 The rule of three is a method of solving proportionality problems. The rule of three originated in India and spread to Islamic and European mathematics. The generic problem solved by the rule of three is the proportionality. If A yields B, then C yields how much? The first description of the rule of three comes to us from Aryabhata in the 6th century, who gave the rule as follows, multiply the fruit by the desire and divide by the measure. The result will be the fruit of the desire. Now this very beautiful and poetic description of the rule of three leaves us wondering what exactly we're supposed to do. So let's think about this. Note that we have one unknown, the fruit of the desire, and three quantities, the fruit, the desire, and the measure. Later commentators explained that the fruit and the measure have to be the same type of quantity, and also the fruit must be associated with the unknown quantity. So let's see if we can figure that out. So here's a typical problem. Five oranges cost three dollars. How many oranges can be purchased for eight dollars? So let's see what was that rule again. We want to multiply the fruit by the desire and divide by the measure. And the fruit and the measure have to be the same type of quantities. Since the fruit and the measure have to be the same, we'll put the dollar amounts in the first and third place. But which goes where? Since the fruit is the value associated with the unknown, we'll put eight in the first place. The other dollar amount will be the measure, and so that leaves the desire to be the last value, five. And so multiplying the fruit by the desire and dividing by the measure gives us the fruit of the desire, 13 and one third. Now Ariabata's formulation is not the clearest, and so fortunately, a little while later, Brahma Gupta, who lived around 600, about a century later, gave a generalization and a new way to look at the rule of three. And Brahma Gupta's approach is much simpler. First, we'll set down our values, then transpose the fruit, and then divide the product of the greater number of terms by the product of the smaller number of terms. And here's an important lesson to learn. Brahma Gupta is organizing this information into a nice table, and as such, it makes it a lot easier to comprehend. So let's take a look at that same problem. Five oranges cost three dollars. How many oranges can be purchased for eight dollars? So what we have are the costs and the number of oranges. Our costs, three and eight dollars, and the number of oranges is five, which goes with this cost of three dollars. Now we'll transpose the fruits. Now, since this five is still a number of oranges, it has to say in the same row. So the only place it can go is in the next column. And now we have a product of a smaller number of things, really just this number three, and a product of a larger number of things, eight times five, which is 40, and so we'll divide the product of the larger number of things by the product of the smaller number of things. 40 divided by three gives us 13 and one-third. Let's take a more complicated problem. 12 yards of silk costs 90 dollars. How much for 20 yards of silk? Now, it'll be convenient to have the thing that we know two values for be the first line. Because we know two lengths, we'll make length the first row. So our lengths are 12 and 20. The only cost that we have is 90, and that goes with the 12 yards. Transpose the fruits and multiply. Now, this first column only has one factor, so the product is just going to be the number. In the second column, we multiply 20 by 90 and divide the product of the larger number of things by the product of the smaller number of things, 1800 divided by 12. We could also go the other way. 12 yards of silk costs 90. How much silk for 200 dollars? And this time, because we know the two costs, we'll make cost the first row. We only have one length, so we'll set it down. Transpose. Find the products. And divide. The advantage to Pramakut's formulation is that it allowed him to express a general rule of odd numbers. This could be used any time the desired quantity is directly proportional to several others. So for example, we might know that volume is proportional to length, width, and height. Cost is proportional to length and width. Or maybe interest is proportional to time and principle. So for example, suppose a principle of 100, borrowed for six months, incurs interest of 30. How much interest would there be for 200, borrowed for eight months? So the trick here is setting up our table, and the important thing here is that our first throws have to have complete information for both situations. So we know two principles. We'll make that the first row. We know two times. So we'll make those the second row. And so the interest is the only quantity we don't know for both situations. Now it's important to remember that in order for the rule of odd numbers to work, the unknown quantity has to be jointly proportional to all of the others. And simple interest is jointly proportional to principle and time. And since we're given five quantities, we can use the rule of five. So we'll transpose the fruits. We'll form our products. The product of the smaller number of things, 100 times 6. The product of the greater number of things, 200 times 8 times 30. And we'll divide the product of the greater number of things by the product of the smaller number of things, 48,000 divided by 600. And that gives our interest about.