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How to solve age word problems

Problem

John is 6 years older than his brother. He will be twice as old as his brother in 4 years. How old is John now?

How to solve it

The two unknowns
John's age now
His brother's age now

The two relationships
Use the following relationships to set up an equation or a system of two equations:

(1)

[John's age now] = [his brother's age now] + 6

Or written as

[his brother's age now] = [John's age now] - 6

(2)

[John's age in 4 years] = 2 · [his brother's age in 4 years]

Depending on how the two relationships are used, the problem can be solved by setting up either:
a system of two equations, or
one equation in one variable.

Solution (by setting up an equation in one variable)

Step 1
Let x represent John's age now.

Step 2
Express the other unknown, his brother's age now in terms of x using relationship (1).
his brother's age now = x - 6

Step 3
Express their ages in 4 years in terms of x.

	now	in 4 years
John's age	x	x + 4
his brother's age	x - 6	x - 6 + 4

Step 4
Substitute their ages in four years obtained in step 3 into relationship (2) to set up an equation.
x + 4 = 2(x - 6 + 4)

Step 5
Solve the equation for x.
x + 4 = 2(x - 6 + 4)
x + 4 = 2(x - 2)
x + 4 = 2x - 4
8 = x
x = 8

So John is 8 years old now.

Alternative Solution (by setting up a system of two equations)

Step 1
Let x represent John's age now.
Let y represent his brother's age now.

Step 2
Express their ages in 4 years in terms of x or y.

	now	in 4 years
John's age	x	x + 4
his brother's age	y	y + 4

Step 3
Use relationship (1) to set up the first equation.
x = y + 6
Substitute ages in 4 years into relationship (2) to set up the second equation.
x + 4 = 2(y + 4)

Step 4
Solve the simultaneous equations for x and y.
x = y + 6 .............................(1)
x + 4 = 2(y + 4) .................(2)

Substituting x = y + 4 into equation (2), we have
y + 6 + 4 = 2(y + 4)

Solving the equation for y, we have
y + 6 + 4 = 2(y + 4)
y + 10 = 2y + 8
2 = y
y = 2

Substituting y = 2 into equation (1), we obtain
x = y + 6 = 2 + 6 = 8

The solution to the simultaneous equation is
x = 8
y = 2

So John is 8 years old now.

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