The average of a set of four numbers is 78. If three of the numbers in the set are 71, 74, and 83, what is the fourth number?

Answer

To find the fourth number in the set, we can follow these steps:

1. **Understand the average formula**: The average of a set of numbers is calculated by dividing the sum of all the numbers by the total count of numbers. In this case, we have four numbers.

2. **Set up the equation**: We know that the average of the four numbers is 78. Therefore, we can write the equation:
\[ \frac{a + b + c + d}{4} = 78 \]
where \(a\), \(b\), \(c\), and \(d\) are the four numbers.

3. **Multiply to eliminate the fraction**: To find the sum of the four numbers, we multiply both sides of the equation by 4:
\[ a + b + c + d = 4 \times 78 \]
This simplifies to:
\[ a + b + c + d = 312 \]

4. **Substitute known values**: We know three of the numbers: \(a = 71\), \(b = 74\), and \(c = 83\). We can substitute these values into the equation:
\[ 71 + 74 + 83 + d = 312 \]

5. **Calculate the sum of the known numbers**: First, we need to find the sum of \(71\), \(74\), and \(83\):
\[ 71 + 74 + 83 = 228 \]

6. **Solve for the fourth number**: Now, we can substitute the sum back into the equation:
\[ 228 + d = 312 \]
To find \(d\), we subtract \(228\) from \(312\):
\[ d = 312 - 228 \]
This gives us:
\[ d = 84 \]

7. **Conclusion**: The fourth number is \(84\).