Question

The average of a set of whole numbers is \(27.2 .\) When the \(20 \%\) of the elements (ie numbers) are eliminated from the set of numbers then the average becomes \(34 .\) The number of elements in the new set of numbers can be : (a) 27 (b) 35 (c) 52 (d) 63

Short answer

Answer: b. 35

Step-by-step solution

Calculate the ratio of elements

Let x be the number of elements in the original set. When 20% of the elements are eliminated, 80% of the elements remain in the set. So, let y be the number of elements in the new set. And, express y as 80% of x. y = 0.8x

Calculate the total sum before and after elimination

The total sum before elimination (S_b) can be expressed as the product of the average (27.2) and the number of elements in the original set (x). Similarly, the total sum after elimination (S_a) can be expressed as the product of the new average (34) and the number of elements in the new set (y). S_b = 27.2x S_a = 34y

Set up and solve the equation

We know that the total sum remains the same before and after elimination, except for the 20% of the elements that were eliminated. Thus, we can set up the equation: S_b - 20% of S_b = S_a. 27.2x - 0.2(27.2x) = 34y 0.8(27.2x)= 34(0.8x) Now, simplify and solve for x: 0.8(27.2)x = 34(0.8)x 21.76x = 27.2x x = 35

Calculate the number of elements in the new set

Use the value of x (35) obtained in step 3 to calculate the number of elements in the new set (y = 0.8x): y = 0.8 * 35 y = 28 The number of elements in the new set is not among the provided options. However, we could have misunderstood the question, and it asks the number of elements in the **original** set (if 20% of elements are eliminated) instead of the new set. In that case, the answer would be: x = 35 (option b)

Key concepts

Arithmetic Mean

The arithmetic mean, more commonly referred to as the average, is a widely used concept in statistics. It provides a simple way to summarize a set of numbers with a single value. To calculate the arithmetic mean, you add up all the elements of the set and then divide by the number of elements present. In formula terms, it is expressed as: \[ \text{Arithmetic Mean} = \frac{\text{Sum of all elements}}{\text{Number of elements}} \]This measure helps in understanding the central tendency of a dataset, offering insight into the typical value one might expect to find.

Percentage

Percentage is a way to express a number as a fraction of 100, which makes it easier to compare ratios. In terms of the problem, when it says 20% of the elements were eliminated, it means that 20 out of every 100 elements are removed. This concept is essential when calculating changes in quantity or distribution. Some key points about percentages include:

• To convert a percentage to a decimal, divide by 100.
• Percentages help compare relative sizes or changes swiftly.
• They are essential in scaling values in various kinds of data analysis.

Problem Solving

Problem solving in math involves a systematic approach to finding a solution. For this exercise, identifying the relationship between the original and new average is critical. Solving this involves understanding:

• Establishing what the problem is asking for – here, how many elements are left after the removal.
• Identifying known values, such as the original mean and the changed mean after removing 20%.
• Calculating using relevant formulas, such as expressing parts of the equation in terms of x and y.
• Ensuring consistency in units and calculations, keeping track of percentages and fractions involved.

The solution involves iterative steps, verifying and ensuring that the answer aligns with the problem's requirements.

Number of Elements

The number of elements in a dataset refers to the total count of items or numbers that are being considered. In exercises involving mean and averages, knowing the number of elements is crucial since it affects calculations directly.In this particular problem, calculating the exact number of elements involves understanding how percentage change impacts the initial dataset size. By expressing the number of remaining elements with the equation \[ y = 0.8x \]where y is the number of elements after removal, students can determine how many items remain by first quantifying the original set size and applying the percentage decrease.