Question
This problem can be solved with Primary
School knowledge using ratios. The
famous Singapore
bar diagramming method can be used to
model the situation, but I prefer my own Distinguished Ratio Units. The former method is good for visualisation
for beginners, while the latter is faster if you want to solve it quickly
without fussing around drawing the perfect diagram. My DRU method is also visual in another way, and
it works with big numbers as well as small numbers. Alternatively, this
can be solved using algebra via simultaneous equations.
Solution 1 (Using my Distinguished Ratio
Units method) [H02]
Explanation: Since the number of white balls is a
multiple of 3, I let “triangle” 3 represent the number of
white balls. I let “heart” 1 represent
the number of red balls. There are 50
more white balls than red balls. [H04]
When the white balls
are removed three at a time, the number of groups of three would be one-third
of the number of white balls, i.e. 1 triangle unit. [H04] This number is less than the number of
red balls (1 heart unit) by 50. So 1
triangle unit plus 50 gives 1 heart unit.
[H04] Following on from the heart to the “triangle” 3, one realises
that 2 “triangle” units is the same as
100. [H05]
So one triangle unit
is 50.
[H11] From here we can solve the rest of the problem.
Solution 2 (using Algebra)
[H13, H05]
w = k + 50 = 3h –––––––––– [1]
r = k = h +
50
–––––––––– [2]
for some
unknown k and h. And then [1] – [2] gives [H10]
w – r = 50 = 2h – 50
so 2h =
50 + 50
h = 50
This
quickly leads to
w = 150
and r
= 100.
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
H13* Use
Equation / write a Mathematical Sentence
Suitable Levels
* Primary School Mathematics (“Ratio”)
* Lower Secondary School (“Simultaneous Linear Equations”)
* other syllabuses that involve ratio or algebra
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