Like all word problems, age problems require writing an equation using the words that define the relationships and actions. Age problems just use more of these words than other types of word problems.

The key is to calmly translate the words into manageable equations. Often each sentence requires a new equation.

The words that translate to the equals sign include “is,” “was” and “equals.”

Common words for other actions and relationships are defined in the following list.

Example | Word | What to do? | As equation |

The sum of John’s age and Steven’s is 19. | sum | addition | j + s = 19 |

The difference between Todd’s age and his younger sister Sandra’s age is 5 years. | difference | subtraction (since Todd is older, it is Todd minus Sandra.) | t – = 5s |

The difference between Todd’s age and Sandra’s age is 5 years. | difference | subtraction with absolute value (you don’t know who is older) | |t – s| = 5 |

John’s age is double Henry’s age. | double | multiplication | j = 2h |

Four years ago, Kim was the same age as Bob is now. | ago | subtraction | k – 4 = b |

Six less than my age is 22. | less than | subtraction | a – 6 = 22 |

The total of Mario’s age and my age is 35. | total | addition | m + a = 35 |

Ten more than my age equals 23. | more than | addition | 10 + a = 23 |

Four times my age is 48. | times | multiplication | 4a = 48 |

The product of my age and 12 is 144. | times | multiplication | a × 12 = 144 |

Correctly translating the words into your own algebraic equation is critical, and it is easy to make mistakes. You can double check yourself by Plugging In your solution or Backsolving from the answer choices.

## Example

Steven is 12 years older than Mary. Three years ago, Steven was 5 times as old as Mary.

How old is Mary?

### Solution

Define the variables.

Mary’s age = *m*

Steven’s age = *s*

There are 2 variables, so you need 2 equations.

**Sentence** **Equation**

Steven is 12 years older than Mary. *s* = *m* + 12

Three years ago, Steven was 5 times as old as Mary. *s* – 3 = 5(*m* – 3)

Simplify the equation. *s* – 3 = 5*m* – 15

*s* = 5*m* – 12

Use the 2 equations and solve.

*s* = *m* + 12 and *s* = 5*m* – 12

*m* + 12 = 5*m* – 12

4*m* = 24

*m* = 6 Mary is 6 years old.

## Example

Ethan is as much older than Harry as Harry is older than Candice. Five years ago, Ethan’s age was double what the age difference between what his age and Harry’s will be 15 years from now.

How old is Candice?

### Solution

Define the variables.

Ethan’s age = *e*

Harry’s age = *h*

Candice’s age = *c*

There are 3 variables, but you only need to solve for one variable,* c*. So you need 3 or possibly just 2 equations.

Sentence

*Ethan is as much older than Harry as Harry is older than Candice.*

Simplify the equation.

Break down each phrase of the next sentence.

Five years ago, Ethan’s age

was

double

difference between

what Ethan’s age and Harry’s will be 15 years from now

Write the complete equation.

Simplify the equation.

**Equation**

*e* –* h* = *h* –* c*

*e* = 2*h* – *c*

*e* – 5

=

× 2

– (subtraction)

*e* + 15 and *h* + 15

*e* – 5 = 2[(*e* + 15) – (*h* + 15)]

*e* – 5 = 2[*e* + 15 – *h* – 15]

*e* – 5 = 2[*e* – *h*]

*e* – 5 = 2*e* – 2*h*

*e* = 2*h* – 5

Use the 2 equations and solve.

*e* = 2*h* – *c*

*e* = 2*h* – 5

2*h* – *c* = 2*h* – 5

*c* = 5 Candice is 5 years old.

In this case you only need two equation

**Before attempting these problems, be sure to review this section on Quantitative Comparison questions.**

https://www.youtube.com/watch?v=oXk7glrVOSk&list=PLD0D070C218D8F5A3&index=12

https://www.youtube.com/watch?v=glnQsBkkEhM&list=PLD0D070C218D8F5A3&index=14