[Pri20150609PCBA] Members of a New Fitness Club

Question

Introduction

     This is a question on percentages.  Percentages are in themselves also units.  One percent (1%) simply means ¹/₁₀₀.  And we can use units (shown circled in the diagrams below) in which each unit is  ¹/₁₀₀  or  1%  of some whole.

     It is useful to think of increases and decreases as multiplying by some percentage.  For example, a decrease by 20%  means multiplying by  100% – 20%  i.e.  80%.  After all, if you work out  100%  of something and subtract  20%  of the same thing,  you will end up with  80%  of that thing.  It is much easier to think of it that way.  Likewise, an increase of  45%  means multiplication by  100% + 45% = 145%.

Solution

     From the information given in the question, we can set up a diagram like this.  I use circles to envelop the percentage units.

We can work out the units in the “after” situation (one year later):-

40 ×  80%  = 40 ×   ⁴/₅  = 32

60 × 145% = 60 × ²⁹/₂₀ = 87

The new total is  119%  or  119 circle units.  The net increase is  19%  or  19 circle units, which we know is equivalent to  228.  Once we got this part, we can work out  1  circle unit  and then  20 circle units, which is the difference between the number of male and female members.  [Remember the check that you are answering the question that was asked.]

Ans: 240

Summary

     We have used a diagram in the form of a ratio-units model [H02].  The ratio unit used in this example happens to be the same as a percentage.  Be careful that other questions may involve different kinds of units with different bases for their percentages.  In other words, in other questions, the “100%” may stand for different things.  In the diagram, we have used the before-after concept [H06].  Increases or decreases in percentages may be re-stated as multiplications by the appropriate percentages, which, in turn, may be thought of as multiplications by fractions [H09].  It is a good idea to be able to inter-convert between fractions and percentages.  By comparison, we found the link between  19 units (or 19%)  and  228 [H11].  Having solved this part of the problem, we are able to answer the original question as asked.

HeuristicsUsed

H02. Use a diagram / model

H06. Use before-after concept

H09. Restate the problem in another way

H11. Solve part of the problem

Suitable Levels

* Primary School Mathematics

* other syllabuses that involve percentages and ratios

[Pri20150308CSR] Charitable Savings Ratios?

Question

Introduction

     Here is another one of those Singapore Mathematics problems that are two-variable simultaneous equations in disguise.  The key to solving this question quickly is to exploit the fact that the amount of donations are the same in this case.

Solution

     First, read the question and translate the information into a diagram [H02. Use a diagram / model].  I use different shapes (e.g. circle and square) to envelop the different types of units.

     Since we have ‘-80’  for both Sharon and Ryan [H04. Look for pattern(s)], we may deduce that 1 ‘circle’ unit  (5 minus 4 ‘circle’ units) is equal to 3 ‘square’ units (10 minus 7 ‘square’ units).  That means 5 ‘circle’ units is 15 ‘square units’.  [ H10. Simplify the problem]

     By comparison again, we realise that 5 square units is 80 [H05. Work backwards] and hence 15 ‘square’ units is 240.

Ans: Sharon’s savings was $240 at first.

Check

Before    $240   $192      5 : 4

After      $160   $112     10 : 7

Solution makes sense.

Heuristics Used

H02. Use a diagram / model

H04. Look for pattern(s)

H05. Work backwards

H10. Simplify the problem

[Pri20150306APC] Apples-with-Pears Comparison

Question

Introduction

     This question involves money and looks rather challenging, because there are a few unknown quantities.  To solve it, we use the concept of unit costs, so that we can make an “apples-to-apples” ... er ... I mean “pears-to-apples” J comparison of the prices.

Solution

     Since absolute dollar amounts are given, we can quickly solve [H11. Solve part of the problem] for the total costs of pears and apples as follows [H02. Use a diagram / model]:-

     Now we know that the total cost (in dollars) of pears is 45 and for the apples it is 50 (5 more than 45).  Although we do not know the absolute numbers of pears and apples, we know their ratio.  So let us write these down as, say, ‘square’ units. 

     Dividing the total costs by the numbers gives the unit costs, which we know only in ratio terms.  So let us use, say, ‘circle’ units to denote these.  But we know that the unit cost (in $) of an apple is 0.50 less than that of a pear.  And that is equivalent to 5 ‘circle’ units.  From here [H05. Work backwards], we quickly work out the cost of a pear (15 ‘circle’ units) as $1.50.

     Ta da!

Heuristics Used

H02. Use a diagram / model

H05. Work backwards

H11. Solve part of the problem

[Pri20150306COH] The Cards of Hearts?

Question

Introduction

     For this question, I shall illustrate my technique of Distinguished Ratio Units to model the situation.  Read the question carefully and translate the information into a diagram [ using heuristic H02 ] like below:-

     I used three different shapes (circle, triangle and square) to envelop the numerical counts of the different kinds ratio units. Although we do not know how many circle units’ worth of cards Kelly had at first, we quickly notice that  9 circle units are equivalent to 3 triangle units, so that one triangle unit is the same as 3 circle units.  So Kelly had 2 circle units’ worth of cards [ heuristic H05 ], as depicted below:-

 

This allows us to answer part (a) of the question already, namely that the required ratio is 10 : 2 i.e. 5 : 1.  With different types of units, it is difficult to compare things.  However, note that in the exchange of cards, the total number of cards remains constant.  Taking the LCM of 12, 4 and 11 which is 132, we can change all the ratio units to a common type of unit [ H09], say ‘heart’ unit, based on the total being 132 ‘heart’ units.  To do that, we can multiply the numerical counts in columns 1 & 2 by 11, multiply column 3 by 33 and multiply column 4 by 12.  This is what we would get:-

     To answer part (b) of the question, we actually do not need to bother about columns 2 and 3.  Just focus on columns 1 and 4.  From column 4 we observe that 84 minus 48 which is 36 ‘heart’ units gives 72, so one ‘heart’ unit corresponds to 2.  The number of cards won by Kelly can be found by comparing the 84 ‘heart’ units and 22 ‘heart’ units highlighted in yellow.


  That means 62 ‘heart’ units and that corresponds to 134.  And we are done!

Answer (a)   5 : 1

              (b)   134

Heuristics Used

H02. Use a diagram / model

H05. Work backwards

H09. Restate the problem in another way

Thinking Back

     In this question, we have used Distinguished Ratio Units to model the given situation.  We worked backwards to find that Kelly’s initial holdings were worth 2 circle units.  Then we converted everything to a common unit (‘heart’ unit) based on the constant total of 132 heart units.  Once again, I © hearts!

[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.