A Special Bar Model Approach to Age Problems

“The esscence of mathematics lies in its freedom”

- Georg Cantor

A Special Bar Model Approach to Age Problems

Ho Soo Thong

Logo
Copyright © AceMath, Singapore. 
First Edition ,Nov 2014, First Revised Edition, July 2016 

Abstract
This article highlights a special bar modelling structure for solving age problems.

Age problems deal with the ages of people now, in the past or in the future. An age problem involves the ages of two or more people and it is solved as an integer problem.

We will see how the age difference between any two people is the key to problem solving with the Bar Model Method.

Example 1

Helen is 12 years older than Kent.
(a) Three year ago, the age of Helen was three times the age of Kent. How old is Kent now?
(b) What will be the age of Kent when the ratio of the age of Helen to that of Kent is 5 : 3 ?
(c) What will be the age of Kent when the ratio of the age of Helen to that of Kent is 8 : 5 ?

Solution

First, we note that the age difference between Helen and Kent is 12 years, that is Kent was born when Helen was 12 years old.
(a) In Figure 1, we construct a comparison bar model for the situations 3 years ago ( Helen’s age was three times the age of Kent) and now.

1re

Kent was 9 years old now.

(b) When their age ratio is 5:3, the comparison bar model for the Future scenario is shown in Figure 2.

2

From the bar model, Kent’s age will be 18.

(c) When their age ratio is 8:5, the comparison bar model for the Future scenario is shown in Figure 3.

3

Kent will be 20 years old.

In the example, we observe that the age difference is always the same when referring to any past, present or future lines and when the bar models are aligned to the Past and Now and Future lines as shown. This is the key feature of Age Problems.

The next example will make use of this feature to apply the Greatest Common Unit Procedure for effective Unitary Method.

 

Example 2

The age of Reyna is 3/5 of the age of her brother. 14 years later, her age is 8/11 of the age of her brother.
What is their age difference ?
How old is Reyna now?

Solution

in Figure 4, we construct a comparison bar model with 3 and 5 equal parts aligned to the Now line and a comparison bar model with 8 and 11 equal parts aligned to the future line 14 years later for the scenarios involving the fractions 3/5 and 8/11 respectively.

For the unknown age difference, we apply the applying Greatest Common Unit Procedure for the Greatest Common Unit U.
Then we consider the age of Reyna’s age 14 years later and obtain a simple algebraic relation in terms of U as shown.

4

(a) Their age difference is 12 years.
(b) Reyna is now 18 ( = 9U ) years old.

A variety of age problems were solved in [1].

Reference
[1] Ho Soo Thong Bar Model Method for Age Problems, Mathematical Medley, Vol 38, No. 1, Dec 2012.