To find the sixth number, we first calculate the total sum of all 11 numbers and the sums of the two subgroups.
Calculate Total Sum of 11 Numbers
The average of 11 numbers is 61.
Total Sum = Average $\times$ Count
Total Sum = $61 \times 11 = 671$
Calculate Sum of First Six Numbers
The average of the first six numbers is 57.
Sum of First Six = Average $\times$ Count
Sum of First Six = $57 \times 6 = 342$
Calculate Sum of Last Six Numbers
The average of the last six numbers is 65.
Sum of Last Six = Average $\times$ Count
Sum of Last Six = $65 \times 6 = 390$
Determine the Sixth Number
The sum of the first six numbers and the sum of the last six numbers includes the sixth number twice. The total sum of all 11 numbers includes the sixth number only once.
Therefore:
Sum of First Six + Sum of Last Six = Total Sum of 11 Numbers + Sixth Number
Substituting the calculated values:
$342 + 390 = 671 + \text{Sixth Number}$
$732 = 671 + \text{Sixth Number}$
Solving for the Sixth Number:
Sixth Number = $732 - 671$
Sixth Number = $61$
The sixth number is 61.
Similar Questions
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- A group of 5 friends went to a restaurant. If 4 of them paid Rs. 200 each and the fifth paid Rs. x, the average bill per person came out to be Rs. 210. What is the value of x?
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Important Questions from Average
The average height of 20 students of class 8 is 152 cm and the average height of 15 students of class 9 is 168 cm. What is the average height (to the nearest cm) of the students of both classes?
The average of 4, 6, 8, 12 and x is 7 and the average of x, 9, 13, 15 and y is 9. What is the value of 2x - 3y?
The average weight of 20 girls in a school was 52 kg. Two new students of weight 54 kg and 50 kg were admitted. The ratio of this new average to the old one is:
If the average of two numbers is 13 and the square root of their product is 12, then the difference between the numbers is:
If the average of 5 consecutive odd integers in increasing order is 11 , then the average of the last 3 of them is:
