Transcription
Welcome to the presentation on more advanced ratio problems. Let's get started with some problems.
So let's say that I have a class, and then--oh, the pen is messed up--OK. So in the class, the total number of students is 57. And I would also tell you that the ratio of girls to boys is equal to 4:15. So now this is the interesting part; so far, it doesn't look too tough. My question is, how many boys need to leave the room, so how many boys need to leave for the ratio of girls to boys to be 4:11? This is fascinating.
So, a good place to start is just to figure out how many girls and how many boys there are in this classroom. And we already learned how to do that in the introduction to ratio problems. We know that the girls plus the boys is equal to 57, right, because there are 57 kids in the room. And we also know, just multiplying--taking this equation--and multiplying both sides by b, we also know that the girls are equal to 4/15 times the boys, right? And then we can just substitute that back into this equation, and then we get 4/15b + b = 57, which is the same thing as 19/15b = 57. Let me clean this up a little bit. That's separate, and then let me go here. And we say b = 57--oh, whoops--it's actually 57 times 15, all of that over 19, right? I just multiply both sides by 15/19. So 57 divided by 19 is 3. So b = 45. And we know there are a total of 57 kids in the class--g + b = 57--so we know that there are 12 girls, right? 57 - 45. Good.
So now we know that the current boys and girls are 45 boys and 12 girls. So let's write that down. So there are 12 girls and 45 boys. Now, the question says, how many boys need to leave for the ratio of girls to boys to equal 4:11? So this is the number of girls right now, 12; this is the number of boys. Let's say x is the number of boys that need to leave the room. So if x boys leave the room, the new ratio will be 12 girls to the 45 boys minus the x boys that leave, right? If that confuses you, sit and look at that for a second. We start off with 12 girls and 45 boys in the room. And we're saying x boys are going to leave, so the new ratio is going to be 12:45 - x. And we know from this part of the problem that that new ratio is going to equal 4:11. There, we just set up an equation with one unknown, and we can solve for x. I hope that doesn't confuse you much. All we did is we figured out how many boys, how many girls are in the room now. We said x is the number of boys that need to leave. And we said the new ratio is going to be girls to the new number of boys, which is 45 - x, and that's going to be equal to the new ratio.
So let's solve for x. Well, 12 times 11 is what, that's 132. 132 = 4 times 45, 160, 180, - 4x. And then if you solve for x, I think you know how to do this right now, and we can say -4x = -48. x = 12. There, we solved it. So we say that if 12 boys left the room, the new ratio of girls to boys would be 4:11. And does that make sense? Well, if 12 boys left the room, then the new ratio of girls to boys would be 12:33, right? Because 45 - 12 = 33. And that's the same thing as if you divide the top and bottom by 3. That's 4:11. So there, we got it right. So what looked like a very hard problem actually wasn't so bad when you just sit down and work through the algebra.
Let's do another problem. Let's say--this thing sometimes malfunctions--OK. Let's say that the ratio of apples to bananas in a basket is equal to 5:19. And when we add 23 bananas, the ratio of apples to bananas--and actually let's write it right now, we now have 23 bananas more--is equal to 10:61. So the question is, what is the total amount of fruit in the basket--amount of fruit--ah, that's so messy--after adding the bananas? So I actually gave you a hint just when I wrote down the initial problem. We're saying the ratio of a to b--so let a equal the number of apples, and b equal the number of bananas--so the ratio of apples to bananas equals 5:19. When I add 23 bananas, now the new ratio's going to be the number of apples to b + 23. The new ratio is 10:61. So how do we solve this? Well, once again we have two equations and two unknowns. We know that--I guess let's take this equation first, because it's a little more complicated--we know if we cross-multiply that 61a = 10b + 230, and if we divide both sides by 61, we know that a = 10/61b + 230/61. Right? And we could take this equation and multiply both sides by b and we could say that a = 5/19b. Right? Well, both of these are equal to a, so we could set them equal to each other. And you get 5/19b = 10/61b + 230/61. And we solve for b. While this might seem complicated to you at first, but it's just a basic linear equation. And for the sake of time, because I only have 2 minutes left in this YouTube, I'm just going to solve for b, and you get b = 38. If b = 38, we know that the initial ratio is 5:19. So that's pretty easy. We just say a = 5/19 times 38 = 10. So the initial number of apples was 10, the initial number of bananas is 38, right? So initially we started off with 48 pieces of fruit, and then we're going to add 23 more pieces of fruit, right? And 48 + 23 = 71 pieces of fruit.
So, let me review real quick what we said. We said the ratio of apples to bananas is 5:19. That's a is the number of apples, b is the number of bananas. When I add 23 bananas, I now have b + 23 bananas; the new ratio of apples to the total number of bananas is 10:61. And I just used both of these equations. Two equations and two unknowns. Solved for a, and then substituted, and I solved for b. Nothing fancy here. I know there are a lot of fractions here, but if you just work through this, the fractions actually work out. And I was able to solve for a and b. Add the 23 pieces, and I got 71 total pieces of fruit. I think you're now ready to try some of the more difficult ratio problems. Have fun!