number = 35.

Verification / Alternative check:
Check the shifted ratio: (20 − 5) : (35 − 5) = 15 : 30 = 1 : 2, correct.

Why Other Options Are Wrong:

• 15 and 20 are the smaller number or an intermediate value.
• 40 does not satisfy the shifted condition when paired with its counterpart.

Common Pitfalls:

• Applying the shift to only one term or forgetting to subtract from both numbers.

Final Answer:
35