Module 7: Notes and Solutions to Practice Problems for Unit 3.
This Unit covers age problems. Most of these problems have a time parameter like “In four years” or
“Three years hence”. The “trick “ to solving these is to set up a table as follows:
Age Now
Age Then
You then make an equation out of the information in the “Age Then” column.
Once again, I will set up each problem so that you can see the process to follow. You must solve the
equation. There is no “trick” to the first 3 problems.
1.
A man is 27 years older than his son. The sum of their ages is 45 years. How old is each?
m=
s + 27
m + s = 45
There are 2 equations: 1) m = s + 27 and 2) m + s = 45
Substitute 1) into 2): s + 27 + s = 45
Solve and answer the question.
2.
A mother is three times as old as her daughter. If the sum of their ages is 52, how old is each?
m=
3d
m + d = 52
There are two equations: 1) m = 3d
and
2) m + d = 52
Substitute 1) into 2): 3d + d = 52
Solve and answer the question.
3.
Find the ages of a man and his son if the father is seven years older than three times the son's age
and the sum of their ages is 51 years.
father is means f =
seven years older means + 7
three times the son's age means 3s
the sum of their ages is 51 means f + s = 51
There are 2 equations: 1) f = 3s + 7 and
2) f + s = 51
Substitute 1) into 2): 3s + 7 + s = 51
Solve and answer the question.
4.
A man is now five times as old as his son. Four years hence, the sum of their ages will be 50
years. How old is each one now?
man is now five times as old as his son means m = 5s
Four years hence means
Age Now
Age Then
Man
5s
5s + 4
Son
s
s+4
sum of their ages will be 50 means 5s + 4 + s + 4 = 50
Solve and answer the question.
5.
Vernon is eight years older than Donald. Two years hence, Vernon will be three times as old as
Donald. How old is each one now?
Vernon is eight years older than Donald means V = D + 8
Two years hence means:
Vernon
Donald
Age Now
D+8
D
Age Then
D + 8 + 2 = D + 10
D+2
Vernon will be means D + 10 =
three times as old as Donald means 3(D + 2) or 3D + 6
We now have the equation: D + 10 = 3D + 6
Solve and answer the question.
6.
A man is now six times as old as his son. In four years, he will be four times as old as his son.
How old is each one now?
man is now six times as old as his son means m = 6s
In four years means:
man
son
Age Now
6s
s
Age Then
6s + 4
s+4
he will be means 6s + 4 =
four times as old as his son means s + 4
We now have the equation: 6s + 4 = s + 4
Solve and answer the question.
7.
A man is six years older than three times his son's age. Five years hence, the sum of their ages
will be 56 years. How old is each one now?
man is six years older than three times his son's age means m = 3s + 6
Five years hence means:
Man
Son
Age Now
3s + 6
s
Age Then
3s + 6 + 5 = 3s +11
s +5
the sum of their ages will be 56 years means 3s + 11 + s + 5 = 56
This is the equation.
Solve and answer the question.
8. John is six times as old as Gregory. Four years hence, John will be twice as old as Gregory. How old is
each one now?
John is six times as old as Gregory means J = 6G
Four years hence means:
John
Gregory
Age Now
6G
G
Age Then
6G + 4
G+4
John will be means 6G + 4 =
twice as old as Gregory means 2(G + 4)
The equation is 6G + 4 = 2(G + 4)
Solve and answer the question.
9.
A man is now twice as old as his son. Eighteen years ago he was five times as old as his son.
Find the age of each.
man is now twice as old as his son means m = 2s
Eighteen years ago means:
man
son
Age Now
2s
s
Age Then
2s – 18
s - 18
he was five times as old as his son means 2s – 18 = 5(s – 18)
Solve and answer the question.
10.
A man's age is seven years more than three times his son's age. Eight years hence, he will be six
times as old as his son was 2 years ago. How old is each now?
man's age is seven years more than three times his son's means m = 3s + 7
Eight years hence applies to the man’s age, 2 years ago applies to the son’s age.
man
son
Age Now
3s + 7
s
Age Then
3s + 7 + 8 = 3s + 15
s-2
he will be means 3s + 15 =
six times as old as his son means 6(s – 2)
The equation is: 3s + 15 = 6(s – 2)
Solve and answer the question.
11. Paul is 15 years older than Peter. In ten years, Paul will be twice as old as Peter. How old are they
now?
Paul is 15 years older than Peter means x = y + 15
In ten years means:
Paul
Peter
Age Now
y + 15
y
Age Then
y + 15 + 10 = y + 25
y + 10
Paul will be twice as old as Peter means y + 25 = 2(y + 10)
Solve and answer the question.
12. John is twice as old as Steve. Eight years ago, John was four times as old as Steve. How old are they
now?
John is twice as old as Steve means J = 2S
Eight years ago means:
John
Steve
Age Now
2S
S
Age Then
2S - 8
S-8
John was four times as old as Steve means 2S - 8 = 4(S - 8)
Solve and answer the question.
13. Judy is 8 years older than Janet. The sum of their ages is 62. How old are they?
x =
y+8
x + y = 62
There are two equations: 1) x = y + 8 and 2) x + y = 62
Substitute 1) into 2) y + 8 + y = 62
Solve and answer the question.
14. Paul is 7 years older than John. If 13 is added to Paul’s age and 5 is subtracted from John’s age, Paul
will be twice as old as John. Find their ages.
Paul is 7 years older than John means P = J + 7
13 is added to Paul’s age and 5 is subtracted from John’s age mean:
Paul
John
Age Now
J+7
J
Age Then
J + 7 + 13 = J + 20
J-5
Paul will be twice as old as John means J + 20 = 2(J - 5)
Solve and answer the question.
15. Norman is five years older than Andy. Five years ago the sum of their ages was 25. How old are they
now?
Norman is five years older than Andy means N = A + 5
Five years ago means:
Age Now
Norman
A+5
Andy
A
Age Then
A+5-5=A
A-5
the sum of their ages was 25 means A + A - 5 = 25
Solve and answer the question.
16. Joe is six years older than Mary. Six years ago, the sum of their ages was 74. How old are they now?
This problem is very similar to # 15.
17. Peter is three times as old as Judy. Eight years ago, Peter was five times as old as Judy was. How old
are they now?
Peter is three times as old as Judy means P = 3J
Eight years ago means:
Age Now
Peter
3J
Judy
J
Age Then
3J - 8
J-8
Peter was five times as old as Judy was means 3J - 8 = 5(J - 8)
Slove and answer the question.
18. Peter is three times as old as Bob. In 20 years, Peter will be twice as old Bob. How old are they now?
This problem is very similar to # 12.
19. David is twice as old as Micheal. In five years, the sum of their ages will be 55. How old are they
now.
David is twice as old as Micheal means D = 2M
In five years means:
Age Now
David
2M
Micheal
M
Age Then
2M + 5
M+5
the sum of their ages will be 55 means 2M + 5 + M + 5 = 55
Solve and answer the question.
20. Ken is three times as old as Andy. In six years, the sum of their ages will be 32. How old are they
now?
This problem is very similar to # 19.