The average of five numbers is 78. The first number is equal to two eleventh of the sum of the other four numbers. What is the value of this first number?

Introduction / Context:
This question combines averages with a fractional relationship between one term and the sum of the remaining terms. It checks whether the learner can work with the definition of average and form equations involving fractions. This type of problem is useful practice for ratio and proportion as well as algebra.

Given Data / Assumptions:

• There are five numbers in total.
• Their average is 78.
• The first number equals two eleventh of the sum of the other four numbers.
• We need to find the value of the first number.

Concept / Approach:
First, use the average to find the total sum of the five numbers. Then denote the first number by a and the sum of the remaining four by S. The condition says a = (2 / 11) * S. From the total sum we know that a + S = 390 because 5 * 78 = 390. These two equations form a system in a and S, which we can solve to get the value of a, the first number.

Step-by-Step Solution:
Average = 78 for five numbers.Total sum of the five numbers = 78 * 5 = 390.Let the first number be a.Let the sum of the other four numbers be S.Then a + S = 390.We are told that the first number is two eleventh of the sum of the other four numbers, so a = (2 / 11) * S.Substitute a from this relation into the total sum expression: (2 / 11) * S + S = 390.Combine like terms: (2 / 11) * S + (11 / 11) * S = (13 / 11) * S.So (13 / 11) * S = 390.Multiply both sides by 11: 13S = 390 * 11 = 4290.Divide by 13: S = 4290 / 13 = 330.Now compute a = (2 / 11) * S = (2 / 11) * 330.This gives a = 2 * 30 = 60.Thus, the first number is 60.

Verification / Alternative check:
Sum of the other four numbers S = 330 and first number a = 60.Check the relation: two eleventh of 330 is (2 / 11) * 330 = 60, which equals a.Check total average: 60 + 330 = 390 and 390 / 5 = 78, which matches the given average.Both conditions hold, confirming the answer.

Why Other Options Are Wrong:
40, 90, and 120 do not satisfy both the total sum and the fractional condition when used as the first number.Only 60 makes the first number exactly two eleventh of the sum of the remaining four numbers while preserving the average of 78.

Common Pitfalls:
Some learners confuse two eleventh of the sum with two eleventh of the average, which is different.Another common mistake is to forget to convert both S and a into a common denominator when adding them, which leads to algebraic errors.Errors in multiplying by 11 or dividing by 13 can also cause incorrect results.

Final Answer:
The value of the first number is 60.