I'm sorry, Facers22 but I still don't understand your way of explaining this. Let me take your example. To quote you, "A ratio of 3/4 using the same class, would mean that for every 4 students that got the wrong answer, 3 would get it right. This would mean that 9 out of 12 students got the answer wrong."

Now if there are 12 students in the class and 9 students got the answer wrong, then there are only 3 students not accounted for (12 - 9 = 3). Did they take the test at all? Did they get the right answers? If they got the right answers, then there were 9 students wrong and 3 students right. That ratio of wrong to right would be 9:3 or (reduced) 3:1. The ratio of right to wrong would be 3:9 or (reduced) 1:3. You wrote "for every 4 students that got the answers wrong, 3 would get it right." And yet, in your own example there are 3 students who got the answers right and 9 - not 4 - who got them wrong.

If you have 3 students who got the answers right and 4 students who got the answers wrong on the same test at the same time, you have a total of 7 students. Or, if there are 4 students wrong per 3 students right, and you have a multiple of 4 or 3, then you could have 8 wrong and 6 right; 12 wrong and 9 right (total of 21 in the class); 16 wrong and 12 right; etc.

I repeat what I said in my earlier post. If we want to use the fraction format to express a ratio, it's important to label the parts accurately. If 3/4 of the students got the answers right, then 3 out of every 4 students who took the test got the answers right. That would leave only 1/4 that got the answers wrong, wouldn't it?

Or, to label it another way: if 3 students are right for every 4 students who are wrong, then we have 3/4 as many right students as there are wrong students. In this case, wrong would be represented by 1. That creates the following ratios of right to wrong: 0.75 : 1 (= 3/4 : 1) or 1.5 : 2 or 2.25 : 3 or 3 : 4.

I look forward to additional discussion and explanation about this.

I am not certain that the third way to express a ratio is by a fraction. But, if it is, the fraction needs to be expressed correctly. In the case of the students who got the answers right, if 3/4 of the students answered correctly, then that means that 3 students answered correctly for every one student who answered incorrectly (1/4 of the students); so in this case, the ratio of accurate students to inaccurate ones would be 3:1 or 3 to 1.

If 3 students are correct for every 4 students who are incorrect, then the ratio is 3:4 or 3 to 4, and the fraction of accurate students is 3/7. The fraction of inaccurate students is 4/7. A fraction expresses a part of a whole.

Another way to use the 3/4 fraction to express a ratio would be to label it differently. That is, if 3/4 of the students answered correctly, then the ratio of students who answered correctly to the number who took the test (or who did the homework, or who answered at all) would be 3:4 or 3 to 4.

Thanks for your very kind response, Facers22 And yes, you are absolutely right about using a fraction to express a ratio. In my first post, I was intending to express my uncertainty about *my* recollection of math rather than skepticism towards your statement, but it didn't come across quite the way I intended it. Since that first post I have done more review/research and feel more confident about the subject than I did at the time.

I had a hard time understanding how to move from the more basic expression of the ratio to the fraction form until I realized (I think) that it can be treated sort of like an equation, in that whatever is done to one side of the equation (or ratio, in this case) is done to the other. So, 3 : 4 (or 3 to 4) can become 3/4 : 4/4 or 3/4 : 1. I'm not sure that is the way a good math teacher would explain it, but it helped me somehow to understand the process better.

A ratio shows the relative size of two or more things against each other. In the case of two integers, a and b, a ratio can be written as:

a:b

a/b (common fraction)

a/b (decimal)

For example 3:4 is the same as 3/4 is the same as 0.75

Percentages such as 75% could be considered as a fourth way to present a ratio but it is really another form of a common fraction e.g. 75/100

In the building industry, ratios of three items, such as cement, sand and aggregate are commonly used to show how they should be mixed. These are usually presented using the first method. For example 1:2:4 would mean a mix consisting of one part cement, two parts sand and four parts aggregate.

You can also write a mathmatical ratio as a fraction. 3/4 is the same as 3 to 4, or 3:4. If you divide out the fraction it will give you the decimal equivalent of the ratio as in this case 3/4 equals .75 as a decimal. If you then change the decimal to its equivalent percentage.

You can say that 3/4 of the students got the answer right, which would mean that 3 out of 4 or 75% of them got the answer right. The ratio of how many student got the right answer would be 3 out of 4, 3/4 or 3:4 meaning that 3 out of every 4 students got the answer correct. In a class of 12 that would mean that 9 students got the answer right and 3 got the wrong answer.

You are right Afic, I worded my explaination wrong. I should have said that the ratio 3/4 referred to 3 students getting the correct answer for every 4 students in the class, not for every 4 students who got the answer wrong. That would give us the 9 students with the right answer which would equal 3/4th of 12 and 75% of 12 and .75 times 12. It would also be equal to the ratio you gave of 3:1.

I was right though that a fraction can be used to denote a ratio. I went back and edited my other postings so that they would be correct. Thank you for catching my mistake.

The ratio 3 to 1 would be the same as the ratio 3/4 where 3 out of every 4 students got the answer right. A ratio of 3 to 1, used to refer to students who got correct answers vs those who did not, would mean that for each 1 student who got the answer wrong, 3 got it right. In a class of 12 if 3 got it wrong then 9 got it right which is what I got in the first answer using the ratio 3/4.