[Pri_20151223WNSV] Unravelling Four Whole Numbers

Problem

If  A,  B,  C  and  D  are whole numbers such that  A × B = 8,  B × C = 28, C × D = 63,  B × D = 36,  find the values of    A,  B,  C  and  D.

Introduction

     This question seems to be taken from a secondary school textbook from a chapter on linear equations.  However, I think a good  primary school pupil could attempt this.

Strategy

     The key to solving the above problem is to make observations.  When you multiply up the first two equations, you get an  A,  a  C  and two  Bs  in the product.  Hmmm ... This doesn’t look promising ...  Ah!  But when you multiply the second and the third equations together, you get an  B,  a  D  and two  Cs  in the product.  This can cancel (via division) with the fourth equation which has one B  and one  D  in the product.

Solution

Remark

     Always cancel as much as possible, to avoid large numbers and reduce chances of making careless mistakes.

     By the way, a whole number is a non-negative (zero or positive) integer that does not contain any fractional part.   As such, the set of whole numbers is {0, 1, 2, 3, 4, ...}.  Thus we do not need to consider the negative square roots.

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H10. Simplify the problem

Suitable Levels

* Primary School Mathematics (Challenge)

* Lower Secondary School Mathematics (Challenge)

* other syllabuses that involve whole numbers

* anyone game for a challenge

[AM_20150302BTRCUK] Binomial with known Ratio of Coefficients but Unknown Power

Question

Introduction

     This Additional Mathematics question is challenging because the index  n  of the binomial power is unknown but you only know the ratio of a pair of coefficients.  Usually the  n  is given and you just plug in the formula and apply the binomial expansion formula.

Reminder

Solution

   

Heuristics Used

H05. Work backwards

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* GCE ‘A’ Levels H2 Mathematics (revision)

* IB Mathematics (revision)

* other syllabuses that involve Binomial Theorem for higher powers

[OlymPri20150526] The Smallest Angle in a Special Trapezium

Question

Introduction

     This looks challenging because we do not seem to be given much information.  For example, we do not know the individual angles of the trapezium (American: trapezoid).  Does this mean that this puzzle cannot be solved?  Are we trapped by this trapezium?  What is the secret key that unlocks the problem?

Solution

     First sketch trapezium  ABCD.  Introduce point  F,  the mid-point of  AD.  All contructed lines and points are shown in grey (American: gray).  Then  DF = FA = BC. 

Now draw a line parallel to  BC  passing through  A  and intersecting  DC  at  E, say.  Note that ÐAED = ÐBCD  (corresponding angles).  Then  ÐADE + ÐAED = ÐADC + ÐBCD = 120°.  Hence the remaining angle  in  DDAE,  ÐDAE = 180° – 120° = 60°.  We can deduce this, even though we do not (currently) know the individual angles  ÐADE  and  ÐAED.  This is the key step.

From here, things get easier.  DAFE  is an isosceles triangle with  ÐAFE = ÐAEF = (180° – 60°) ¸ 2 = 60°.  So  DAFE  is in fact an equilateral triangle.  That means  FE = FA = FD.  So  DFDE  is an isosceles triangle.

Note that  ÐAFE  is an exterior angle of the triangle  DFDE.  If you know that exterior angle of a triangle is the sum of the interior opposite angles,  then from  ÐAFE = ÐFED + ÐFDE,  we easily see that  ÐADE = ÐFDE = 60° ¸ 2 = 30°.  If you do not know the theorem about exterior angles, you can still quickly work out that  ÐDFE = 120°,  and then use  ÐADE = (180° – 120°) ¸ 2  and arrive at the same conclusion.  ©

Remarks

     We can now see that actually  ÐBCD  = ÐAED  = 90°,  but we do not need to rely on that (or on accurate drawing) to deduce the answer.  From solving this question, we learn that even though we do not know the individual angles, by construction and using the sum of angles in a triangle, it is possible to solve for an important angle, namely  ÐAFE.  The rest of the solving uses isosceles triangles and equilateral triangles, which are part of the common repertoire of tactics.

HeuristicsUsed

H02. Use a diagram / model

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H12* Think of a related problem   (Draw construction lines)

Suitable Levels
* Primary School Mathematics

* other syllabuses that involve area of triangles and circles