Answer :
To find the smallest number among the 8 consecutive numbers whose average is 107.5, we will go through the following steps:
1. Calculate the total sum of the 8 numbers:
- Given that the average of the 8 consecutive numbers is 107.5.
- We use the formula: [tex]\(\text{sum} = \text{average} \times \text{number of terms}\)[/tex].
- [tex]\(\text{total sum} = 107.5 \times 8 = 860.0\)[/tex].
2. Express the sum in terms of the smallest number:
- Let's denote the smallest number in this sequence as [tex]\(x\)[/tex].
- Since the numbers are consecutive, they can be written as [tex]\(x, x+1, x+2, x+3, x+4, x+5, x+6, x+7\)[/tex].
- The sum of these consecutive numbers is: [tex]\(x + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6) + (x+7) = 8x + 28\)[/tex].
3. Set up the equation based on the total sum:
- We know the sum of these 8 consecutive numbers should equal 860.0.
- Hence, we write the equation: [tex]\(8x + 28 = 860.0\)[/tex].
4. Solve for [tex]\(x\)[/tex]:
- Subtract 28 from both sides of the equation: [tex]\(8x = 860.0 - 28\)[/tex].
- [tex]\(8x = 832.0\)[/tex].
- Divide both sides by 8: [tex]\(x = \frac{832.0}{8}\)[/tex].
- [tex]\(x = 104\)[/tex].
So, the smallest number among the 8 consecutive numbers is 104.
Therefore, the correct answer is:
b. 104
1. Calculate the total sum of the 8 numbers:
- Given that the average of the 8 consecutive numbers is 107.5.
- We use the formula: [tex]\(\text{sum} = \text{average} \times \text{number of terms}\)[/tex].
- [tex]\(\text{total sum} = 107.5 \times 8 = 860.0\)[/tex].
2. Express the sum in terms of the smallest number:
- Let's denote the smallest number in this sequence as [tex]\(x\)[/tex].
- Since the numbers are consecutive, they can be written as [tex]\(x, x+1, x+2, x+3, x+4, x+5, x+6, x+7\)[/tex].
- The sum of these consecutive numbers is: [tex]\(x + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6) + (x+7) = 8x + 28\)[/tex].
3. Set up the equation based on the total sum:
- We know the sum of these 8 consecutive numbers should equal 860.0.
- Hence, we write the equation: [tex]\(8x + 28 = 860.0\)[/tex].
4. Solve for [tex]\(x\)[/tex]:
- Subtract 28 from both sides of the equation: [tex]\(8x = 860.0 - 28\)[/tex].
- [tex]\(8x = 832.0\)[/tex].
- Divide both sides by 8: [tex]\(x = \frac{832.0}{8}\)[/tex].
- [tex]\(x = 104\)[/tex].
So, the smallest number among the 8 consecutive numbers is 104.
Therefore, the correct answer is:
b. 104
