The average of 5 numbers is 4 . A sixth number is added to the set which increases the mean to 7 .What is the 6th number

Accepted Solution

Answer:The sixth number is 22. Surprisingly, the exact values of the first five numbers do not matter. See the explanation.Step-by-step explanation:Let the sixth number be [tex]x[/tex]. The average of [tex]n[/tex] numbers is equal to the sum of these number divided by [tex]n[/tex]. That is:[tex]\displaystyle \text{Average} = \frac{\text{Sum of Entries}}{\text{Numbers of Entries}}[/tex].Working backwards (multiply both sides by the denominator, [tex]n[/tex]) to find a formula for the sum of the numbers:[tex]\text{Sum of Entries} = n \cdot \text{Average of Entries}[/tex].Apply this formula to find the sum of the first five numbers:[tex]\underbrace{4}_{\text{Avg. of}\atop\text{entries}} \times \underbrace{5}_{\text{Num. of}\atop \text{entries}} = 20[/tex].If [tex]x[/tex] represents the value of the sixth number, the sum of the first six numbers will be equal to:[tex]20 + x[/tex].The average of the first six number will thus be equal to:[tex]\displaystyle \text{Average of the first six numbers} = \frac{\text{Sum of Entries}}{\text{Numbers of Entries}}= \frac{20 + x}{6}[/tex].However, the question states that the average of the first six numbers is equal to [tex]7[/tex]. In other words,[tex]\displaystyle \frac{20 + x}{6} = 7[/tex].Multiply both sides by the denominator, [tex]6[/tex]:[tex]20 + x = 42[/tex].[tex]x = 22[/tex].In other words, the sixth number is [tex]22[/tex].