If the integer 76 is divided into four parts that are proportional to 7 : 5 : 3 : 4, what is the numerical value of the smallest part obtained from this division?

Introduction / Context:
In many aptitude exams, questions involving division of a number into parts in a given ratio are very common. Such questions test a candidate's understanding of ratios, proportional division and basic arithmetic manipulation. In this problem, the total number 76 is split into four parts that are proportional to 7, 5, 3 and 4 respectively. We are asked to determine the smallest of these four parts, which is a straightforward application of the concept of ratios and proportional allocation of a total.

Given Data / Assumptions:
• Total value to be divided = 76. • The four parts are in the ratio 7 : 5 : 3 : 4. • Parts are directly proportional to these ratio values. • All parts are positive real numbers.

Concept / Approach:
The key idea is that if a quantity S is divided into parts in the ratio a : b : c : d, then the actual parts are S * a / (a + b + c + d), S * b / (a + b + c + d), and so on. Therefore, we first compute the sum of the ratio terms, then find the common multiplying factor, and finally compute each individual part. The smallest ratio term will correspond to the smallest part, provided the total is positive.

Step-by-Step Solution:
Step 1: Sum of ratio terms = 7 + 5 + 3 + 4 = 19. Step 2: Common multiplying factor = total / sum of ratios = 76 / 19. Step 3: Compute 76 / 19. Since 19 * 4 = 76, the factor is 4. Step 4: Actual parts = 7*4, 5*4, 3*4, 4*4 = 28, 20, 12, 16. Step 5: Among these, the smallest part is 12.

Verification / Alternative check:
We can quickly verify by adding the four parts: 28 + 20 + 12 + 16 = 76, which matches the original total, so the proportional division is correct. Also, the ratio 28 : 20 : 12 : 16 simplifies to 7 : 5 : 3 : 4 if each term is divided by 4. This confirms both the correctness of the factor and the parts.

Why Other Options Are Wrong:
• 16 is one of the parts but it is not the smallest; 12 is smaller. • 19 does not appear as any part; it is only close to the ratio sum. • 15 is not part of the computed distribution and does not satisfy the ratio.

Common Pitfalls:
A frequent mistake is to divide 76 by 4, assuming there are four parts of equal size, ignoring the given ratio. Another common error is to use the smallest ratio term (3) directly as the answer without scaling it by the common factor. Missing the step of summing the ratio terms can also lead to incorrect factors and wrong final parts.

Final Answer:
Hence, the smallest part obtained when 76 is divided in the ratio 7 : 5 : 3 : 4 is 12.