What is the average of all numbers from 1 to 150 that end in 5 and 7?
Answers
Answer:
75
Step-by-step explanation:
The average (arithmetic mean) of a set of n numbers is equal to the sum of the numbers divided by n. The sum from 1…n = n(n+1)/2.
1+2+…+150 = 150(151)/2. Assuming that the first natural number is 1, then the average = 150(151)/[(2)(150)] = 75.5.
(If you count 0 as a natural number, then the average is (150(151)/[(2)(151)] = 75.
Given,
The number range is = 1 to 150
To find,
The average of all numbers in given range that end in 5 and 7
Solution,
We can simply solve this mathematical problem by using the following mathematical process.
Instead of counting the total number of the numbers that end in 5 and 7, we can simply use the AP formula also.
Now, for numbers end in 5 :
First number (a) = 5
Second number = 15
Last number or nth term (aₙ) = 145
Common difference (d) = Second number - First number = 15-5 = 10
Number of terms (n) = ?
Now,
aₙ = a + (n-1)d
145 = 5 + (n-1)×10
145 = 5+10n-10
145 = 10n-5
10n = 145+5
10n = 150
n = 15
Now, for the numbers end in 7 :
First number (a) = 7
Second number = 17
Last number or nth term (aₙ) = 147
Common difference (d) = Second number - First number = 17-7 = 10
Number of terms (n) = ?
Now,
aₙ = a + (n-1)d
147 = 7 + (n-1)×10
147 = 7+10n-10
147 = 10n-3
10n = 147+3
10n = 150
n = 15
Now,
In case of numbers that end with 5 :
n = 15
a = 5
d = 10
So, Sum of all the numbers that end with 5 :
= n/2 [2a+(n-1)×d]
= 15/2 [2×5+(15-1)×10]
= 15/2 (10+140)
= 15/2 × 150
= 1125
In case of numbers that end with 7 :
n = 15
a = 7
d = 10
So, Sum of all the numbers that end with 5 :
= n/2 [2a+(n-1)×d]
= 15/2 [2×7+(15-1)×10]
= 15/2 (14+140)
= 15/2 × 154
= 1155
Now,
Total sum = (1125+1155) = 2280
So, there are 15 numbers that end with 5 and 15 numbers that end with 7
Thus, Total numbers = (15+15) = 30
Average = (2280÷30) = 76
Hence, the average will be 76