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The average of five consecutive numbers is n. If the next two numbers are also included the average will
A) increase by 1.4
B) increase by 2
C) increase by 1
D) remain the same
Answer
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Verified
Verified
Hint: Assume the number to be $x$ and then note the next numbers and then accordingly find the new average. Apply an average formula.
Complete step-by-step answer:
Let us assume the first number to be $x$. Therefore the next four consecutive numbers are $x + 1,x + 2,x + 3,x + 4$. Average is the sum of observations divided by the number of observations.
Average = Sum of Observations/ Number of Observations
Applying this concept, we get $\dfrac {{x + x + 1 + x + 2 + x + 3 + x + 4}}{5} = \dfrac {{5x + 10}}{5} = n$
Therefore $5x + 10 = 5n$ and after solving this equation we get $x = n - 2$. When we add the next two numbers, we get $5x + 10 + x + 5 + x + 6 = 7x + 21$. Therefore, the new average is $\dfrac {{7x + 21}}{7} = x + 3$. Substituting the value of $x = n - 2$, the value of the new average in terms of n is $n - 2 + 3 = n + 1$.
The previous average of numbers is $n$ and the new average is $n + 1$. So, the average increases by 1.
So, the correct option is option C.
Note: We could have easily solved the problem by assuming any five consecutive numbers but, this was only possible in this case because they have given five numbers. It would have been a big problem for us if it were fifty, hundred, two hundred or even a thousand numbers or more than that. Thus, using algebra by assuming a variable would help us greatly in this field because it would help us reduce the load of our calculations and give the answer quickly.
Complete step-by-step answer:
Let us assume the first number to be $x$. Therefore the next four consecutive numbers are $x + 1,x + 2,x + 3,x + 4$. Average is the sum of observations divided by the number of observations.
Average = Sum of Observations/ Number of Observations
Applying this concept, we get $\dfrac {{x + x + 1 + x + 2 + x + 3 + x + 4}}{5} = \dfrac {{5x + 10}}{5} = n$
Therefore $5x + 10 = 5n$ and after solving this equation we get $x = n - 2$. When we add the next two numbers, we get $5x + 10 + x + 5 + x + 6 = 7x + 21$. Therefore, the new average is $\dfrac {{7x + 21}}{7} = x + 3$. Substituting the value of $x = n - 2$, the value of the new average in terms of n is $n - 2 + 3 = n + 1$.
The previous average of numbers is $n$ and the new average is $n + 1$. So, the average increases by 1.
So, the correct option is option C.
Note: We could have easily solved the problem by assuming any five consecutive numbers but, this was only possible in this case because they have given five numbers. It would have been a big problem for us if it were fifty, hundred, two hundred or even a thousand numbers or more than that. Thus, using algebra by assuming a variable would help us greatly in this field because it would help us reduce the load of our calculations and give the answer quickly.
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