Problem
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Danny saved some
50-cent coins and $1-coins in
his coin box. The total value of
the 50-cent coins to the total value
of the $1-coins he had was in the
ratio 2 : 5. After
$14 worth of 50-cent coins and an equal value of $1-coins were added to the coin box, the
ratio of the total value of 50-cent
coins to the total value of $1-coins
became 5 : 9. How many coins of each type did Danny have
in the end?
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Introduction
Here is a “Singapore math”
coin problem that can be befuddling for kids and even for adults. To rub salt to the wound (or pour oil to the
fire?), the value of a collection of
coins is different than its number. Whilst a $1-coin obviously has a value of one
dollar, you would need two 50-cent coins to make up a dollar.
Strategy
Notice that
after adding $14 worth of coins to both types of coins, the difference
in the total value of the two types of coins remains the same. Some people call this a “constant difference” problem.
But how do we exploit this constant difference, when the type of ratio
units used in 2 : 5 are most likely not the same as those used
in 5 : 9? Well, we need to bring them to a common unit!
[H09] How?
Read on!
Solution [H02, H06]
Ans: Danny had 54 $1-coins and
60 50¢-coins in the end.
Commentary
I am using
Distinguished Ratio Units in my presentation.
This makes it clear that the ratio units are of different types. In the the “before” stage [H06], the
difference in the value of the two sets of coins is 3
circle units. In the the “after”
stage, the difference in the value of the two sets of coins is 4 square
units. But we know these two differences
refer to the same numerical number. The
Lowest Common Multiple of 3 and
4 is 12. So
both of them must me equal to 12 common units (which I envelop with
triangles). We multiply the numbers
inside the circle units by 4 and we multiply the numbers in square units
by 3.
I put these multiplications in quotation marks because we are not really
changing the numbers of coins. We are merely
changing the type of units used. I am
saying that each square unit is the same as
3 triangle units and each circle
unit is the same as 4 triangle units.
Once we
bring everything to common units (triangle units), we can see the $14 added
corresponds to 7 triangle units. Henceforth the whole problem unravels easily. [H11, H05]
H02. Use a diagram / model
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 independent learner who is interested
