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# Age Word Problems

Lesson Plans > Mathematics > Algebra > Word Problems

## Slide Show

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John's age now is twice what it was ten years ago. How old is John now?

Georgia age is ten more than three times what it was 30 years ago. How old will she be in ten years?

Mark is 12 years older than David was twenty years ago, and David's age now is how old Mark will be in  eight years. How old is David?

Ollie's age in two years will be twice the sum of Evan's age and Evan's age ten years ago. In eighteen years, Ollie will be twice as old as Evan. How old is Ollie?

Joan is 5 years older than Betty was last year. In twenty four years, Betty will be twice as old as Joan is now. What is the sum of their ages?

James is half as old now as he will be when he is 60 years older than he was six years before he was half as old as he is now.  How old will James be when his age is twice what it was 10 years after he was half his current age?

## Age Word Problems One of the things I love about "age" problems is that they require students to not just convert sentences to equations, but also to identify different time frames in which parts of the problem may occur. It is helpful for students to recognize that for any given person in a problem, there may be multiple "unknowns." We might not assign a separate variable for each unknown, but it's helpful to write out each one. For example, in the following statement, we have one person, but two unknowns:

John's age now is twice what it was ten years ago. How old is John now?

Unkowns
John's age NOW: J
John's age TEN YEARS AGO: J - 10

Now we use this to write an equation:

J = 2(J - 10). From here, students can solve the simple equation

Here is another example with one person, but with three unknowns:

Georgia age is ten more than three times what it was 30 years ago. How old will she be in ten years?

Unknowns
Georgia's age NOW: G
Georgia's age 30 YEARS AGO: G - 30
Georgia's age IN TEN YEARS: G + 10

G = 10 + 3(G - 30)
G = 10 + 3G - 90
2G = 80
G = 40

Therefore, in ten years, she will be 40 + 10 = 50 years old.

The following problem involves two people:

Mark is 12 years older than David was twenty years ago, and David's age now is how old Mark will be in  eight years. How old is David?

Unknowns
Mark's age NOW: M
Mark's age IN EIGHT YEARS: M + 8
David's age NOW: D
David's age  TWENTY YEARS AGO: D - 20

The equations, then, are M = 12 + D - 20 and D = M + 8

Combining like terms in the first equation makes it evident that these are actually different forms of the same equation, which results in an indeterminate system.

Ollie's age in two years will be twice the sum of Evan's age and Evan's age ten years ago. In eighteen years, Ollie will be twice as old as Evan. How old is Ollie?

Have students take a close look at the last statement. Are both of those ages eighteen years in the future? Or just one? Since a time frame is not specified for Evan's age, we assume it to be the same as the time frame for Ollie's age - 18 years in the future.

Unknowns
Ollie's age NOW: L
Ollie's age IN TWO YEARS: L
Ollie's age IN TEN YEARS: L + 18
Evan's age NOW: E
Evan's age TEN YEARS AGO: E - 10
Evan's age IN EIGHTEEN YEARS: E + 18

From this, we can write our equations:

L + 2 = 2(E + (E - 10)) or L = 4E - 22
L + 18 = 2(E + 18) or L = 2E + 18

This gives Ollie's age as 58 years.

Joan is 5 years older than Betty was last year. In twenty four years, Betty will be twice as old as Joan is now. What is the sum of their ages?

Students should note that, intstead of asking for either Joan's age or Betty's age, we're being asked for the sum of their ages, or B + J.

Unknowns
Joan's age NOW: J
Betty's age NOW: B
Betty's age LAST YEAR: B - 1
Betty's age in TWENTY FOUR YEARS: B + 24

So our equations are J = (B - 1) + 5 = B + 4
B + 24 = 2J

Solving these equations gives B = 16; J = 20. Thus, the answer is 36.

Much more interesting problems can be obtained by including time frames which are unknown. For example, in the sentence, "John's age is twice what it was when he was as old as Bob is now," not only do we not know Bob's age and John's age, we also don't know how long ago John was as old as Bob is now. This introduces another unknown into the problem. This a good topic to explore in a more advanced lesson.

NOTE: The final problem in the slide show above is intended as a JOKE. Students who are learning to solve the problems in this lesson will not be solving "The Infamous James Problem" (as my students call it).

Lesson by Mr. Twitchell

## How old is James?

James is half as old now as he will be when he is 60 years older than he was six years before he was half as old as he is now.  How old will James be when his age is twice what it was 10 years after he was half his current age?

## Solutions

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## Age Problems

List each unkown, and solve the problem.

Example:    Troy's age is two more than twice what it was 25 years ago. How old is Troy?
Solution:     Troy's age: T; Troy's age 25 years ago: T - 25
T = 2 + 2(T - 25); T = 48

1. Freida is four times as old as she was 12 years ago. How old is Frieda?

2. Wally's age twelve years ago, plus Wally's age last year, is Wally's age next year. How old is Wally?

3. Karl is twice as old Bob. Nine years ago, Karl was three times as old as Bob. How old is each now?

4. Ten years ago, Terrence's and Walter's ages added to 53.  Walter is three years older than Terrence. How old is Terrence?

5. Mark's age now, plus Linus's age in twenty years, is equal to three more than twice Mark's age now. In twenty six years, their ages will add to 99. How old will Mark be in twenty years?

6. Edie is twice as old as Stewart.  Their ages in ten years will add to 59. How old is each now?

7. Twelve years ago, Lester was twice as old as Hester. Lester is 21 years older than Hester. What will the sum of their ages be in 5 years?

8. Bobby is a year younger than Sue, and Lambert is two years older than Sue.  In 12 years, their ages will add to 73. How old will Lambert be next year?

9. Nat is four years older than he was when he was twice as old as Kat. Nat is seventeen years older than Kat. How old is Nat?

10. Pete's age ten years ago was two more than twice what his age was 15 years ago. In 10 years, Pete's age and George's ages will add to 71. How old is George now?

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