Math  /  Algebra

QuestionIn fourteen years' time, a mother will be twice as old as her son. Four years ago, the sum of their ages was 30 years. Find how old the mother was, when the son was born.

Studdy Solution

STEP 1

1. Let the current age of the mother be M M .
2. Let the current age of the son be S S .
3. In fourteen years, the mother will be twice as old as her son.
4. Four years ago, the sum of their ages was 30 years.
5. We need to find the age of the mother when the son was born.

STEP 2

1. Set up equations based on the problem statements.
2. Solve the equations to find the current ages of the mother and son.
3. Calculate the age of the mother when the son was born.

STEP 3

Set up equations based on the problem statements.
From the first statement, in fourteen years, the mother will be twice as old as her son:
M+14=2(S+14) M + 14 = 2(S + 14)
From the second statement, four years ago, the sum of their ages was 30:
(M4)+(S4)=30 (M - 4) + (S - 4) = 30

STEP 4

Solve the equations to find the current ages of the mother and son.
First, simplify the equation from the first statement:
M+14=2S+28 M + 14 = 2S + 28 M=2S+14 M = 2S + 14
Now simplify the equation from the second statement:
M4+S4=30 M - 4 + S - 4 = 30 M+S=38 M + S = 38
Substitute the expression for M M from the first equation into the second equation:
2S+14+S=38 2S + 14 + S = 38
Combine like terms:
3S+14=38 3S + 14 = 38
Subtract 14 from both sides:
3S=24 3S = 24
Divide by 3:
S=8 S = 8
Substitute S=8 S = 8 back into the equation for M M :
M=2(8)+14 M = 2(8) + 14 M=16+14 M = 16 + 14 M=30 M = 30

STEP 5

Calculate the age of the mother when the son was born.
The age of the mother when the son was born is the current age of the mother minus the current age of the son:
Age of mother when son was born=MS \text{Age of mother when son was born} = M - S =308 = 30 - 8 =22 = 22
The mother was 22 \boxed{22} years old when the son was born.

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