Find the mean of the median of the first five multiples of 5 and the first 4 multiples of 4 .
The Correct Answer and Explanation is :
Solution:
Step 1: List the first five multiples of 5 and the first four multiples of 4.
- First five multiples of 5: ( 5, 10, 15, 20, 25 )
- First four multiples of 4: ( 4, 8, 12, 16 )
Step 2: Find the median of each set.
- For the multiples of 5 (( 5, 10, 15, 20, 25 )):
- Arrange the numbers (already sorted): ( 5, 10, 15, 20, 25 ).
- The median is the middle number since there are 5 terms.
Median = 15 - For the multiples of 4 (( 4, 8, 12, 16 )):
- Arrange the numbers (already sorted): ( 4, 8, 12, 16 ).
- Since there are 4 terms (even number of terms), the median is the average of the two middle numbers:
[
\text{Median} = \frac{8 + 12}{2} = 10
]
Step 3: Find the mean of the two medians.
- The two medians are ( 15 ) and ( 10 ).
- Mean of the medians:
[
\text{Mean} = \frac{\text{Median of multiples of 5} + \text{Median of multiples of 4}}{2}
]
[
\text{Mean} = \frac{15 + 10}{2} = 12.5
]
Final Answer:
The mean of the medians is 12.5.
Explanation (300 Words):
This problem involves finding the central tendency of two sets of numbers and combining the results mathematically.
To begin, the multiples of 5 (( 5, 10, 15, 20, 25 )) and the multiples of 4 (( 4, 8, 12, 16 )) represent arithmetic sequences with fixed differences. The median of a set provides the central value (or average of central values for even-sized sets) when data is sorted.
In the set of multiples of 5, since there are 5 elements, the middle number, ( 15 ), is the median. In the set of multiples of 4, which has an even count of elements, the median is calculated by averaging the two middle numbers (( 8 ) and ( 12 )), resulting in ( 10 ).
The mean of two values, ( 15 ) and ( 10 ), is the arithmetic average of these two medians. Adding these medians and dividing by 2 yields ( 12.5 ). This approach shows the simplicity and logical structure of arithmetic and statistical methods, combining medians and means for insight into the data.
This exercise highlights the importance of understanding medians and means in analyzing numerical sequences and showcases their utility in practical scenarios.