Finding the Fifth Number Using Average

The problem asks us to find the value of the fifth number when we know the average of five numbers and the values of four of those numbers.

We are given:

• The average of five numbers is 56.
• Four of the numbers are 35, 42, 32, and 26.

Understanding the Average Formula

The average (or mean) of a set of numbers is calculated by dividing the sum of the numbers by the count of the numbers.

The formula is:

$$ \text{Average} = \frac{\text{Sum of Numbers}}{\text{Count of Numbers}} $$

Calculating the Total Sum of Five Numbers

Using the average formula, we can find the total sum of the five numbers. We rearrange the formula to:

$$ \text{Sum of Numbers} = \text{Average} \times \text{Count of Numbers} $$

In this case:

$$ \text{Sum of the five numbers} = 56 \times 5 $$

Let's calculate this:

$$ 56 \times 5 = 280 $$

So, the total sum of all five numbers must be 280.

Calculating the Sum of the Four Known Numbers

Next, we need to find the sum of the four numbers that are given:

$$ \text{Sum of four numbers} = 35 + 42 + 32 + 26 $$

Let's add these numbers:

• $35 + 42 = 77$
• $77 + 32 = 109$
• $109 + 26 = 135$

The sum of the four known numbers is 135.

Determining the Fifth Number

The fifth number is the difference between the total sum of the five numbers and the sum of the four known numbers.

$$ \text{Fifth Number} = (\text{Sum of the five numbers}) - (\text{Sum of four numbers}) $$

Substituting the values we calculated:

$$ \text{Fifth Number} = 280 - 135 $$

Calculating the difference:

$$ 280 - 135 = 145 $$

Therefore, the fifth number is 145.

Verification

We can check our answer by calculating the average of the five numbers: 35, 42, 32, 26, and 145.

Sum = $35 + 42 + 32 + 26 + 145 = 135 + 145 = 280$.

Average = Sum / Count = $280 / 5 = 56$.

This matches the given average, confirming our result.