Three numbers are in the ratio 5 : 6 : 8 and their total sum is 380. What is the value of the smallest of these three numbers?

Introduction / Context:
This aptitude question on ratio and proportion tests how to convert a given ratio of three numbers into actual values when their total sum is known. It also checks basic understanding of how to identify the smallest number once all the individual values are calculated.

Given Data / Assumptions:

• The ratio of the three numbers is 5 : 6 : 8.• The sum of the three numbers is 380.• The numbers are positive real numbers and follow the given ratio exactly.• We are asked to find the smallest of these three numbers.

Concept / Approach:
The core idea is that if three numbers are in the ratio a : b : c, then they can be written as a*k, b*k, c*k for some common multiplier k. Their sum is then (a + b + c) * k. By equating this to the given total, we can solve for k. Once k is known, each individual number can be obtained by multiplying the ratio parts by this k. Finally, we compare the three values to identify the smallest number.

Step-by-Step Solution:
Step 1: Let the three numbers be 5k, 6k and 8k.Step 2: Their sum is 5k + 6k + 8k = 19k.Step 3: We are told that the sum of the three numbers is 380, so 19k = 380.Step 4: Solve for k by dividing both sides of the equation by 19: k = 380 / 19 = 20.Step 5: Now compute each number:• First number = 5k = 5 * 20 = 100.• Second number = 6k = 6 * 20 = 120.• Third number = 8k = 8 * 20 = 160.Step 6: Compare the three values. The smallest among 100, 120 and 160 is 100.

Verification / Alternative check:
We can quickly verify the result by checking the ratio and the sum. The three numbers we found are 100, 120 and 160. Their ratio is 100 : 120 : 160. Dividing each by 20 gives 5 : 6 : 8, which matches the given ratio. Also, the sum is 100 + 120 + 160 = 380, which matches the given total. Both the ratio and the sum are correct, so the calculation is consistent and the smallest number must indeed be 100.

Why Other Options Are Wrong:
• 80: If 80 were the smallest number, the other numbers scaled from the given ratio would not sum to 380 and the ratio would not remain 5 : 6 : 8.• 120: This is actually the middle number, not the smallest. The ratio and total confirm that 120 is the second value.• 140: This value does not appear among the three numbers obtained from the correct multiplier and therefore cannot be the smallest number.

Common Pitfalls:
Students sometimes try to guess numbers instead of using the multiplier method. Another common mistake is to divide the total by the wrong sum of ratio parts or to miscalculate the sum as something other than 19. Some learners also forget that the ratio parts represent relative sizes and must all be multiplied by the same k. Finally, a few learners might accidentally identify the largest or middle number instead of carefully choosing the smallest one after calculation.

Final Answer:
The smallest of the three numbers is 100.