Tuesday, June 9, 2015
[Pri20150609PCBA] Members of a New Fitness Club
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%.
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.]
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.
H02. Use a diagram / model
H06. Use before-after concept
H09. Restate the problem in another way
H11. Solve part of the problem
* Primary School Mathematics* other syllabuses that involve percentages and ratios