Answer :
Alright, let's tackle each part of the question step-by-step.
### Part (1)
The mean of the numbers [tex]\(3, 4, 8, 9, x\)[/tex] is 7.
The mathematical formula for the mean is:
[tex]\[ \text{Mean} = \frac{\text{Sum of all numbers}}{\text{Total number of numbers}} \][/tex]
Thus:
[tex]\[ 7 = \frac{3 + 4 + 8 + 9 + x}{5} \][/tex]
First, calculate the sum of the known numbers:
[tex]\[ 3 + 4 + 8 + 9 = 24 \][/tex]
Now substitute the sum into the equation:
[tex]\[ 7 = \frac{24 + x}{5} \][/tex]
To solve for [tex]\(x\)[/tex], multiply both sides by 5:
[tex]\[ 35 = 24 + x \][/tex]
Now, isolate [tex]\(x\)[/tex]:
[tex]\[ x = 35 - 24 \][/tex]
Thus:
[tex]\[ x = 11 \][/tex]
### Part (2)
The median of five consecutive natural numbers is 12.
For five consecutive natural numbers, the median will be the middle number. If the median is 12, then the five numbers are:
[tex]\[ 10, 11, 12, 13, 14 \][/tex]
The mean of these numbers can be calculated as:
[tex]\[ \text{Mean} = \frac{10 + 11 + 12 + 13 + 14}{5} \][/tex]
[tex]\[ \text{Mean} = \frac{60}{5} \][/tex]
[tex]\[ \text{Mean} = 12 \][/tex]
### Part (3)
The numbers [tex]\(4, 6, 8, 9, x\)[/tex] are arranged from smallest to largest, and the mean and median are equal.
First, find the median of these numbers when arranged:
Since there are five numbers, the median will be the third number when they are in order. Given [tex]\(4, 6, 8, 9, x\)[/tex], the median is:
[tex]\[ 8 \][/tex]
Since the mean and median are equal, we set the mean to 8:
[tex]\[ 8 = \frac{4 + 6 + 8 + 9 + x}{5} \][/tex]
Solving for [tex]\(x\)[/tex], multiply both sides by 5:
[tex]\[ 40 = 27 + x \][/tex]
Subtract 27 from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[ x = 40 - 27 \][/tex]
Thus:
[tex]\[ x = 13 \][/tex]
### Part (4)
The mean of five numbers is 27, and they are in the ratio [tex]\(1:2:3:4:5\)[/tex].
Let the numbers be [tex]\(a, 2a, 3a, 4a, 5a\)[/tex]. The mean is given by:
[tex]\[ \text{Mean} = \frac{a + 2a + 3a + 4a + 5a}{5} = 27 \][/tex]
Sum of the numbers is:
[tex]\[ 15a = 27 \times 5 = 135 \][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[ 15a = 135 \][/tex]
[tex]\[ a = \frac{135}{15} = 9 \][/tex]
Now, calculate the five numbers:
[tex]\[ a = 9, \quad 2a = 18, \quad 3a = 27, \quad 4a = 36, \quad 5a = 45 \][/tex]
So, the numbers are:
[tex]\[ 9, 18, 27, 36, 45 \][/tex]
### Part (5)
We need three sets of five numbers such that the median is 4, the mean is 5, and the mode is 3.
Set 1: The median is 4 (middle number), mean is 5 (average) and mode is 3 (most frequently occurring number).
[tex]\[ 3, 3, 4, 5, 10 \][/tex]
Set 2: Similarly,
[tex]\[ 2, 3, 3, 4, 13 \][/tex]
Set 3: Similarly,
[tex]\[ 3, 3, 3, 6, 10 \][/tex]
### Part (6)
The mean of six numbers is 44, and the mean of five of these numbers is 46.
First, find the total sum of the six numbers:
[tex]\[ \text{Total sum of six numbers} = 44 \times 6 = 264 \][/tex]
Next, find the total sum of the five numbers:
[tex]\[ \text{Total sum of five numbers} = 46 \times 5 = 230 \][/tex]
The sixth number can be found by subtracting the sum of the five numbers from the sum of the six numbers:
[tex]\[ \text{Sixth number} = 264 - 230 = 34 \][/tex]
Hence, the sixth number is:
[tex]\[ 34 \][/tex]
### Part (1)
The mean of the numbers [tex]\(3, 4, 8, 9, x\)[/tex] is 7.
The mathematical formula for the mean is:
[tex]\[ \text{Mean} = \frac{\text{Sum of all numbers}}{\text{Total number of numbers}} \][/tex]
Thus:
[tex]\[ 7 = \frac{3 + 4 + 8 + 9 + x}{5} \][/tex]
First, calculate the sum of the known numbers:
[tex]\[ 3 + 4 + 8 + 9 = 24 \][/tex]
Now substitute the sum into the equation:
[tex]\[ 7 = \frac{24 + x}{5} \][/tex]
To solve for [tex]\(x\)[/tex], multiply both sides by 5:
[tex]\[ 35 = 24 + x \][/tex]
Now, isolate [tex]\(x\)[/tex]:
[tex]\[ x = 35 - 24 \][/tex]
Thus:
[tex]\[ x = 11 \][/tex]
### Part (2)
The median of five consecutive natural numbers is 12.
For five consecutive natural numbers, the median will be the middle number. If the median is 12, then the five numbers are:
[tex]\[ 10, 11, 12, 13, 14 \][/tex]
The mean of these numbers can be calculated as:
[tex]\[ \text{Mean} = \frac{10 + 11 + 12 + 13 + 14}{5} \][/tex]
[tex]\[ \text{Mean} = \frac{60}{5} \][/tex]
[tex]\[ \text{Mean} = 12 \][/tex]
### Part (3)
The numbers [tex]\(4, 6, 8, 9, x\)[/tex] are arranged from smallest to largest, and the mean and median are equal.
First, find the median of these numbers when arranged:
Since there are five numbers, the median will be the third number when they are in order. Given [tex]\(4, 6, 8, 9, x\)[/tex], the median is:
[tex]\[ 8 \][/tex]
Since the mean and median are equal, we set the mean to 8:
[tex]\[ 8 = \frac{4 + 6 + 8 + 9 + x}{5} \][/tex]
Solving for [tex]\(x\)[/tex], multiply both sides by 5:
[tex]\[ 40 = 27 + x \][/tex]
Subtract 27 from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[ x = 40 - 27 \][/tex]
Thus:
[tex]\[ x = 13 \][/tex]
### Part (4)
The mean of five numbers is 27, and they are in the ratio [tex]\(1:2:3:4:5\)[/tex].
Let the numbers be [tex]\(a, 2a, 3a, 4a, 5a\)[/tex]. The mean is given by:
[tex]\[ \text{Mean} = \frac{a + 2a + 3a + 4a + 5a}{5} = 27 \][/tex]
Sum of the numbers is:
[tex]\[ 15a = 27 \times 5 = 135 \][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[ 15a = 135 \][/tex]
[tex]\[ a = \frac{135}{15} = 9 \][/tex]
Now, calculate the five numbers:
[tex]\[ a = 9, \quad 2a = 18, \quad 3a = 27, \quad 4a = 36, \quad 5a = 45 \][/tex]
So, the numbers are:
[tex]\[ 9, 18, 27, 36, 45 \][/tex]
### Part (5)
We need three sets of five numbers such that the median is 4, the mean is 5, and the mode is 3.
Set 1: The median is 4 (middle number), mean is 5 (average) and mode is 3 (most frequently occurring number).
[tex]\[ 3, 3, 4, 5, 10 \][/tex]
Set 2: Similarly,
[tex]\[ 2, 3, 3, 4, 13 \][/tex]
Set 3: Similarly,
[tex]\[ 3, 3, 3, 6, 10 \][/tex]
### Part (6)
The mean of six numbers is 44, and the mean of five of these numbers is 46.
First, find the total sum of the six numbers:
[tex]\[ \text{Total sum of six numbers} = 44 \times 6 = 264 \][/tex]
Next, find the total sum of the five numbers:
[tex]\[ \text{Total sum of five numbers} = 46 \times 5 = 230 \][/tex]
The sixth number can be found by subtracting the sum of the five numbers from the sum of the six numbers:
[tex]\[ \text{Sixth number} = 264 - 230 = 34 \][/tex]
Hence, the sixth number is:
[tex]\[ 34 \][/tex]
