The arithmetic mean of three numbers is 23. The second number is six more than the first and the sum of the second and third terms is 54. Calculate the numbers.
- print Print
- list Cite
Expert Answers
calendarEducator since 2008
write3,662 answers
starTop subjects are Math, Science, and Social Sciences
Let the numbers be:
a, b, c
Then:
(a+b+c)/3 = 23
==> a+ b + c = 69 .......(1)
But we know that:
b = a+ 6
==> a = b-6 .........(2)
also:
b+c = 54.........(3)
Now let us substitute (2) and (3) in (1):
a+ b + c = 69
(b-6) + 54 = 69
==> b = 69 - 54 + 6 = 21
==> b= 21
==> a = b-6 = 21-6 = 15
==> a= 15
==> b+c = 54
==> c = 54- 21 = 54- 21 = 33
==> c= 33
Related Questions
- The first term of a G.P. is 1. The sum of the third and fifth term is 90. Calculate the common...
- 1 Educator Answer
- The sum of the first two terms of an arithmetic progression is 18 and the sum of the first four...
- 2 Educator Answers
- The sum of the first three terms of G.P. is 7 and sum of their squares is 21. Calculate first...
- 1 Educator Answer
- the sum of three numbers in an arithmetic progression is 15 and their product is 105. Find the...
- 1 Educator Answer
- Find the number of terms in the arithmetic series 49 + 42 + 35 + … = 189.Arithmetic sequences and...
- 1 Educator Answer
Let the 1st term or number of the AP be x .
The second number of the AP = x+6 by data.
By another condtion, sum of the 1st 2nd and third numbers = 54.
Threfore the 3rd number = 54 -(first +second) = 54 -(x+x+6) = 54-2x-6 = 48-2x.
Since x , x+6 and 48-2x are inAP, they must have the equal common increment between any two successive terms. Or
2nd term -1st term = 3rd term -2nd term
x+6-x = 48-2x -(x+6)
6 = 42 -3x.
3x= 42-6 =36
3x/x = 36/3
x = 12.
So the three numbers in AP are x, x+6 and x+12
and they are 12,18, 24. Obviously they satisfy the condtion. of their sum = 54 .
We'll note the numbers as a,b,c.
We'll calculate the arithmetic mean:
(a+b+c)/3= 23
We know from enunciation that "The second number is six more than the first" and we'll translate it mathematically as:
b = a + 6 (1)
We also know that "the sum of the second and third terms is 54".
b + c = 54 (2)
Now, we'll substitute the relations (1) and (2) into the arithmetic mean.
(a + 54)/3 = 23
We'll cross multiply:
a+54 = 69
We'll subtract 54 both sides:
a = 69-54
a = 15
We'll substitute a in (1):
b = 15 + 6
b = 21
We'll substitute b in (2):
b+c = 54
21 + c = 54
c = 54 - 21
c = 33
Student Answers