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Example for age word problems

Problem

Carol is five times as old as her brother. She will be three times as old as her brother in two years. How old is carol now?

How to solve it

The two unknowns
Carol's age now
Her brother's age now

The two relationships

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

(1)

[Carol's age now] = 5 · [her brother's age now]

(2)

[Carol's age in 2 years] = 3 · [her brother's age in 2 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 her brother's age now.

Step 2
Express the other unknown, Carol's age now in terms of x using relationship (1).
Carol's age now = 5x

Step 3
Express their ages in 2 years in terms of x.

	now	in 2 years
Her brother's age	x	x + 2
Carol's age	5x	5x +2

Step 4
Substitute their ages in two years obtained in step 3 into relationship (2) to set up an equation.
5x +2 = 3(x + 2)

Step 5
Solve the equation for x.
5x +2 = 3(x + 2)
5x + 2 = 3x + 6
2x = 4
x = 2

Carol's age now is 5x = 5 · 2 = 10.

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

Step 1
Let x represent her brother's age now.
Let y represent Carol's age now.

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

	now	in 2 years
Her brother's age	x	x + 2
Carol's age	y	y +2

Step 3
Use relationship (1) to set up the first equation.
y = 5x
Substitute their ages in 2 years into relationship (2) to set up the second equation.
y + 2 = 3(x + 2)

Step 4
Solve the simultaneous equations for x and y.
y = 5x .............................(1)
y + 2 = 3(x + 2) .................(2)

Substituting y = 5x into equation (2), we have
5x + 2 = 3(x + 2)

Solving the equation for x, we have
5x +2 = 3(x + 2)
5x + 2 = 3x + 6
2x = 4
x = 2

Substituting x = 2 into equation (1), we obtain
y = 5x = 5 · 2 = 10

The solution to the simultaneous equation is
x = 2
y = 10

So Carol's age now is 10.

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