Transcription
Let's do another ratio problem that's a bit of a classic. So, let's say that I start off with a certain number of fruit. So, I have some number of fruit. But I'll get back to that in a second. But we know that the ratio. We know that the ratio--we know that the ratio of apples--I'll do apples in this reddish color. The ratio of apples to oranges. To oranges, those colors are very similar. Let's say, it is equal to--let's say it is equal to 5:8. So it's equal to 5:8. 5:8. Or I could also write it as 5:8, just like that. That's the ratio of apples to oranges.
Now, the next thing that I know about this problem is that when I take away 15--when I take away 15 apples. So, take away 15 apples, and the ratio becomes 1:4. The ratio of apples to oranges becomes 1:4. How many apples--let me write it this way. How much total fruit do I have after taking away the 15 apples? The 15 apples? How much total fruit do I have? So let's do a little bit of algebra. Let's say that x is the number, x is equal to the number of fruit that we start with. How much fruit are we going to end up with at the end? We're taking away 15 apples. We're not doing anything to the oranges. So we know that x minus 30, I'll do that in a different color, x minus 15, we take away 15 apples, x minus 15 is equal to the number of fruit we end up with. We know these two things are true.
And now, we know that we start off with a ratio of 5:8. So we know that the ratio of apples to oranges is equal to 5:8 that we start off with. Or, that's just another way of saying that for every group--assuming that we can divide our group of fruit into even--or we could divide it into a set of groups, and every group has the same number of apples and oranges. That means that for every group of, what, 13 fruits, 13 fruits, there are 5 apples and 8 oranges. That's what this tells me. For every group of 13 fruits, there are 5 apples and 18 oranges. And how many groups of 13 fruit are there? Well, I have x total fruit. So if I divide x divided by 13, if I take x divided by 13, that is equal to--this is 13 thirteen right here. This is the 3. I have 13 fruits per group. I have x total fruit. So this is the total number of groups. So this is equal to the number of groups. Now, for every group, I have 5 apples. So how many apples am I starting off with? This is the number of groups. For every group, I have 5 apples. So I'm starting off with 5 times that apples. 5x over 13 apples. 5x over 13 apples. Now, for every one of these groups, I also have 8 oranges. So, how many oranges do I have? I have 8 times that. Because for every one of these, I have 8 oranges. So 8 times that is 8x over 13 oranges.
Fair enough. That's what I start off with. Now, when I'm done we said that the ratio of apples to oranges becomes 1:4. Becomes 1:4. Or, another way of saying that is that, for every 5--right, if you add these two numbers together, for every 5 fruit there are 1 apple and 4 oranges. Fair enough. So, how many groups of 5 fruit do we have? Well, this is the total number of fruit we're left with after we took 15 away. So how many groups of 5 do we have? We take the total number of fruit. x minus 15. Divide it by how many fruit we have per group. And this is how many groups of--this is equal to the groups of 5 fruit. Of 5 fruit. Now, for each of these groups, I'm going to have 1 apple and 4 oranges. So how many apples and oranges do I now have after I took away 15 of the apples? I took away 15 fruit? Let's think about this a little bit. How many apples am I going to have? I have 1 apple for every 1 of these groups. So, that's how many--so the number of groups is the number of apples I have. So, I'm going to have x minus 15 over 5 apples. And then how many oranges am I going to have? For every one of these groups, I'm going to have 4 oranges. So, I'm going to have 4 times this many oranges. So we could write that as 4/5 times x minus 15 oranges. That's just 5 times the number of groups of 5 fruit.
Now, what doesn't change? I'm messing with the number of apples, but I never changed the number of oranges. So, this number which is the number of oranges we started off with, has to equal this number. The number of oranges we ended up with. There's many ways to do this problem, but this is the way that's popping out of me that might be straightforward. So let's see if we can solve for x. This has to be equal to that. That never changed. So you could say 8x over 13 is equal to 4/5 times x minus 15. And, maybe, the best way to do this, let's multiply both sides of this equation times 5/4. So, times 5/4 times 5/4. These cancel out. That just becomes 1. What does this become? This becomes, you get a 4, divide it by 4, you get a 2 there. This becomes 10x over 13. 10x over 13, that's the left-hand side, is equal to x minus 15. Now, let's subtract x from both sides. And we get 10x over 13 minus x is equal to minus 15. Let me scroll down a little bit. And so, what is this? This is the same thing as 10x over 13 minus 13x over 13. That's just x. Is equal to minus 14. And then, 10 minus 13, this is equal to--so this is minus 3x over 13, is equal to minus 15. Now, we can just multiply both sides of this equation by minus 13 over 3. Multiply this side by minus 13 over 3. I picked that number so that this and that cancel out. And that and that cancel out, and the negatives cancel out. So I'm just left with x on that side. And then, the right-hand side becomes what? The negatives cancel out. I can divide the 15 and the 3 by 3, so that becomes a 5. And then this becomes a 1. And I'm just left with 5 times 13. This was just, times plus 13, I wrote that just to make sure that you understood that the negatives canceled out. So, 5 times 13 is what? 5 times 12 is 60, so x is equal to 65.
But that is not our answer. Remember what x was. x was the number of fruit we started with. We just figured out that this is equal to 65. But what the question asked, the question wanted to know is, how much total fruit do I have after taking away the 15 apples. So the question wants to know, how many after taking the 15 apples. If I started with 65, I give away 15 of the fruit, all of them are apples, then I'm left with 65 minus 15, which is equal to? 50 pieces of fruit. 50 pieces of fruit. That's what they're asking for. 50 pieces of fruit. If you want, you can go on and figure out how many apples you started with, you finished with, the number of oranges, or whatever else, but this is what they're asking. How much total fruit, not the number of apples, total fruit, do I have after taking away the 15 apples?