Age ProblemAn application of Solving linear

equations involving two variables

Recall:

We have learned the two ways in solving for a system of linear equations.

Two ways

By substitution

By elimination

Age Problem

• Age problems are algebra word problems that deal with the ages of people currently, in the past or in the future.

Example 1

Danny is 5 years older than Nila. Seven years ago, Danny was twice as old as Nila. What are their ages now?

Solution (1)

Representation:

We let x be Danny’s age and y be Nila’s age.

From the table, we get a system of two equations in x and y.

x = y+5 (1)

x-7 = 2(y-7) (2)

Simplifying equation (2), we have

x-7 = 2y-14(3)

Age 7 years ago Age Now

Danny

Nila

Relation

x-7

y-7

x-7 = 2(y-7)

x

y

x = y+5

… Solution (1)

Substitue y+5 for x in equation (3).

x-7 = 2y-14 (3)

y+5-7 = 2y-14

y-2 = 2y-14

y = 12

Substitue y = 12 in equation (1) to solve for x.

x = y+5 (1)

x = 12+5

x = 17

Answer (1)

Danny is 17 years old and Nila is 12 years old now.

(In solving problem # 1, we use the method of substitution.)

Example 2

The sum of Aiza’s present age and her grandfather’s present age is 68. In three years, Aiza's grandfather will be six times as old as Aiza was last year. How old is each one now?

We let x be Aiza’s age and y be her Grandpa’s age.

From the problem, we get a system of two equations in x and y.

x+y = 68 (1)

y+3 = 6(x-1) (2)

Simplifying equation (2), we have

y+3 = 6x-6 (3)

6x-y = 9 (4)

A year ago Present In three years

Aiza

Granpa

x-1

y-1

x

y

x+3

y+3

Solution (2)

We can use equations (1) and (4) to eliminate one of the variables. Take for instance, we will eliminate the variable y.

x+y = 68 (1)

6x-y = 9 (4)

We will just add the 2 equations to eliminate y. After adding, we have,

7x = 77

x = 11

Substitue x = 11 in equation (1) to solve for y.

x+y = 68 (1)

11+y = 68

y = 68-11

y = 57

Answer (2)

Aiza is 11 years old and her grand father is 57 years old.

(We use the elimination method in problem # 2.)

Evaluation

Henry is one more than three times as old as Cheryl. In 5 years, the sum of their ages will be 63. How old is Henry now?

Ken is 3 years older than Carla. 9 years ago, Ken was twice as old as Karla. What are their ages?

Jane is two more than two times as old as Ruben. In 25 years, the sum of their ages will be 106. How old is Jane now?

Key (Evaluation)

Henry is 40 years old now.

Ken is 15 years old and Carla is 12 years old.

Jane is 38 years old now.

Assignment

Tony is 3 times as old as Ted. In 5 years, Tony’s age will be 4 years more than 2 times as old as Ted. How old is Ted?

The sum of the ages of a mother and her daughter is 59. The mother’s age is 11 more than thrice the daughter’s age. Find their ages.

Prepared By:

Angeline Diamance Dator

III-B Math