What single number should be added to each of the numbers 35, 115, 53 and 165 so that the resulting four numbers are in proportion, that is, 35 + x : 115 + x = 53 + x : 165 + x?

Introduction / Context:
This question asks for a constant to be added to four given numbers so that they form a proportion. In proportion notation, a : b = c : d means a/b = c/d. When the same number x is added to all of them, we want (35 + x) : (115 + x) = (53 + x) : (165 + x). This type of problem tests setting up and solving a proportion equation in one variable.

Given Data / Assumptions:

• We have four base numbers: 35, 115, 53 and 165.
• We add the same number x to each: 35 + x, 115 + x, 53 + x, 165 + x.
• These must be in proportion: (35 + x) : (115 + x) = (53 + x) : (165 + x).
• We must find the value of x.

Concept / Approach:
When four numbers A, B, C and D are in proportion, A/B = C/D. So we set (35 + x)/(115 + x) equal to (53 + x)/(165 + x) and cross-multiply to get a linear equation in x. Solving this equation gives the required number that must be added to all four original numbers to make them proportional.

Step-by-Step Solution:

Verification / Alternative check:
Substitute x = 10 back. New numbers: 45, 125, 63, 175. Check proportion: 45/125 = 0.36 and 63/175 = 0.36. Since both ratios are equal, the four numbers 45, 125, 63 and 175 are proportional, confirming that x = 10 is correct.

Why Other Options Are Wrong:

• Values 12, 8 and 6 do not make the two ratios equal when substituted for x.
• For example, if x = 8, we get 43/123 and 61/173, which are clearly not equal.

Common Pitfalls:
Common errors include mis-expanding the products when cross-multiplying or incorrectly simplifying terms, leading to a wrong value of x. Some students may also misinterpret the phrase “continued proportion” and try to impose extra conditions. The key is to stick to the clear proportional equation and perform algebra carefully.

Final Answer:
The number that must be added is 10.