[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