## Thursday, February 18, 2016

### [P6_20160217RTTU] Books on Bookshelves

Problem

Introduction
Here we have a numerically challenging problem that involves ratios, and it ultimately reduces to an algebraic problem with two unknowns.  Nevertheless, we are spoilt for choice as regards to methods of solution:-
(1)   Bar Diagram Modelling
(2)   explicit letter-symbolic Algebra
(3)   “p” and “u”  (parts and units)
(4)   Distinguished Ratio Units
Despite the fact that Bar Diagram Modelling made “Singapore mathematics” famous, let us remember that it is only one of the ways of solving problem by diagramming, which is just one of the eleven Primary School heuristics recommended by the Singapore Ministry of Education.
The methods have a lot in common, and they differ mainly in the form of presentation.  However, standard Bar modelling is impractical under high-stakes high-stress examination conditions for this problem, not least because one would have to cut the bars into many pieces.  One should not cut off one’s feet just so as to fit the shoes (削足适履), as one Chinese saying goes.  We need to be flexible and open-minded.  I present a solution using my own Distinguished Ratio Units.

Solution
Ans:  735 books

Commentary
First off, we need to equalise the numerators of  2/5  and  11/4 = 5/4  and put them ratio form.   This is because the  “2”  in the  2/5  represents the same quantity as the  “5”  in  5/4.
We do this adjustment by multiplying the former through by  5  and the latter through by  2.  Thus we deduce that the original number of books in A and in B are  25  and  8  “heart” units respectively.
Next, we add on the  2  and  3  “triangle” units.  By doing a comparison, we can figure out that  1  “triangle” unit must be  45  more than  17  “heart” units.  So  2  “triangle” units must be equal to  34  “heart” units plus  90.  Replacing the  2  “triangle” units (shown in yellow) with their equivalent, we now know that  59  “heart” units plus 90 gives  444.  This allows us to figure out that  1  “heart” is actually  6.  Thus, we can work out what  1  “triangle” unit, and then what  5 “triangle” units are worth.

Final Remarks
Due to the difficulty of the numbers, the solution presented above is about as streamlined as I can make it to be.
There is another variation that can be used – equalising the “triangle” units (akin to the technique of elimination in standard algebra).  What we do is we multiply the group with total  444  by  3  and to multiply the group with total  489  by  2.  This would give  6  triangle units on each side.  Then we can compare the “heart” units and continue from there.  This way of proceeding is not for those who fear 4-digit numbers.
If there are nicer or more elegant ways to tackle this question, I would definitely love to hear from you.

H01. Act it out
H04. Look for pattern(s)
H05. Work backwards
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 whole numbers and ratios
* any problem solver who loves a challenge