Average of Five numbers is 20. If Each number is multiplied by 2, What will be the New Average?

Average of Five numbers is 20. If Each number is multiplied by 2, What will be the New Average?

The average of a set of numbers is the sum of the numbers divided by the count of the numbers.

Average = Sum of all numbers / n

The average of five numbers is given as 20.

This means the sum of these five numbers can be calculated using the average formula.

Given the average of five numbers is 20

Sum of the five numbers = Average × n

Sum = 20 × 5 = 100

If each number in the set is multiplied by 2, the new set of numbers will be twice the original numbers.

New numbers: 2x₁, 2x₂, 2x₃, 2x₄, 2x₅

Sum of the new numbers = 2x₁ + 2x₂ + 2x₃ + 2x₄ + 2x₅

This can be factored to 2(x₁ + x₂ + x₃ + x₄ + x₅)

From the previous sum calculation x₁ + x₂ + x₃ + x₄ + x₅ = 100

Therefore 2 x (100) = 200

The new average will be the sum of the new numbers divided by the count of the numbers

New average = Sum of new numbers / n

New average = 200 / 5 = 40

Thus, each number is multiplied by 2, and the new average is 40.

Multiplying Numbers and their Average

When dealing with averages, certain operations can significantly change the result. Understanding how these operations affect the average can be crucial in various contexts, such as statistics, mathematics, and everyday calculations.

Concept of Average

• The average of a set of numbers is a single value that represents the central point of the set.
• It is calculated by dividing the sum of all numbers by the count of the numbers.

Initial Average Calculation

• Consider a problem where we have a set of five numbers with an average of 20.
• To find the sum of these numbers, multiply the average by the number of values.

Effect of Multiplying Each Number

• If each number in the set is multiplied by 2, each value doubles.
• This operation affects the sum of the numbers but not the count.
• When each number is doubled, the sum of the new set of numbers is also doubled.

Determining the New Average

• The new average is found by dividing the new sum by the same count of numbers.
• Multiplying each number in a set by a constant factor multiplies the average by the same factor.