What smallest number must be added to each of the numbers 94, 24, 100 and 26 so that the four resulting numbers are in proportion?

Introduction / Context:This question checks the concept of forming a proportion by adding the same number to several given quantities. The goal is to find a constant k such that the modified numbers are in proportion. Here, four numbers are given, and we require that after adding k to each, the first two and the last two form equal ratios. Problems like this are standard in the ratio and proportion chapter of quantitative aptitude.

Given Data / Assumptions:

• Original numbers are 94, 24, 100 and 26.
• The same number k is added to each of them.
• The resulting numbers 94 + k, 24 + k, 100 + k and 26 + k must be in proportion in the sense that 94 + k : 24 + k = 100 + k : 26 + k.
• We assume all numbers are real and that k is chosen so the ratios are well defined.

Concept / Approach:If four numbers a, b, c and d are in proportion, then a : b = c : d. This can be converted into the equation a * d = b * c. In this question, after adding k, our four numbers become a = 94 + k, b = 24 + k, c = 100 + k and d = 26 + k. Setting a : b = c : d and cross multiplying gives us one algebraic equation in k, which we solve to obtain the required constant. Once k is found, we can quickly check if the condition holds.

Step-by-Step Solution:Let k be the number added to each given number.Then the new numbers are 94 + k, 24 + k, 100 + k and 26 + k.For them to be in proportion, (94 + k) : (24 + k) = (100 + k) : (26 + k).Cross multiply: (94 + k) * (26 + k) = (24 + k) * (100 + k).Expand the left side: (94 + k) * (26 + k) = 94 * 26 + 94k + 26k + k^2.This is 2444 + 120k + k^2.Expand the right side: (24 + k) * (100 + k) = 24 * 100 + 24k + 100k + k^2.This is 2400 + 124k + k^2.Set them equal: 2444 + 120k + k^2 = 2400 + 124k + k^2.Cancel k^2 from both sides and simplify: 2444 + 120k = 2400 + 124k.2444 - 2400 = 124k - 120k gives 44 = 4k, so k = 11.

Verification / Alternative check:Add k = 11 to each number: 94 + 11 = 105, 24 + 11 = 35, 100 + 11 = 111, 26 + 11 = 37.Check the ratios: 105 : 35 = 3 : 1 and 111 : 37 = 3 : 1 as well, since 105 / 35 = 3 and 111 / 37 = 3.Therefore, the four new numbers are indeed in proportion, confirming that k = 11 is correct.

Why Other Options Are Wrong:If we try k = 10, 9, 8 or 12, the resulting ratios do not match exactly on both sides. For example, with k = 10, we would get 104 : 34 and 110 : 36; these fractions are not equal when simplified. Only k = 11 gives equal ratios, so other options are incorrect.

Common Pitfalls:Some students mistakenly form the equation using only three numbers or treat “continued proportion” as requiring (24 + k)^2 = (94 + k)(100 + k), which is a different condition. Others may make algebraic errors while expanding products or forget to cancel k^2 correctly. Always write the proportion clearly, cross multiply carefully, and simplify the resulting linear equation step by step.

Final Answer:The required number to be added to each term is 11.