[MUSIC] A ratio is a quotient of two quantities. It is normally used to compare the quantities. The ratio of number a to number b can be written as a to b. A divided by b or a/b or are pronounced as a to b or a is to b. This last way is the most common in mathematics. The writing a divided by b is more used in business instead. Let us see an example the ratio of five hours to two hours is 5 to 2. If we need to find the ratio of three hours to three days, first, we need to convert the three days to hours. 24 hours in one day, and then three days is equal to three times 24, which is equal to 72 hours. Therefore, the ratio of three hours to three days is the quotient of three and 72 that is three to 72. Another example, we use a ratio between two numbers in order to compare two amounts. If two ratios are equal, such as, 3 to 4 and 15 to 20, then we can tell that there is proportion. We can see that by using the methods of cross products. And then we have that three times 20 is equal to 60 and four times 15 is equal to 60 then, there is a proportion. According to rule of proportion a to b is equal to c to d if the cross product a times d and b times c are equal that is if ad = bc. For example decide if the following proportions are correct. 3 to 5 and 12 to 20, 3 x 20 = 5 x 12 is that correct? Yes, because 3 x 20 = 60 and 5 x 12 = 60 as well. 2/3 equal to 9 to 16, 2 times 16 is equal to 9 times 3, is that correct? No, because 32 is different than 27 A proportion is composed of two ratios, therefore, of four numbers. If we know three of our four numbers, then any three numbers, what is unknown? But still we can solve our proportion by finding default unknown number. We can use alternatively, two methods to solve the proportion. We can either multiply both sides by the product of the two denominators, or use the method of cross products. Notice, that whatever the method used, we get the same equation. If we use method one, we need to multiply both sides of the equation by the product of the two denominators. Then if we have 3/5 equal to x to 40 then we have 3/5 x (5 x 40) = x/40 times five times 40 and we get 120 equal to 5x. If we use the method two, we have (3x40)=(5 x X) and then we get again 120 = 5x. Now we can solve the equation. In this case in order to isolate our variable which is x, on the one side of the equation in a numbers in all the sides of the equation we need to get rid of five. We can do it just by applying the operation, which is the inverse of the multiplication, that is the division. Therefore, we solve the equation by dividing both sides by five. And then we have 120/5 = 5x/5 our result is x = 24. [MUSIC]