The problem asks us to find the average of $(p + q)$ numbers given specific averages for $p$ numbers and $q$ numbers.
Calculating Average of (p + q) Numbers
Step 1: Find the sum of the first p numbers.
We are given that the average of $p$ numbers is $q^2$.
Using the formula: Sum = Average × Count
Sum of the first $p$ numbers = $p × q^2 = pq^2$.
Step 2: Find the sum of the next q numbers.
We are given that the average of $q$ numbers is $p^2$.
Sum of the next $q$ numbers = $q × p^2 = qp^2$.
Step 3: Calculate the total sum and total count.
The total sum of all $(p + q)$ numbers is the sum of the two groups:
Total Sum = $pq^2 + qp^2$
The total count of numbers is $p + q$.
Step 4: Calculate the average of (p + q) numbers.
Average = Total Sum / Total Count
Average = $(pq^2 + qp^2) / (p + q)$
Factor out $pq$ from the numerator:
Average = $pq(q + p) / (p + q)$
Simplify by cancelling the $(p + q)$ term:
Average = $pq$
Therefore, the average of $(p + q)$ numbers is $pq$.
Important Questions from Average (Notes)
- The average age of a group of boys and girls is 18 years. If the average age of boys is 20 years and that of girls is 15 years, then what is the percentage of boys in the group?
- The average of 101 consecutive odd numbers is 303. Find the largest number.
- For three consecutive years, the cost of a product were Rs. $134$ per litre, Rs.$268$ per litre and Rs.$335$ per litre respectively. If a common man spends an average of Rs. $14740$ per year on that product, then what is the average cost of that product per litre for the three years? (In Rs. - upto two decimals)
- If the average of six consecutive even numbers is 13, then the average of the next three even numbers is:
- Raju obtained 75 and 85 marks in mathematics and physics, respectively. Find the marks he must get in chemistry to have an average of 80 marks in all the three subjects.
