Combining Ratios
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Combining ratios, sometimes between different subgroups in a collection, a problem will give two separately expressed ratios, and we will have to relate these either to the whole or to the absolute quantities, the absolute counts in the situations. So, let's look at this, though it can be a tricky situation. On a certain high school team, the ratio of sophomores to juniors is two to three and the ratio of juniors to seniors is five to six.
Sophomores are what fraction of the whole team. So, I recommend that you pause the video here and try this problem. Okay, here's what we're gonna do. Identify the commonality. Here, the common element is juniors. Find equivalents of both ratios so that the common element is equal.
So, these are the original ratios. We have three representing juniors in the first ratio. We have five representing juniors in the second ratio. We'd like those two numbers to be equal. So the easiest thing to do is to figure out the least common multiple which, of course, is 15.
That means we have to multiply the top ratio by five and the bottom ratio by three. So, we have ratios at 10 to 15 and 15 to 18. Now that the common ratio is the same we can put everything together. Sophomores, to juniors to seniors. 10 to 15 to 18.
Well, now we can just use proportioning to solve. So, the whole is, of course, 43 parts, juniors are, sophomores are 10 parts out of that 43. So 10 out of 43 is the fraction. Here's another problem. In a certain company, the ratio of programmers to marketers is three to eight, and the ratio of customer service reps to marketers is two to three.
If there are 27 programmers, there are how many customer service reps? So, again pause the video here. And try this. Here we're looking for an absolute count. So we another strategy available.
We can solve for the absolute quantity in each term. So right now we know how many programmers, and we can relate programmers to marketers. So programmers to marketers, three to eight. We have 27 programmers. We'll use M for marketers.
Notice in this proportion, we can do some horizontal cancellation in the numerator. Cancel that to one and nine. From there, we can cross multiply, we get 72 marketers. Incidentally, if the idea of horizontal cancellation is something new to you, I would suggest watching the operations with proportions video. So now, we have the number of marketers, from this we can figure out the customer service reps, marketers to customer service reps is 2 to 3, marketers count as 72.
We can again do some horizontal cancellation here. In that denominator 3 goes into 72, 24 times from here we can cross multiply and there are 48 customer service reps. So, here we just sort of ratcheted our way through the ratios, figuring out an absolute quantity for each group, one at a time. Here's a third practice problem, pause the video and try this.
So in this one, we don't have any absolute quantities. So, we'll set up these ratios and we want to make an equivalent ratio. We want number of cookies to be the same, 8 and 12, where the least common multiple is 24. So we're going to multiply the top ratio by 2 the bottom one by 3.
So then, we can relate everything, butter to sugar to cookies, 2 to 3 to 24, and at this point, because we're talking about a difference, I'm going to express these in terms of scale factors. So, in other words, butter is 2n, sugar is 3n, cookies are 24n, and so the difference between sugar and butter is simply n. Well, we know that that difference should be five cups, so in other words n equals 5.
It means we had 10 cups of butter and 15 cups of sugar for the number of cookies. We multiply 24 times 5 and an easy way to do this, we'll divide the 24 by 2 and give that factor 2 to the 10. So dividing 24 by 2 we get 12, times 10 is 120. In summary, one method we can use is find equivalents of the common terms are equals.
Both rations. Then combine the ratios into one big ratio. Another method, if possible, solve for the individual value and then use ratios to solve for each one of the individual values. So, there's not a general rule about which one is better. Different problems will be better suited for one or the other.
It's good to know more than one way to solve a problem. In fact, this can be very helpful. Even if you solve the problem one way, later on try and solve it the other way. And get a sense of how these two methods compare.
Read full transcriptSophomores are what fraction of the whole team. So, I recommend that you pause the video here and try this problem. Okay, here's what we're gonna do. Identify the commonality. Here, the common element is juniors. Find equivalents of both ratios so that the common element is equal.
So, these are the original ratios. We have three representing juniors in the first ratio. We have five representing juniors in the second ratio. We'd like those two numbers to be equal. So the easiest thing to do is to figure out the least common multiple which, of course, is 15.
That means we have to multiply the top ratio by five and the bottom ratio by three. So, we have ratios at 10 to 15 and 15 to 18. Now that the common ratio is the same we can put everything together. Sophomores, to juniors to seniors. 10 to 15 to 18.
Well, now we can just use proportioning to solve. So, the whole is, of course, 43 parts, juniors are, sophomores are 10 parts out of that 43. So 10 out of 43 is the fraction. Here's another problem. In a certain company, the ratio of programmers to marketers is three to eight, and the ratio of customer service reps to marketers is two to three.
If there are 27 programmers, there are how many customer service reps? So, again pause the video here. And try this. Here we're looking for an absolute count. So we another strategy available.
We can solve for the absolute quantity in each term. So right now we know how many programmers, and we can relate programmers to marketers. So programmers to marketers, three to eight. We have 27 programmers. We'll use M for marketers.
Notice in this proportion, we can do some horizontal cancellation in the numerator. Cancel that to one and nine. From there, we can cross multiply, we get 72 marketers. Incidentally, if the idea of horizontal cancellation is something new to you, I would suggest watching the operations with proportions video. So now, we have the number of marketers, from this we can figure out the customer service reps, marketers to customer service reps is 2 to 3, marketers count as 72.
We can again do some horizontal cancellation here. In that denominator 3 goes into 72, 24 times from here we can cross multiply and there are 48 customer service reps. So, here we just sort of ratcheted our way through the ratios, figuring out an absolute quantity for each group, one at a time. Here's a third practice problem, pause the video and try this.
So in this one, we don't have any absolute quantities. So, we'll set up these ratios and we want to make an equivalent ratio. We want number of cookies to be the same, 8 and 12, where the least common multiple is 24. So we're going to multiply the top ratio by 2 the bottom one by 3.
So then, we can relate everything, butter to sugar to cookies, 2 to 3 to 24, and at this point, because we're talking about a difference, I'm going to express these in terms of scale factors. So, in other words, butter is 2n, sugar is 3n, cookies are 24n, and so the difference between sugar and butter is simply n. Well, we know that that difference should be five cups, so in other words n equals 5.
It means we had 10 cups of butter and 15 cups of sugar for the number of cookies. We multiply 24 times 5 and an easy way to do this, we'll divide the 24 by 2 and give that factor 2 to the 10. So dividing 24 by 2 we get 12, times 10 is 120. In summary, one method we can use is find equivalents of the common terms are equals.
Both rations. Then combine the ratios into one big ratio. Another method, if possible, solve for the individual value and then use ratios to solve for each one of the individual values. So, there's not a general rule about which one is better. Different problems will be better suited for one or the other.
It's good to know more than one way to solve a problem. In fact, this can be very helpful. Even if you solve the problem one way, later on try and solve it the other way. And get a sense of how these two methods compare.