### Solving Challenging Problems

### Articles of Learning

In 2014, we built the website barmodelhost.com to illustrate the visual Bar Model Method with Basic Algebra in solveing a wide rang of challenging problems at *Primary Olympiad Level*.

Recently, we built a new site alpha-psle.com (To appear) for solving a wide range of Singapore PSLE *Word Problems* with

Finally, the website alpha-beyond.sg will solve complex problems and focus on comparing the *bar model approach* with the *traditional algebraic approach.*

**Bar Model Method**

This article aims to illustrate how to solve problems involving mixtures of two or more different compositions following steps:

1. use the ratio approach as a heuristic system in the *Bar Model Method*; and

2. use the *Unitary Method* for simple *Algebr*a.

The first step uses bar models with identical bars to depict the ratio situations for a clearer overview of the problem. This allows for the effective use of the Unitary Method for simple algebra employment in the second step. For comparison, we present the traditional approach vis-à-vis all solutions.

**A Mixture of Liquid Solutions**

A chemist has a solution with an acidic concentration of 20% and another with an acidic concentration of 25%. What amount of each solution should be used in order to make 300 ml of a 22% acidic solution?

**Intriguing Speed Problems**

The bar modelling view of motion in kinematics helps to realise the vast potential and capacity of Mathematics.

‘The bicycles and the Fly’ is a well-known speed problem imagined by Martin Gardner and this kinematic problem gives rise to a learning attitude of intrigue towards the basic concepts of time and distance in kinematics.

Similar examples and adapted problems of this nature can be seen in our earlier works (see [1], [2],[3]).

Here, we focus on a kinematic problem at the Primary Olympiad Level . Problems such as the below example will be henceforth known as ‘*Hitch-hikers*’ Problems’.

A motorist wants to ferry some hitch-hikers travelling from one place to another.

He only has one back seat to carry one passenger at a time.

How can he plan the journey so that all the hitch-hikers will arrive at their destination at the same time?

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**Introduction**

The following classic word problem – involving chickens and rabbits in a cage – was posed in the ancient Chinese book *Sunzi Suanjing* :

There are chickens and rabbits in a cage.

Look at the top of the cage – there are 35 heads.

Look at the bottom of the cage – there are 94 legs.

How many chickens and how many rabbits are there in the cage?

A problem-solving strategy is to create a hypothetical or supposed situation, and compare it with the actual situation using mathematical deduction from the *Distributive Law*.

*Actual situation* : There are 35 heads and 94 legs.

*Supposed situation* : There are 35 chickens and 70 ( = 2×35 ) legs.

We construct a comparison bar model for the two situations:

Therefore we have 12 rabbits and 23 chickens.

The above problem solving approach is known as the *Chicken-Rabbit Approach.*

Next, we proceed to apply the *Chicken-Rabbit Approach* to some challenging problems.

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