## Sunday, May 17, 2015

### [Pri20150510PEP] Percentages of Erasers and Pens

Question

Introduction
This is a question about percentages, which are really fractions based upon 100 as denominator.  For example,  60%  just means  60/100.    It is possible to solve this using some sort of algebraic approach based on  100  units for a percentage.  However it is more convenient to use fractions in their lowest terms.  I present a solution based on my Distinguished Units Method, which is a proto-algebraic approach.

Solution
Note that  60% = 60/100 = 3/5  and  25% = 1/4.  The Lowest Common Multiple (LCM) of the denominators is  20.  I use  20 “square” units for the original number of erasers and  20  “circle” units for the original number of pens.  This makes the units easy to divide.  It does not matter what shape you use to envelop the different units, as long as different shapes are used for different types of units.

Ans:  There were  240  pens at first.
Since the question asks for the original number of pens, it is a good idea to equalise the eraser’s “square” units.  Multiplying the first row numbers by  8/20  gives 8  “square” units for the third rows.  This serves as a stepping stone to connect the “circle” units.  See the part highlighted in yellow.  From  8  “circle” units to  15  “circle” units, the difference is  84.  This allows us to deduce the value of  1  “circle” unit.  The original number of pens is represented by  20  “circle” units corresponds to  240,  which is the answer we want.

Final Remarks
It is a good idea to know the fractions of some of the more common percentages.  For example,
25% = 1/4,   50% = 1/2,   75% = 3/4
20% = 1/5,   40% = 2/5,   60% = 3/5 ,   80% = 4/5
The usage of  LCM of the denominators is very effective for making calculations easy.

H02. Use a diagram / model
H04. Look for pattern(s)
H06. Use before-after concept
H09. Restate the problem in another way
H10. Simplify the problem
H11. Solve part of the problem

Suitable Levels
Primary School Mathematics

* other syllabuses that involve whole numbers and ratios