The ages

The ages of the four sons make an arithmetic sequence, the sum of which is the father's age today. In three years, the father's age will be the sum of the ages of the three eldest sons, and in the next two and three months, the father's age will be the sum of the ages of the three youngest sons. What are the ages of all five people today?

Correct answer:

a =  6
b =  8
c =  10
d =  12
f =  36
x =  2

Step-by-step explanation:


b=a+x
c = b+x
d = c+x
f = a+b+c+d
f+3 = (b+3)+(c+3)+(d+3)
f+5 = 2+(a+5)+(b+5)+(c+5)

a-b+x = 0
b-c+x = 0
c-d+x = 0
a+b+c+d-f = 0
b+c+d-f = -6
a+b+c-f = -12

Row 4 - Row 1 → Row 4
a-b+x = 0
b-c+x = 0
c-d+x = 0
2b+c+d-f-x = 0
b+c+d-f = -6
a+b+c-f = -12

Row 6 - Row 1 → Row 6
a-b+x = 0
b-c+x = 0
c-d+x = 0
2b+c+d-f-x = 0
b+c+d-f = -6
2b+c-f-x = -12

Pivot: Row 2 ↔ Row 4
a-b+x = 0
2b+c+d-f-x = 0
c-d+x = 0
b-c+x = 0
b+c+d-f = -6
2b+c-f-x = -12

Row 4 - 1/2 · Row 2 → Row 4
a-b+x = 0
2b+c+d-f-x = 0
c-d+x = 0
-1.5c-0.5d+0.5f+1.5x = 0
b+c+d-f = -6
2b+c-f-x = -12

Row 5 - 1/2 · Row 2 → Row 5
a-b+x = 0
2b+c+d-f-x = 0
c-d+x = 0
-1.5c-0.5d+0.5f+1.5x = 0
0.5c+0.5d-0.5f+0.5x = -6
2b+c-f-x = -12

Row 6 - Row 2 → Row 6
a-b+x = 0
2b+c+d-f-x = 0
c-d+x = 0
-1.5c-0.5d+0.5f+1.5x = 0
0.5c+0.5d-0.5f+0.5x = -6
-d = -12

Pivot: Row 3 ↔ Row 4
a-b+x = 0
2b+c+d-f-x = 0
-1.5c-0.5d+0.5f+1.5x = 0
c-d+x = 0
0.5c+0.5d-0.5f+0.5x = -6
-d = -12

Row 4 - 1/-1.5 · Row 3 → Row 4
a-b+x = 0
2b+c+d-f-x = 0
-1.5c-0.5d+0.5f+1.5x = 0
-1.333333d+0.333333f+2x = 0
0.5c+0.5d-0.5f+0.5x = -6
-d = -12

Row 5 - 0.5/-1.5 · Row 3 → Row 5
a-b+x = 0
2b+c+d-f-x = 0
-1.5c-0.5d+0.5f+1.5x = 0
-1.333333d+0.333333f+2x = 0
0.333333d-0.333333f+x = -6
-d = -12

Row 5 - 0.33333333/-1.33333333 · Row 4 → Row 5
a-b+x = 0
2b+c+d-f-x = 0
-1.5c-0.5d+0.5f+1.5x = 0
-1.333333d+0.333333f+2x = 0
-0.25f+1.5x = -6
-d = -12

Row 6 - -1/-1.33333333 · Row 4 → Row 6
a-b+x = 0
2b+c+d-f-x = 0
-1.5c-0.5d+0.5f+1.5x = 0
-1.333333d+0.333333f+2x = 0
-0.25f+1.5x = -6
-0.25f-1.5x = -12

Row 6 - Row 5 → Row 6
a-b+x = 0
2b+c+d-f-x = 0
-1.5c-0.5d+0.5f+1.5x = 0
-1.333333d+0.333333f+2x = 0
-0.25f+1.5x = -6
-3x = -6


x = -6/-3 = 2
f = -6-1.5x/-0.25 = -6-1.5 · 2/-0.25 = 36
d = 0-0.33333333333333f-2x/-1.33333333 = 0-0.33333333 · 36-2 · 2/-1.33333333 = 12
c = 0+0.5d-0.5f-1.5x/-1.5 = 0+0.5 · 12-0.5 · 36-1.5 · 2/-1.5 = 10
b = 0-c-d+f+x/2 = 0-10-12+36+2/2 = 8
a = 0+b-x = 0+8-2 = 6

a = 6
b = 8
c = 10
d = 12
f = 36
x = 2

Our linear equations calculator calculates it.

Try another example


Did you find an error or inaccuracy? Feel free to write us. Thank you!







Tips for related online calculators
Do you have a system of equations and are looking for calculator system of linear equations?
Do you want to convert time units like minutes to seconds?

You need to know the following knowledge to solve this word math problem:

Units of physical quantities:

Grade of the word problem:

Related math problems and questions: