In Ms. Matthews' classroom, there
are 12 boys and 15 girls. In Mr. Lopez's classroom, there are
8 boys and 6 girls. In Ms. Waddell's classroom, there are 4 boys
and 5 girls.
two classrooms have the same boys-to-girl ratio?
one occasion Ms. Matthews' class joined Mr. Lopez's class. What
was the resulting boys-to-girls ratio?
another occasion Ms. Waddell's class joined Mr. Lopez's class.
What was the resulting boys-to-girls ratio?
your answers to the two questions above equivalent? What does
this tell you about adding ratios?
Ms. Louvin's class has
a boy-to-girl ratio of 5 to 6. At the end of the 1st, 2nd, and
3rd quarters, the class gets larger by two boys and one girl.
So, by the end of the year, Ms. Louvin has 6 extra boys and 3
extra girls in her class since the beginning of the year. If the
class never has more boys than girls, then what is Ms. Louvin's
largest possible class size at the beginning of the year?
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Submit your idea for an investigation
Ms. Matthews' class ratio is 12:15 (which
is also eqivalent to 4:5).
Mr. Lopez's class ration is 8:6
(which is also equivalent to 4:3).
Ms. Waddell's class ratio is 4:5.
Ms. Matthews' and Ms. Waddell's
class ratio are the same...4:5.
When Ms. Matthews' class joined
Mr. Lopez's class, the resulting class ratio was (12 + 8)/(15
+ 6) = 20:21
Looking back at the reduced ratios,
Ms. Matthews had a ratio of 4:5 and Mr. Lopez had a ratio of 4:3.
Now when I combined these, I had
(4+4)/(5+3) = 8:8, which reduced
to a 1:1 ratio!!! Well, that cannot be because combining the original
numbers actually led me to a 20:21 ratio, so it is quite obvious
that the reduction did not lead to the same ratio. Simply put,
the sum of reduced ratios does not ALWAYS give a correct ratio
for the original number.
Let's look at this in another way
using fractions. Okay, combining Ms. Matthews' and Mr. Lopez's
classes, using the reduced fractions gives 8/8 or 1/1...AND...combining
Ms. Matthews' and Mr. Lopez's classes, again using the reduced
fractions gives 8/8 or 1/1. When you cross multiply these two,
they say they are equal. Now, is this truly correct? NO! The reason
is in the original combined numbers, 20/21 and 12/11. The first
thing I saw is that one is a common fraction and the other is
an improper fraction....dingdingding!!! and when I cross multiply
these they prove they are not equal!
When Ms. Waddell's class joined
Mr. Lopez's class, the resulting class ratio was (8 + 4)/(6 +
5) = 12:11
Looking back at the reduced ratios
of Ms. Waddell (4:5) and Mr. Lopez (4:3), which gives me (4+4)/(5+3),
again leading to an 8:8 or 1:1 ratio. However, the original numbers
in the ratio were (4:5) and (8:6), respectively. Combining the
classes lead to an (4+8)/(5+6) or 12:11 ratio.
Looking at the answers from questions
2 and 3 give different answers, although Ms. Matthews' ratio could
be reduced. It seems as if the higher number of items you are
working with, the more change will occur in the ratio. Let's look
for some research on this:
According to Contemporary
Mathematics for elementary teachers (1966), "ratio
was defined in terms of the measures of two segments, or of two
magnitudes with the same unit of measure. We, can however, think
of any two numbers as being possible measures, and hence define
their quotient to be a ratio...a ratio of numbers a
to b is the quotient a/b, which is a number r.
...A statement that two ratios are equal is called a proportion.
Thus, a/b = c/d is a proportion in which the terms are a, b, c,
d. It is customary to call the terms a and d the extremes and
b and c the means of this proportion. Thus we have from the formula
(above): Two ratios are in proportion if and only if the product
of the means is equal to the product of the extremes." (p.
Now, according to this statement,
this gives the reasoning why the two (Ms. Matthews' and Ms. Waddell's)
ratios are equal, but upon further examination, I found
that when I combined the numbers in the groups, the new ratios
could be found from the sums of those groups...i.e. you CANNOT
add reduced ratios and get correct answers.
Now it is YOUR turn!!! Why don't
you try the extension above.