Sure, let's solve these problems step by step.
### 1. Finding the Number Added to the Group
Given:
- The mean of 9 numbers is 16.
- One more number is added, and the mean remains 16.
Steps:
1. Calculate the total sum of the original 9 numbers.
[tex]\[
\text{Total Sum} = \text{Mean} \times \text{Number of Numbers} = 16 \times 9 = 144
\][/tex]
2. With the addition of one more number, the total number of numbers becomes 10. Since the mean remains 16, this is also the new mean.
3. Calculate the new total sum with the additional number.
[tex]\[
\text{New Total Sum} = \text{New Mean} \times \text{New Number of Numbers} = 16 \times 10 = 160
\][/tex]
4. Find the added number by subtracting the original total sum from the new total sum.
[tex]\[
\text{Added Number} = 160 - 144 = 16
\][/tex]
So, the number that has been added is 16.
### 2. Finding the Sum of the 3rd and 4th Numbers among Seven Consecutive Multiples of 3
Given:
- The mean of seven consecutive multiples of 3 is 30.
Steps:
1. Let the seven consecutive multiples of 3 be [tex]\(3x, 3(x+1), 3(x+2), 3(x+3), 3(x+4), 3(x+5), 3(x+6)\)[/tex].
2. The mean of these numbers is given by:
[tex]\[
\text{Mean} = \frac{\text{Sum of the Numbers}}{\text{Number of Numbers}} = 30
\][/tex]
3. Calculate the sum of these seven multiples of 3.
[tex]\[
\text{Sum} = 3x + 3(x+1) + 3(x+2) + 3(x+3) + 3(x+4) + 3(x+5) + 3(x+6)
\][/tex]
[tex]\[
= 3(x + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6))
\][/tex]
[tex]\[
= 3(7x + (0+1+2+3+4+5+6))
\][/tex]
[tex]\[
= 3(7x + 21)
\][/tex]
[tex]\[
= 21x + 63
\][/tex]
4. Equate the sum to the total sum using the given mean:
[tex]\[
\frac{21x + 63}{7} = 30
\][/tex]
[tex]\[
21x + 63 = 210
\][/tex]
[tex]\[
21x = 147
\][/tex]
[tex]\[
x = 7
\][/tex]
5. Substitute [tex]\(x = 7\)[/tex] to find the specific numbers:
[tex]\[
3x = 3(7) = 21
\][/tex]
[tex]\[
\text{Multiples are: } 21, 24, 27, 30, 33, 36, 39
\][/tex]
6. Identify the 3rd and 4th numbers:
[tex]\[
\text{3rd Number: } 27 = 3(x+2)
\][/tex]
[tex]\[
\text{4th Number: } 30 = 3(x+3)
\][/tex]
7. Find the sum of the 3rd and 4th numbers:
[tex]\[
\text{Sum} = 27 + 30 = 57
\][/tex]
So, the sum of the 3rd and 4th numbers is 57.
