[Pri20150302BGP] Boys, Girls and Party

Question

A group of pupils met at a party. Each child exchange phone number with everyone else. Melissa exchange phone number with 5 times as many boys as girls. Peter exchange phone numbers with 4 times as many boys as girls. How many boys and girls were at the party?

Solution

In the set up stage, we read the question carefully and transfer the information into a good representation.  I am using my Distinguished Ratio Units method i.e. using different shapes for different types of ratio units.We set up as follows:  One row for boys (B) and one row for girls.  Reserve one column for ‘original’.  In the first scenario, if you subtract 1 (Melissa) from the girls, you get 1 unit (let’s use a circle to mark it) and then the boys would be 5 circle units.  In the second scenario, if you subtract 1 (Peter) from the boys, you would get 4 units (let’s use triangle) for the boys as compared to 1 triangle unit for the girls.

Going back to the ‘original’ column, we use circle 5 to represent the number of boys (as this was unchanged in the first scenario), and we use triangle 1 to represent the number of girls (as this was unchanged in the 2^nd scenario).

With the set-up done, now my plan is to try to match up one of the types of units (either circle or triangle units).  I choose to match up the triangle units.  Take the relation indicated in yellow for Melissa that says ‘1 triangle unit minus 1 gives 1 circle unit’.  When we multiply this by 4, we get ‘4 triangle unit minus 4 gives 4 circle units’ and then we link this up the the existing triangle 4 (highlighted in yellow on the right).  We see that from circle 5, when you subtract 1 and then subtract 4, you get circle 4. 

This means that 1 circle unit is 5.  Adding 1, we see that 1 triangle unit is 6.  5 circle units makes 25.  The total number being 5 circle units and 1 triangle unit, we add up 6+25 and get 31, which is the answer. 


Commentary

In this word "problem", you start of with two original quantities and when you add/subtract something from one of the quantities, you get a certain ratio, and when you add/subtract from the other quantity you end up with another ratio.  This is a typical difficult type of question that stumps many students and many parents who trying to help them.

Many word "problems" in Singapore's Primary School Mathematics curriculum are just linear simultaneous equations in two variables in disguise.  The pupils will revisit this sort of maths problems in their secondary school years under algebra.  For primary school, simple algebra is taught but not emphasised.  Singapore is famous for her "Bar Model Method".  This method is good for visualisation for pupils at the lower primary levels.  However, at primaries 5 and 6 (~ roughly equivalent to grades 5 and 6), the numbers get bigger and the types of problems get harder.  You often have to cut or draw many bars.  It is especially challenging to draw bars to a suitable estimated length and if you draw the bars wrongly, you often have to erase the whole thing and redraw.  In their year 6, pupils take the Primary School Leaving Examination (PSLE), a high-stakes examination that determines the secondary school and stream (track) that they will go to. .  Many anxious parents and tutors cope by teaching the kids algebra.  However, algebra seems to be socially frowned upon and used only as a last resort.

Last year, reflecting on the above-mentioned challenges, and upon realising that many people use the same letter "u" (for unit) or same bar to represent different types of units and end up getting confused, I cooked up my Distinguished Ratio Units method.  It is powerful and avoids the need to draw and redraw the diagram, hence allowing the pupil to concentrate on the thinking and solving, rather than wasting time with drawing.  I admit that my method is actually algebra in disguise.  The bridging tactic where you equalise one unit to allow the other type of unit to connect is like substitution and elimination.  One triangle unit and one circle unit, for example, can be interpreted as variables x and y, preparing the pupils for secondary school.  The triangle and circle units also link with the concept of ratios which they have learnt in their middle primary years.  By the way, you do not need to use triangles and circles.  You can use squares, diamonds, heart shape etc.  Who says you cannot be creative in mathematics?

I hope you like my new method.

Here is a related problem.