The average of 'x' numbers is 24. If 1/4 of the numbers are increased by 6 each and remaining are decreased by 4 each then what is the new average?

Options

A. `21.5 `
B. `22.5`
C. can't be determined
D. 23

Answer

To solve the problem step by step, we will follow the logical process outlined in the video transcript.

### Step 1: Understand the Given Information
We know that the average of 'x' numbers is 24. This means that the total sum of these numbers can be expressed as:
\[ \text{Sum} = \text{Average} \times \text{Number of Observations} = 24 \times x \]

### Step 2: Assume a Value for 'x'
To simplify calculations, we can assume \( x = 100 \). This is a convenient number because it allows us to easily calculate fractions of 'x' without dealing with decimals.

### Step 3: Calculate the Total Sum
Using our assumption:
\[ \text{Sum} = 24 \times 100 = 2400 \]

### Step 4: Determine the Number of Numbers Increased and Decreased
According to the problem, \( \frac{1}{4} \) of the numbers are increased by 6, and the remaining \( \frac{3}{4} \) are decreased by 4.

- The number of numbers increased:
\[ \frac{1}{4} \times 100 = 25 \]
- The number of numbers decreased:
\[ \frac{3}{4} \times 100 = 75 \]

### Step 5: Calculate the New Sum After Increases and Decreases
**For the 25 numbers increased by 6:**
- Original average of these numbers = 24
- New value after increase = 24 + 6 = 30
- New sum for these 25 numbers:
\[ \text{Sum}_{\text{increased}} = 30 \times 25 = 750 \]

**For the 75 numbers decreased by 4:**
- Original average of these numbers = 24
- New value after decrease = 24 - 4 = 20
- New sum for these 75 numbers:
\[ \text{Sum}_{\text{decreased}} = 20 \times 75 = 1500 \]

### Step 6: Calculate the Total New Sum
Now, we can find the total new sum after the increases and decreases:
\[ \text{Total New Sum} = \text{Sum}_{\text{increased}} + \text{Sum}_{\text{decreased}} = 750 + 1500 = 2250 \]

### Step 7: Calculate the New Average
Finally, we can find the new average by dividing the total new sum by the total number of observations:
\[ \text{New Average} = \frac{\text{Total New Sum}}{x} = \frac{2250}{100} = 22.5 \]

Thus, the new average is **22.5**.

### Summary of Steps
1. Understand the average and sum relationship.
2. Assume a convenient value for 'x'.
3. Calculate the total sum based on the average.
4. Determine how many numbers are increased and decreased.
5. Calculate the new sums after increases and decreases.
6. Find the total new sum.
7. Calculate the new average.