Algebra Help: More Age Word Problems

Over a year ago I wrote an article "Algebra Help: How to Solve Age Word Problems" in which I gave three examples of Algebra word problems centered around two people's ages (often talking about their ages in the past and/or in the future). This article has become quite popular, so I decided to provide three more solved examples to help any Algebra students that may still be struggling with these kinds of math word problems. The key to solving these is to be very careful in how you convert English statements of the word problem into the language of Algebra.

Age Algebra Word Problem #1:
John is 5 years older than his brother. Three years ago he was twice as old. How old is each?

Solution:
As with most Algebra word problems, the choice of variables is very important. Here we need two variables:

J = age of John now
B = age of John's brother now

If it helps you to solve these problems easier, you may want to even write the above statements at the beginning of your solution, to remind you what the variables mean. When we are working with times in the past and the future it can get confusing.

We have two equations that we can write based on the word problem. The first statement looks at the present, the second statement looks at the past.

"John is 5 years older than his brother" can be represented with an Algebraic equation as:

J = 5 + B

"Three years ago he was twice as old" can be represented in Algebra language as:

J - 3 = 2*(B-3)

It is in this second statement that the student can get confused, as we are representing three years in the past, which means you have to take 3 years from both John's age (J) and his brother's age (B).

Simplify, substitute and solve:

J = 3 + 2*B - 6
J = 2*B - 3

5 + B = 2*B - 3
8 = B
J = 5 + 8 = 13

John is currently 13 years old, while his brother is currently 8 years old.

Age Algebra Word Problem #2:
Frank is 6 years younger than his brother. Five years ago he was only one-fourth as old as his brother. Find their ages.

Solution:
F = age of Frank now
B = age of Frank's brother now

Take the two statements and convert them into Algebra equations:

F = B - 6
F - 5 = (1/4)*(B - 5)

Again, the tricky part of these kinds of problems is remembering that when you are working in the past (or the future) you have to subtract or add the same number of years from both variables to get the equation to work.

Substitute, simplify, and solve:

B - 6 - 5 = (1/4)*B - (5/4)
B - 11 = (1/4)*B - (5/4) ---> { multiply both sides by 4 }
4*B - 44 = B - 5
3*B = 39
B = 13
F = 13 - 6 = 7

Frank is currently seven years old, while his brother is thirteen years old.

Age Algebra Word Problem #3:
In three years Nell will be twice as old as Mary is now. Two years ago the sum of their ages was 20. Find their ages.

N = age of Nell now
M = age of Mary now

In this problem, you must be sure to read the problem carefully. The first statement says "as Mary is now", so we do not add any years to M for our first statement, though we do to N.

N + 3 = 2*M

In the second statement, we do subtract the same number of years from both variables.

(N - 2) + (M - 2) = 20

Simplify, substitute, and solve:

N = 2*M - 3
N + M = 20 + 4 = 24
2*M - 3 + M = 24
3*M = 27
M = 9
N = 2*9 - 3 = 15

Mary is currently nine years old, while Nell is currently fifteen years old.

Blessings!

Source
Virgil S. Mallory. A First Course in Algebra (1943)