Age Problem

Word problem solving in Algebra: Age Problem

Similar to number problem in the next page, here is another worded problem about age. Our link will help us to remind the possible translation to use. Just click it out and you will be guided. 

Age problems usually involves father's age, son, daughter, mother but not limited to. This is another type of worded problem appears in many classroom scene. This is one of the "hard one" if the students is not really exposed to the said. Below were few examples of how to deal with it.

If John's father is twice as old as him(John), How do we translate this statement into Algebraic one?

Well, if John's current age is x,then his father is twice that x, or 2x. Now how do we write about their current ages 7 years ago? So it is by,

John's current age 7 years ago is by, 

x - 7, and 

2x - 7 for his father's. 

If the statement goes like," seven years ago, the father was three times as old as his son, John".

Then we can draw this,

2x - 7 = 3(x - 7), and this can be your working equation. Just find the vale of "x" with Algebraic manipulation. To make it more clear, let's have an actual example:

Example Problem 1:

"John's Dad is three times as old as John. Six years ago he was four times as old. How old are they now?"

So it is stated here that John's Dad has age tripled than John's. And after six years, his Dad was 4 times older than him. If we don't know how to deal with this, it'll became harder. See how vague the statement was. But we are provided link  for the easy translation. So let's proceed for the solution.

Given: 4, 6, the word three times, unknown age

Required: their current ages

Solution:

Let:

x = John's age

3x = his Dad's age

x - 6 = John's age 6 years ago

3x - 6 = his Dad's age 6 years ago.

3x - 6 = 4(x - 6), our working equation and this satisfies to the statement "Eight years ago he was 6 times as old". The word "he" actually refers to Dad's.

So then, 

3x - 6 = 4(x - 6)

3x - 6 = 4x - 24

24- 6 = 4x -3x

18 = x

x = 18, so John is 18 years old,  and

3x = 3(18) = 54, Dad's age.

So their current ages are, John is 18 years old, and his dad is 54 years old.

Example Problem 2:

"Hanna is 6 years older than Amy. In three years, Hanna will be twice as old as Amy. How old is Hanna now?"

There's almost similarities in this problem compared above. A bit of twist can be observed as we go along. The unclear given here is Amy's unknown age which we can represent as "x".

Given: 6, 3, the word twice, unknown age

Required: their current ages

Solution:

Let:

x = Amy's age
x + 6 = Hanna's age. She's 6 years older than Amy. See this link.

x + 3 = Amy's age 3 years from now.

(x + 6) + 3 = Hanna's age 3 years from now.

(x + 6) + 3 = 2(x + 3), working equation. Satisfies the statement "In three years, Hanna will be twice as old as Amy".

To solve for this, we have;

(x + 6) + 3 = 2(x + 3)

x + 6 + 3 = 2x + 6

x +9 = 2x + 6

9 - 6 = 2x + x

3 = x,  or

x = 3, So Amy is 3 years's old. But the question is only for Hanna's age. So;

x + 6 = 3 + 6 = 9, Hanna's age is 9 years.

Example Problem 3:

"Jennifer is 6 years older than Cindy. Six years ago, she was twice as old as her. Find their present ages."

The problem was almost similar to problem number 2. But in this case, we are talking "age years ago", the previous was "age years from now". Although the presentation is quite different, the approach seems the same. Let's solve it then.

Given: 6 years older,6 years ago , the word twice, unknown age

Required: their current ages

Solution:

Let:

x = Cindy's age
x + 6 = Jennifer's age. She's older than Cindy by 6 years.
x - 6 = Cindy's age 6 years ago.
(x + 6) - 6 = Jennifer's age 6 years ago.
(x + 6) - 6 = 2(x - 6), working equation. This satisfies the statement "Six years ago, she was twice as old as her."

Let's solve;

(x + 6) - 6 = 2(x - 6)

x + 6 - 6 = 2x - 12

x = 2x - 12

x = 12 . So x represent Cindy'd age which is found to be 12. So their present ages are;

Cindy: x = 12 years old

Jennifer: x + 6 = 12 + 6 = 18 years old. (18 is 6 more than 12)

Example Problem 4:

"Chaw is twice as old as Lee. If Lee's age will be diminished by 8, and 4 augmented to Chaw's, Chaw will be then four times as old as Lee. What are their ages now?"

This problem is a combination of addition and subtraction based on Algebraic translation. Diminished means "minus" and augmented means "plus". We can solve this one by the same pattern as we did before in previous examples.

Let:

x = Lee's age
2x = Chaw's age
x - 8 = Lee's age diminished by 8
2x + 4 = Chaw's age augmented by 4
2x + 4 = 4(x - 8), working equation. Satisfies the statement "If Lee's age will be diminished by 8, and 4 augmented to Chaw's, Chaw will be then four times as old as Lee".

So,

2x + 4 = 4(x - 8)
2x + 4 = 4x - 32
32 + 4 = 4x - 2x
36 = 2x
18 = x, or
x = 18.

We asked for their present ages, and x represent to Lee's age, so;

Lee's age: x = 18 years old
Chaw's age: 2x = 2(18) = 36 years old.

Pop Quiz: 

Solve the following using this link as guide to your solution:

1. A father is four times as old as his son Mark. In 4 years, Mark's father is three times as old as him. How old are they now?

2. Sean is twice as old as Ricky. If 6 is added to Ricky's age,and Sean subtracted by 16, their ages will be the same. How old are they now?

3. Joan is twice as old as her brother Flynn. After 10 years, the sum of their ages was 46. How old is her brother?

4. Jack's age 20 years from now will be the same as Jill's current age. After ten years, Jill will be twice the age of Jack. What are their ages?

5. Todd's age is three times his cousin Mike. If 20 is added to Mike's age and such number is subtracted from Todd's, they will have the same age. How old are they?

As human, sometimes errors happen. Help me find it and post it here. And click here to post your possible answers. 

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