It will be totally illogical to open a discussion on ratio problems when you have no or little idea about the term ‘’ratio’’ itself. It is nearly impossible to find the roots of the concepts of ‘’ratio’’ or ‘’ratio problems’’ because of the reason that the ideas from which they developed would have been related to preliterate cultures. Really, the idea that one house may be twice as large as the second one is so basic that it could have been understood even in prehistoric society. However, we can trace the origin not of the concept but of the word ‘’ratio’’ much more easily. This word is closely related to the Ancient Greek word ‘’λόγος’’(logos). Early translators rendered this word into Latin as ratio meaning ‘’reason’’ while medieval writers often used the word proportion for indicating ratio and proportionalities. Today in math, specialists are extremely interested in ratio problems considering ratio to be a relationship between two numbers showing or indicating how many times the first number contains the second one. For example, if the bowl of fruit contains eight apples and six oranges, it means that the ratio of apples to oranges is eight to six (by the way 8:6 is equivalent to 4:3). So the ratio can be a fraction and not a whole number. Also, note that in this specific example the ratio of oranges to apples is six to eight where 6:8 is equivalent to 3:4. Of course, these are just the basics for solving ratio problems later.

To have a better understanding of ratio problems it is necessary to get acquainted with the main types of ratio. We bet you think that there is only one type of ratio because this term is used just to compare one thing with the other one, right? But we are here to blow your mind by changing your idea about ratio problems and telling you there are three main types of ratio: part-to-part, part-to-whole and finally rates. In fact, absorbing information on ratio problems and ratio types is not that difficult because sometimes the perfect examples are right around the corner.

- Imagine that the room is full of people where there are likely to be several women and men. The way we are going to represent the number of women to men is done via part-to-part ratio. For example, if there are 4 women and 5 men in the room our ratio will be 4/5. If you are required to use a sentence to explain what this ratio means then you should say that for every 5 men in the room there are 4 women. As you have probably noticed dealing with ratio problems can be easy and even fun if you are determined to learn this topic thoroughly and with pleasure.
- The second type of ratio is the already mentioned part-to-whole one. This type of ratio is necessarily a fraction and is surely used when solving easy and complicated ratio problems. In this case, we put two parts, women and men, together finding out that there are 9 people in the room. This means the ratio of women to people will be 4/9 as there are 4 women and 4+5=9 people altogether. Note, that in the case of part-to-whole ratio the bottom number is called denominator whereas in the part-to-part ratio this number is not the denominator simply because it does not tell us how many equal parts are in the whole. This is vital information when you start analyzing ratio problems.
- Let’s analyze ratio problems in a deeper level by discovering the last type of ratio. The third type of ratio is called rate. Rates are used for the comparison of two quite different things. One good example of a rate can be considered a speed limit which compares miles to hours or miles per hour. In other words, a rate is such kind of a ratio that is used to compare things that seem impossible to be compared at a first glance. For example, if one is familiar with ratio problems it will not be difficult for him to compare pizza and people. Yes, learning ratio problems can turn a mathematician into a real magician! The main difference between the part-to-part ratio and a rate is that in the part-to-part ratio the parts will always maintain their relationship and in the case of being added together they will definitely equal the whole. When talking about rates and ratio problems in general, it is necessary to remember that a unit rate tells us how many per unit we have. Here is an example: miles per hour tells us how many miles one goes with 1 hour of travel.

When talking about ratio problems and types of ratio it is essential to pay attention to some important facts. For example, someone may look at a group of people, count their hats referring to the ’’ratio of men to women’’. Please, pay attention to the fact that in this expression the word men came first. This order is vital for solving ratio problems and must be definitely respected. The thing is that whichever word was used first its number must come first too. Suppose there are 15 men and 20 women in the house. The ratio of men to women will be 15 to 20 while the ratio of women to men will certainly be 20 to 15. Simple and easy, right? Ratio problems disappear when you treat them with logic!

Let’s assume our task is related to ratio problems and looks like this: Ann and Jack climbed the hill to pick apples and pears. Ann was able to pick 10 apples and 15 pears while Jack was the one to pick 20 apples and some pears. Keep in mind that the ratio of apples and pears picked by Ann and Jack were the same. Now it is time to solve our ratio problems and determine how many pears Jack was able to pick. In order to solve this task, let’s have a closer look at the chart. We have

Apples Pears

Ann 10 15

In order to obtain the 20 apples picked by Jack we just need to double the 10 apples that Ann picked herself thus multiplying our chart by two, getting closer and closer to the solution of ratio problems.

Apples Pears

Ann 10 15

Jack 20 30

As we can see from the chart Jack picked 30 pears. Congratulations! Your ratio problems are solved in no time!

Now let’s pass on analyzing another task regarding ratio problems. Imagine that a sample of 96 light bulbs involved 4 defective ones. Now assume that today the batch of 6000 light bulbs has quite the same proportion of defective bulbs as in the sample. Can you assist us in solving these ratio problems by determining the total number of the defective bulbs that were made today? By the way for handling this task related to ratio problems you do not need to pay attention to the fact that the assumption of exact proportionality is quite unlikely. Here we will show you the chart so that it is easier for you to deal with ratio problems and start the work!

Number of Defective Bulbs Total Bulbs

4 96

Are you ready to solve these kinds of ratio problems as well? Here we go! Having ‘’the same proportion’’ means that we have the right to multiply the chart by a number. We already know that there will be 6000 bulbs in total so what number do we need to multiply 96 by so that we get the number 6000 as a result? To prove our solid knowledge in the sphere of ratio problems we will divide 6000 by 96 getting number 62.5. The answer is that we should multiply the chart by 62.5. Ratio problems are solved once again!

Number of Defective Bulbs Total Bulbs

4 96

62.5 x 4 = 250 62.5 x 96 = 6000

As you can see we have faced all kinds of ratio problems and already have the answer which is 250 defective bulbs.

Already having pretty good knowledge on ratio problems don’t you think it is time to make our knowledge even firmer by doing one more exercise? Take a deep breath and go on reading our next challenging task! In this case, we have a very small sample of light bulbs consisting of 4 defective and 96 good ones. As in the previous example, you should assume that today’s batch of 6000 light bulbs has again the same proportion of defective bulbs as we have in our sample. Our task is to face ratio problems by determining the total amount of defective bulbs made today. At first sight, this task seems quite similar with the previous one but the difference is that in this case 96 is the number of good bulbs and not the total number. Here is what kind of picture we have regarding these ratio problems.

Number of Defective Bulbs Number of Nondefective Bulbs Total Bulbs

4 96 100

As in the previous example, the fact of having ‘’the same proportion’’ enables us to multiply the chart by a specific number. Here it is more preferable to multiply it by 60 as 60 x 100 is 6,000.

Number of Defective Bulbs Number of Nondefective Bulbs Total Bulbs

4 96 100

60 x 4 = 240 60 x 96 60 x 100 = 6000

We bet you have already solved the task even before we did and we are more than sure you have got the number 240 indicating the number of defective bulbs out of a total of 6000 ones. Of course, these kinds of ratio problems should be presented only after students or people interested in math are fluent with the simpler ratio problems otherwise multiplying a chart may seem like a magical process to them. So make sure to analyze this topic of ratio problems thoroughly before passing on to even more complicated exercises.

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