What number must be added to each term of the ratio 7 : 13 so that the ratio becomes 2 : 3?

Introduction / Context:
To transform one ratio into another by adding the same number k to both terms, set up an equation: (7 + k)/(13 + k) = 2/3, then solve for k. This is a direct proportion manipulation.

Given Data / Assumptions:

• Original ratio: 7 : 13
• Target ratio: 2 : 3
• Same k added to both terms.

Concept / Approach:
Cross-multiply and solve the resulting linear equation in k. Ensure integer arithmetic is clean to avoid mistakes.

Step-by-Step Solution:
(7 + k)/(13 + k) = 2/33(7 + k) = 2(13 + k)21 + 3k = 26 + 2k ⇒ k = 5

Verification / Alternative check:
New ratio = (7 + 5) : (13 + 5) = 12 : 18, which reduces to 2 : 3. Works perfectly.

Why Other Options Are Wrong:
1, 2, 3, or 6 do not convert 7 : 13 to 2 : 3 when added to both terms.

Common Pitfalls:
Setting 7/13 = 2/3 directly (ignoring the added k); small arithmetic slips in cross-multiplication.

Final Answer:
5