[S1_20150516AELI] Apples and Pears on a Table?

Question

Michael has enough money to buy either 12 pears or 36 apples. if he intends to buy equal number of pears and apple, how many of each fruit can he buy with the money

Introduction

     This is an interesting algebra problem meant for secondary 1 students (» grade 7) that has the potential to lead to a system of complicated simultaneous equations in two variables.  Fortunately, there are some interesting and simpler approaches.  I present three of them below.

Solution 1 (using a Table and Unit Costs)

Solution 2 (Insight from proportion)

     This is perhaps the intended algebraic approach.  Observe that since  12 pears are as expensive as  36  apples,  1  pear is “equivalent” to  3  apples.  Michael’s budget is  36  apples-worth of money.  If there are  n  pears and  n  apples,  the  n  pears can be exchanged for  3n  apples.  We arrive at an equation as in solution  1.

  

Solution 3 (Acting it Out)

     You can role-play this with your friend using toy-fruits.  Your friend is the fruit-seller and you are the buyer.  No toy-fruits?  Well, use some counters, bottle caps, Lego bricks, ... whatever to represent the apples and pears.  If you are a lonely person, or if your friend is too busy, or your mother has thrown away all your toys since you are (sort of) more grown up already, maybe just do a thought experiment.  Imagine, at first, you took  36  apples to the check-out counter.  Then you saw some pears, and you grabbed  12 of them, ditching the apples.  You figure out that  1  pear is equivalent to  3  apples.  Just then, you change your mind.  You decide that you want an equal number of apple and pears.  OK, so you replace  1  pear with  3 apples.  You get  11  pears,  3  apples.  Keep on swapping pears for apples.   You get  10  pears,  6  apples.  Then  9  pears,  9 apples.  Bingo!

Final Remarks

     There is no magic potion for mathematics (although heuristics is a start).  There is also no one fixed method for you to memorise to solve problems.  As they say, you have more than one way to skin the cat.

Heuristics Used

H01. Act it out

H02. Use a diagram / model

H03. Make a systematic list

H04. Look for pattern(s)

H05. Work backwards

H06. Use before-after concept

H07. Use guess and check

H08. Make suppositions

H09. Restate the problem in another way

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* Lower Secondary Mathematics (Secondary 1)

* GCE ‘O’ Level “Elementary” Mathematics (revision)

* other syllabuses that involve ratios or algebra

[Pri20150306APC] Apples-with-Pears Comparison

Question

Introduction

     This question involves money and looks rather challenging, because there are a few unknown quantities.  To solve it, we use the concept of unit costs, so that we can make an “apples-to-apples” ... er ... I mean “pears-to-apples” J comparison of the prices.

Solution

     Since absolute dollar amounts are given, we can quickly solve [H11. Solve part of the problem] for the total costs of pears and apples as follows [H02. Use a diagram / model]:-

     Now we know that the total cost (in dollars) of pears is 45 and for the apples it is 50 (5 more than 45).  Although we do not know the absolute numbers of pears and apples, we know their ratio.  So let us write these down as, say, ‘square’ units. 

     Dividing the total costs by the numbers gives the unit costs, which we know only in ratio terms.  So let us use, say, ‘circle’ units to denote these.  But we know that the unit cost (in $) of an apple is 0.50 less than that of a pear.  And that is equivalent to 5 ‘circle’ units.  From here [H05. Work backwards], we quickly work out the cost of a pear (15 ‘circle’ units) as $1.50.

     Ta da!

Heuristics Used

H02. Use a diagram / model

H05. Work backwards

H11. Solve part of the problem