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 applesworth
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 roleplay this with your friend using toyfruits. Your friend is the fruitseller and you are the
buyer. No toyfruits? 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 checkout 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.
H01. Act it out
H02. Use a diagram / model
H03. Make a systematic list
H04. Look for pattern(s)
H05. Work backwards
H06. Use beforeafter 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
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