Ratios and Rates (Advanced) - By Anywhere Math
Transcript
| 00:0-1 | Welcome anywhere . Math . I'm Jeff Jacobson . And | |
| 00:01 | today we're gonna talk about ratios and rates . Let's | |
| 00:05 | get started . Yeah . Okay . Today's lesson on | |
| 00:24 | ratios and race is gonna be a little more advanced | |
| 00:27 | uh than the previous lesson . If you're new to | |
| 00:31 | ratios and rates , check out this video up here | |
| 00:33 | . That's when I did earlier . And that's kind | |
| 00:36 | of covering the basics of ratios and rates . Okay | |
| 00:38 | , before we get in our first example , just | |
| 00:40 | a quick recap on ratios and rates and unit rates | |
| 00:44 | . So first a ratio , that's just a way | |
| 00:46 | of comparing two quantities . You don't have units . | |
| 00:49 | Uh And there's three ways you can write it . | |
| 00:51 | We could write it . If we're doing the ratio | |
| 00:53 | of 3 to 4 , we could write 3 to | |
| 00:57 | 4 , spelling it out , We could right 3-4 | |
| 01:00 | like that with a colon . Or we could write | |
| 01:03 | it as a as a fraction where three would be | |
| 01:06 | in the numerator and four would be in the denominator | |
| 01:09 | . Those are all ratios and now rates . Those | |
| 01:12 | are ratios of two quantities , but they have different | |
| 01:15 | units . So for example , uh instead of just | |
| 01:18 | 3-4 , I could say uh three km to four | |
| 01:25 | hours . Okay . Or again , I could write | |
| 01:28 | it this way , three km to four hours . | |
| 01:33 | Or as a fraction three km every four hours . | |
| 01:39 | With rates the most common way to see it is | |
| 01:44 | like a fraction When we're doing rates most of the | |
| 01:46 | time , they're gonna write it as a fraction . | |
| 01:48 | Now , last unit rate . Unit rate is a | |
| 01:51 | special kind of rate where the denominator is one . | |
| 01:55 | So if you look here , if we write it | |
| 01:57 | as a fraction the denominators for , that's why this | |
| 01:59 | is just a rate and not a unit rate . | |
| 02:02 | But common unit rates are things like if you go | |
| 02:05 | to the grocery store and they say , You know | |
| 02:08 | , the meat is $3.25 per pound , that's per | |
| 02:12 | £1 . Right ? So that would be a unit | |
| 02:15 | rate or a speed 60 mph . We would write | |
| 02:20 | that at 60 mph one hour . That's a unit | |
| 02:25 | rate . The denominator is one . Um so that's | |
| 02:28 | just a quick recap . Let's get to our first | |
| 02:30 | example . Okay example one use the graph to find | |
| 02:34 | the speed of a subway car . So if we | |
| 02:36 | look at this graph , the title is speed of | |
| 02:38 | a subway car . The y axis is distance in | |
| 02:43 | miles . The X axis is time in minutes . | |
| 02:46 | And if you look at that line it's straight now | |
| 02:50 | . What that means that line is showing the speed | |
| 02:54 | and because it's a straight line it means the speed | |
| 02:57 | is constant . The speed is not changing if the | |
| 03:00 | line was like this and then like this and then | |
| 03:03 | like this and then like that then the speech is | |
| 03:05 | changing all the time . But because it's a straight | |
| 03:08 | line means that that speed is constant and that's important | |
| 03:13 | because what that allows us to do is pick any | |
| 03:16 | point on that line and use that point to be | |
| 03:20 | able to find the speed . It doesn't matter which | |
| 03:23 | point we choose because it's a straight line , it's | |
| 03:26 | always gonna be the same . The speed will always | |
| 03:28 | be the same along that line . So we can | |
| 03:32 | choose any point we want . Let's just choose that | |
| 03:34 | first point . That first point is one half 1/4 | |
| 03:39 | . And if we look this is the X coordinate | |
| 03:42 | and like we said earlier on the X axis , | |
| 03:44 | that's the time . And this is the why . | |
| 03:48 | And that is the distance . Now . To find | |
| 03:52 | speed , you need to know how do we find | |
| 03:55 | speed ? And speed is pretty simple if you think | |
| 03:58 | of speeds like MPH or kilometers per hour , It's | |
| 04:04 | just calculated by the distance distance traveled over the time | |
| 04:09 | the time it takes to go . So when someone | |
| 04:11 | says 60 mph that means It takes it takes one | |
| 04:16 | hour to go 60 miles . So here is our | |
| 04:19 | little formula for speed . Here's our our data point | |
| 04:23 | . This is what we're gonna use . So distance | |
| 04:27 | 1/4 . So I'm gonna put that in the numerator | |
| 04:31 | divided by time , which is one half , 1/4 | |
| 04:34 | divided by one half . Now we need to simplify | |
| 04:38 | this right here is a special kind of fraction . | |
| 04:42 | If you notice it looks weird because you have a | |
| 04:44 | fraction within a fraction and we call this um All | |
| 04:49 | right here a complex fraction . That's a complex fraction | |
| 04:54 | . When you have a fraction within a fraction . | |
| 04:56 | But to solve it or to simplify it , it's | |
| 04:59 | really quite simple . So if I remember this line | |
| 05:02 | right here means division . So this is 1/4 divided | |
| 05:05 | by one half . Well divided by a fraction . | |
| 05:08 | 1/4 divided by one half , divided by a fraction | |
| 05:13 | of the same thing as multiplying by the reciprocal . | |
| 05:15 | So I can rewrite this has 1/4 times to over | |
| 05:20 | one . And if I simplify that , that's going | |
| 05:23 | to give me one half and the units are miles | |
| 05:28 | per minute . So there is my speed . Let's | |
| 05:33 | try another example . Alright example to write the ratio | |
| 05:36 | as a fraction in simplest form . So here we | |
| 05:40 | have some ratios . Uh they're not written as fractions | |
| 05:43 | so we gotta do that first . Remember we can | |
| 05:45 | do that with ratios . So 63-28 , I'm gonna | |
| 05:48 | write that as 63/28 . Remember the first number that | |
| 05:52 | goes into your numerator ? Uh And now I just | |
| 05:55 | have to simplify well , common factor of 63 28 | |
| 06:01 | 7 would be a common factor . So if I | |
| 06:03 | divide by seven divided by seven I'll get 9/4 . | |
| 06:08 | They want as a fraction in simplest form , that's | |
| 06:11 | a fraction . I won't change it to a mixed | |
| 06:13 | number . And that is in simplest form part B | |
| 06:18 | , two and one third feet to 4.5 ft . | |
| 06:21 | Now here we've got mixed numbers , so my first | |
| 06:24 | step is going to change them to improper fractions . | |
| 06:28 | So this becomes seven thirds . I don't need to | |
| 06:31 | feed anymore because we're just doing a ratio , we're | |
| 06:34 | just doing the quantities , so no units to nine | |
| 06:38 | half . And again I'm gonna write that as a | |
| 06:40 | fraction . Seven thirds over nine halves . That's a | |
| 06:46 | complex fraction . Again fraction within a fraction . that | |
| 06:49 | just means 7/3 divided by nine halves . And that | |
| 06:56 | is the same thing as 7/3 times the reciprocal , | |
| 07:02 | which is tonight . Nothing to simplify their , unfortunately | |
| 07:06 | . So seven times two is 14 and three times | |
| 07:10 | nine is 27 and there are no common factors there | |
| 07:17 | . So that isn't simplest form . Here's something to | |
| 07:20 | try on your own example , three find the unit | |
| 07:29 | rates . So 21 3/4 meters , 2 , 2.5 | |
| 07:32 | hours notice we have different units , which is important | |
| 07:35 | for rates . Um And for a unit rate , | |
| 07:38 | this needs to become one hour instead of 2.5 . | |
| 07:41 | So what I'm gonna do first is I don't want | |
| 07:44 | to deal with mixed numbers , you could change these | |
| 07:46 | two decimals , that would be easy , but I'm | |
| 07:49 | going to change them just to improper fractions . So | |
| 07:51 | this is going to become 87/4 uh meters to five | |
| 07:59 | half hours . And now I'm gonna rewrite this as | |
| 08:02 | a complex fraction , 87 force divided by five halves | |
| 08:08 | . And now I'm just gonna do that division . | |
| 08:10 | Just simplify 87 force divided by five halves , which | |
| 08:15 | is the same thing as multiplying by two fists . | |
| 08:19 | Simplify their simplify there and I get 87/10 , Which | |
| 08:27 | is really nice because I'm just going to change that | |
| 08:29 | to a decimal 8.7 . And now I got to | |
| 08:32 | remember my units . This is 8.7 in the numerator | |
| 08:36 | . Right when we simplify that , that becomes my | |
| 08:39 | first number 8.7 m per one hour . And there | |
| 08:44 | is my una rate for a Now let's look at | |
| 08:46 | B seven kilometers to 0.25 hours . Now with this | |
| 08:50 | one again I'm gonna write it seven kilometers over 0.25 | |
| 08:56 | hours . Remember the goal is to get This denominator | |
| 09:00 | to one . So instead of dividing I could divide | |
| 09:04 | by 0.25 . But instead of doing that There's a | |
| 09:08 | little trick maybe you're noticing it . If I multiply | |
| 09:11 | this by four That would give me to one hour | |
| 09:14 | . And if I anything I do to the denominator | |
| 09:17 | I should do the same thing to the numerator to | |
| 09:19 | make sure it stays equivalent . So a quick little | |
| 09:22 | tip here is just multiplied by 4/4 Which is gonna | |
| 09:27 | give me 28 km/ one . Our and there is | |
| 09:35 | my unit right there . Here's some more to try | |
| 09:38 | on your own as always . Thank you so much | |
| 09:43 | for watching and if you like this video please subscribe | |
| 09:45 | at . Mhm . |
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