Solving the Average Age Problem
This problem involves finding the original average age of a group of students after a new person joins and changes the average.
Identify Key Information
- Number of students: 10
- Teacher's age: 30 years
- Increase in average age: 1 year
- Total number of people after teacher joins: 10 + 1 = 11
Define Variables
Let the original average age of the 10 students be A.
Calculate Total Ages
- Original total age of students: $10 \times A$
- Total age after the teacher joins: $(10 \times A) + 30$
Set Up the Equation
The new average age is the total age of the 11 people divided by 11. This new average is also equal to the original average plus 1 year ($A + 1$).
Therefore, the equation is:
$ \frac{10A + 30}{11} = A + 1 $Solve for the Original Average Age (A)
- Multiply both sides by 11: $ 10A + 30 = 11(A + 1) $
- Distribute the 11 on the right side: $ 10A + 30 = 11A + 11 $
- Subtract $10A$ from both sides: $ 30 = A + 11 $
- Subtract 11 from both sides: $ 30 - 11 = A $ $ 19 = A $
Conclusion
The original average age of the students was 19 years.
Similar Questions
- The average of a group of 8 consecutive even numbers is 41. What is the sum of the smallest and largest numbers in the group?
- The average of x and y is 30. The average of x, y, and z is 40. What is the average of x, y, and 2z?
- The average weight of 10 boxes is 15.5 kg. If the heaviest box weighs 21 kg and the lightest weighs 10 kg, what is the average weight of the remaining boxes?
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- The average of a certain number of quantities is 45. If 5 is added to each quantity, what will be the new average?
- 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 of 28 numbers is 77. The average of first 14 numbers is 74 and the average of last 15 numbers is 84. If the 14 th number is excluded, then what is the average of remaining numbers? (correct to one decimal places)
24 students collected money for donation. The average contribution was Rs. 50. Later on, their teacher also contributed some money. Now the average contribution is Rs. 56. The teacher’s contribution is:
Out of 6 numbers, the sum of the first 5 numbers is 7 times the 6 th number. If their average is 136, then the 6 th number is:
The average of five numbers is 30. If one number is excluded, then average becomes 31. What is the excluded number?
The average weight of 49 students in a class is 39 kg. Seven of them whose average weight is 40 kg leave the class and other seven students whose average weight is 54 kg join the class. What is the new average weight (in kg) of the class?
