Associated Ratios and the Value of a Ratio 05:44 minutes

Video Transcript

Transcript Associated Ratios and the Value of a Ratio

Meet Preston Leopold Waltz, the world’s foremost orchestral composer. Not only does he decide how many of each instrument family make up his orchestra, he also orchestrates the orchestras he orchestrates. Preston, as he’s known in the classical music world, always comes up with the perfect ratio of woodwinds to brass, to strings, to percussion. To better understand Preston’s genius, let’s take a look at associated ratios and their values. For the perfect sounding orchestra, Preston chooses 5 woodwind instruments, 4 brass, 20 strings and 1 percussion instrument. Let's take a look at the different ratios of instruments to help us better understand Preston's musical genius. One ratio we can see is the number of woodwinds to the number of brass instruments. This can be written as 5 to 4. But how many brass to woodwind instuments are there? We can find it by using the associated ratio. 4 brass to 5 woodwinds. These ratios are associated because they are related to one another. This means, given one ratio, we can find the other. But, as you can see, the order in which we write the ratio makes a difference. Ratios can be written in fraction form as well. The first number always becomes the numerator and the second always becomes the denominator. But let’s say we have a different mixture of instruments: like brass instruments to stringed instruments. We already know two ways of writing ratios: with a colon or as a fraction. If we replace these icons with their respective numbers, as we did before, we get the same ratio expressed in two ways! We can read this as 4 brass to 20 stringed instruments, or for every 4 brass instruments in the orchestra, there are 20 stringed instruments. Since one of the ratios is written as a fraction, you might be thinking that we can reduce ratios, just like a fraction, and you’d be correct! When expressing ratios, it’s always good to express them in the most reduced form, just like a fraction. How do we do that? First, we have to look for the greatest common factor of 4 and 20. In this case, the GCF is 4. Just like when reducing a fraction, we have to divide both sides by 4, giving us the ratio 1 to 5. Are there other associated ratios we can find with our given information? What about expressing a family of instruments to the orchestra as a whole? That’s no problem! Let’s see how many woodwinds there are compared to the total number of instruments in the orchestra. We know how many woodwinds there are, 5. So, to find the total we sum all the instruments in the orchestra. Since we have 5 woodwinds, 4 brass, 20 stringed instruments and 1 percussion, we have a total of 30 instruments in the orchestra, giving us a final ratio of 5 to 30. But wait! Don’t leave it like that! You must reduce the ratio to its simplest form by canceling out the greatest common factor! What's the GCF of 5 and 30? 5! So, dividing both sides by 5, we get our final ratio of 1 to 6. Which means 1 out of every 6 instruments is a woodwind. Could we find out how many instruments are NOT woodwinds? We can subtract 6 minus 1, since 6 represents total instruments and 1 represents the woodwind. This means 5 are NOT woodwinds. Therefore, we can write the associated ratio of 5 out of every 6 intruments are NOT woodwinds. To review associated ratios… Remember that ratios can be written different ways: with a colon and as a fraction. Associated ratios are the ratios that can be found from the given ratios or information. We must always remember to write ratios in their simplest form, and don't forget the order in which you present the ratio matters! Let’s get back to our maestro, Preston Leopold Waltz. He’s once again constructed his orchestra with the perfect ratio of instruments. What a performance! Take a bow, Preston!