Boys and Girls in the class


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.

Which two classrooms have the same boys-to-girl ratio?
On one occasion Ms. Matthews' class joined Mr. Lopez's class. What was the resulting boys-to-girls ratio?
On another occasion Ms. Waddell's class joined Mr. Lopez's class. What was the resulting boys-to-girls ratio?
Are 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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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

20 boys

21 girls

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

12 boys

11 girls

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.