Question

The average of 9 numbers is 11 . If each of these 9 numbers is multiplied by 5 and then 5 is added to each of these resultant numbers, then the riew average is : (a) 20 (b) 30 (c) 60 (d) 50

Short answer

Answer: The new average of the 9 numbers is 60.

Step-by-step solution

Find the sum of the original 9 numbers

Since the average of the 9 numbers is 11, we can find their sum by multiplying the average by the number of terms (9). Sum = Average × Number of terms = 11 × 9 = 99.

Find the new numbers after multiplying by 5 and adding 5

To find the new numbers, we will first multiply each of the original numbers by 5 and then add 5 to each of them. To find the new sum, we can multiply the sum of original numbers (99) by 5 and then add 5 to each number (which is the same as adding 5 × 9 to the sum). New sum = (99 × 5) + (5 × 9) = 495 + 45 = 540.

Calculate the new average

To calculate the new average, we will divide the new sum by the number of terms (9). New average = New sum ÷ Number of terms = 540 ÷ 9 = 60. So, the new average of the numbers is 60. The correct answer is (c) 60.

Key concepts

Arithmetic Mean

The arithmetic mean, commonly known as the average, is a fundamental concept in the field of statistics and everyday mathematics. The idea is quite straightforward: you add up a set of numbers and then divide the total by the count of those numbers. For example, consider you have nine different numbers. To find their average, you would sum them all up and then divide that sum by nine. The formula is:

\[\begin{equation}Arithmetic Mean = \frac{{\text{{Sum of numbers}}}}{{\text{{Number of terms}}}}\end{equation}\]
It's a measure of central tendency, showcasing a central value within a dataset. Improving your understanding of calculating the average equips you with a versatile tool applicable across various domains—from evaluating performance grades to analyzing data sets in scientific research.

Mathematical Transformations

Understanding mathematical transformations is crucial for manipulating and analyzing data. In the context of the given problem, a transformation involves two steps: multiplying a set of numbers by a constant factor and then adding a constant value to each product. These are linear transformations because they can be described as functions of the form

\[\begin{equation}\text{{f(x)}} = ax + b\end{equation}\]
where 'a' represents the multiplicative factor and 'b' represents the value added to each term. It's important to recognize how these transformations affect statistical measures such as the mean: multiplying by 'a' scales the average by 'a', and adding 'b' increases the average by 'b'. Familiarity with these transformations allows for quick adjustments to data sets without recalculating each term individually—an essential skill in problem-solving and data analysis.

Problem Solving

Problem-solving is a multi-step process that involves understanding the problem, devising a plan, carrying out the plan, and then looking back to check and extend the work. When improving problem-solving skills, especially in mathematics, it’s beneficial to familiarize oneself with diverse strategies and to practice them regularly.

In the given exercise, the problem-solving strategy involved:

• Understanding that 'average' refers to the arithmetic mean.
• Devising a plan which includes recognizing the initial mean calculation and predicting how a linear transformation will change the mean.
• Executing the plan through simple mathematical operations (multiplication and addition) applied consistently to the set of numbers.
• Checking the results by verifying the transformed average corresponds with one of the provided options.

Problem-solving advice: Always read the problem carefully and look for keywords that indicate specific mathematical operations. Practice varied sets of problems to become adept at recognizing patterns and mathematical relationships. This will help you swiftly apply the correct methods for efficient and accurate problem-solving.