Understanding the Average Problem

This problem asks us to find the value of 'q' based on how the average of a set of numbers changes when two new numbers are added. We are given two conditions involving averages, which we can use to solve for the unknown values.

Let's first understand what an average is. The average (or mean) of a set of numbers is calculated by summing up all the numbers and then dividing the sum by the total count of numbers.

Mathematically, the average is given by:

$$\text{Average} = \frac{\text{Sum of numbers}}{\text{Total count of numbers}}$$

Step 1: Finding the Value of 'p'

The first part of the problem gives us the average of six numbers: 5, 3, 9, 11, 29, and (p+1). The average of these numbers is 12.

The six numbers are: 5, 3, 9, 11, 29, p+1.

The total count of these numbers is 6.

The average is given as 12.

Let's find the sum of these numbers:

Sum = $$5 + 3 + 9 + 11 + 29 + (p+1)$$

Sum = $$(5 + 3 + 9 + 11 + 29) + p + 1$$

Sum = $$57 + p + 1$$

Sum = $$58 + p$$

Now, using the average formula:

$$\text{Average} = \frac{\text{Sum of numbers}}{\text{Total count of numbers}}$$

$$12 = \frac{58 + p}{6}$$

To solve for p, multiply both sides by 6:

$$12 \times 6 = 58 + p$$

$$72 = 58 + p$$

Subtract 58 from both sides:

$$p = 72 - 58$$

$$p = 14$$

So, the value of p is 14. This means the sixth number in the original set is p+1 = 14+1 = 15. The original set of numbers is 5, 3, 9, 11, 29, 15.

Step 2: Finding the Value of 'q'

The problem states that when two new numbers, (q – 5) and (q +11), are included, the average increases by 2. The original average was 12, so the new average is 12 + 2 = 14.

The original set had 6 numbers. Two new numbers are added, so the new total count of numbers is $$6 + 2 = 8$$.

The new set of numbers includes the original 6 numbers plus (q – 5) and (q + 11). The sum of the original 6 numbers was $$58 + p$$. Since we found p = 14, the sum of the original 6 numbers is $$58 + 14 = 72$$.

The new sum of 8 numbers is:

New Sum = (Sum of original 6 numbers) + (q – 5) + (q + 11)

New Sum = $$72 + (q - 5) + (q + 11)$$

New Sum = $$72 + q - 5 + q + 11$$

New Sum = $$72 - 5 + 11 + q + q$$

New Sum = $$67 + 11 + 2q$$

New Sum = $$78 + 2q$$

Now, using the average formula for the new set:

$$\text{New Average} = \frac{\text{New Sum of numbers}}{\text{New total count of numbers}}$$

$$14 = \frac{78 + 2q}{8}$$

To solve for q, multiply both sides by 8:

$$14 \times 8 = 78 + 2q$$

$$112 = 78 + 2q$$

Subtract 78 from both sides:

$$112 - 78 = 2q$$

$$34 = 2q$$

Divide both sides by 2:

$$q = \frac{34}{2}$$

$$q = 17$$

Final Answer for q

The value of q is 17.

Revision Table: Key Concepts in Average Problems

Additional Information: Understanding Average Calculations

Average problems are common in mathematics and quantitative aptitude. They often involve finding missing values or understanding how changes in the data set affect the average.

• Calculating Sum: If you know the average and the count of numbers, you can always find the sum using the formula: Sum = Average × Count. This is a powerful tool in these types of problems.
• Working with Variables: When the numbers include variables (like p+1 or q-5), treat them as expressions that you add or subtract like any other number when calculating the sum.
• Changes in Average: An increase in average indicates that the numbers added were, on average, greater than the original average. Conversely, a decrease would mean the added numbers were, on average, less than the original average. In this problem, the average increased from 12 to 14.
• Solving Equations: These problems usually reduce to solving simple linear equations once the sums and counts are set up correctly based on the average formula. Carefully apply algebraic steps to isolate the unknown variable.

By breaking down the problem into steps and using the basic definition and formula of the average, we can systematically find the unknown values.