Systems and rate problems

Aug 15, 2023

Systems and rate problems

U06_L2_T1_we1 Systems and rate problems
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Content

0.77 -> Jason bicycled from home to the train station at an

4.08 -> average speed of 10 miles per hour.

7.12 -> Then he boarded a train and traveled into the city at an

9.91 -> average speed of 50 miles per hour.

12.1 -> The entire distance was 30 miles; the

14.36 -> entire trip took 1 hour.

16.44 -> How many miles did Jason travel by train?

20.36 -> So let's write that down as a variable.

21.98 -> So let's say distance, and we'll write a little small t

24.62 -> right here, a little subscript t.

26.06 -> This is the distance by train, so train distance.

32.84 -> Right there.

33.72 -> And let's have a d with a little bit of a

36.28 -> little b right there.

37.25 -> That is the bike distance.

42.18 -> Now, they give us one piece of information, that the total

44.99 -> distance, the entire distance, was 30 miles.

48.92 -> So that means that the distance by train plus the

51.2 -> distance by bike is 30 miles.

54.03 -> So I could write that right here.

55.14 -> The distance by train plus the distance by bike

59.68 -> is equal to 30 miles.

61.91 -> That's what that constraint tells us.

63.65 -> Now the next one, they tell us that the

64.81 -> entire trip took 1 hour.

67.15 -> So the time by train plus the time by bike took 1 hour.

71.98 -> And you might be thinking, hey, gee, that's going to

73.45 -> introduce two new variables.

75.04 -> But let's think a little bit about whether we can express

77.76 -> the time by train and the time by bike in terms

80.97 -> of these two variables.

82.54 -> So just as a bit of review-- I think this is review-- you're

86.56 -> probably familiar with distance is equal to rate

90.63 -> times time.

91.96 -> Or if you divide both sides of this equation by rate, you get

96.78 -> time is equal to distance divided by rate.

102.89 -> So let's think about the situation for

104.7 -> each of these guys.

105.8 -> What is the time by train?

107.34 -> I'll write it like this.

108.2 -> The time by train is going to be equal to the distance by

112.13 -> train divided by the rate that the train was going at.

116.22 -> And they gave us that information.

118.2 -> They told us that the train traveled into the city at an

121.14 -> average speed of 50 miles per hour.

123.2 -> That is the rate of the train.

125.35 -> So the time of the train was the distance of the train

129.259 -> divided by 50.

131.32 -> Same exact argument.

133.42 -> The time of the bicycle, the time traveled on the bicycle,

136.99 -> will be the distance traveled on the bicycle divided by the

139.95 -> speed of the bicycle.

141.05 -> And they give us the speed right there,

142.34 -> 10 miles per hour.

143.51 -> And everything in this problem we're assuming is going to be

145.53 -> in either miles or hours.

147.05 -> There's not any major unit conversion.

149.35 -> So that's there right there.

150.61 -> So the second statement is that the

153.69 -> entire trip took 1 hour.

156.11 -> So the time by train plus the time by bicycle is

159.47 -> going to be 1 hour.

161 -> Actually, let me do that in a different color.

163.765 -> The entire trip took 1 hour.

166.57 -> So this time plus this time is 1 hour.

169.21 -> And I'm going to write it in terms of the distances so I

171.1 -> only have two unknowns.

173.21 -> So the time by train is the distance by train divided by

177.78 -> 50 plus the time by bicycle is the distance by bicycle

183.05 -> divided by 10.

184.51 -> And then that is equal to 1 hour.

187.25 -> That's what this statement right here is telling us,

189.02 -> although we had to use that information and that

192.2 -> information to divide by the 10 and to divide by the 50.

196.15 -> Now we have two equations of two unknowns.

198.81 -> And our whole goal, the whole purpose of this problem, they

201.57 -> want to know how many miles did Jason travel by train.

205.51 -> So we want to figure out that variable, or we want to figure

209.44 -> out that variable.

210.73 -> Now, the easiest way to do this is if we can just

212.71 -> eliminate the distance by bicycle and then solve for the

216.02 -> distance by train.

217.05 -> Then we're done.

217.64 -> We would have solved the problem.

219.63 -> Now, the easiest way I can think of doing that is this

222.53 -> one is pretty simple already.

223.99 -> Let me just rewrite it to the right here.

226.02 -> So the distance by train plus the distance by bicycle is

229.7 -> equal to 30.

231.23 -> And I want to cancel out the distance by bicycle.

233.5 -> So if I could make this into just a negative distance by

236.33 -> bicycle, then I'm all set.

238.03 -> And if I add the two equations, they'll cancel out.

239.96 -> So the easiest way to just make this a negative distance

241.95 -> by bicycle is multiply both sides of this equation by

245.21 -> negative 10.

250.1 -> Because you multiply negative 10 times this, the 10's cancel

252.36 -> out, this just becomes a negative distance by bicycle.

254.92 -> So let's write it over here.

255.74 -> Negative 10 times the distance by train over 50.

259.18 -> 10 divided by 50 is 1/5, so it's negative

261.42 -> 1/5 distance by train.

267.05 -> And then negative 10 times this expression right here,

269.62 -> the 10's cancel out.

270.47 -> It's just negative distance by bicycle is equal to-- forgot

275.16 -> that parentheses-- is equal to negative 10.

278.89 -> The whole point here was so that this becomes the negative

281 -> version of that.

281.7 -> So when I add these two equations, they'll cancel out.

284.46 -> So let's do that.

285.83 -> Let's add these two equations.

288.41 -> So if you add the left-hand side, these guys cancel out.

291.47 -> You have 1 dt minus 1/5 dt.

294.76 -> 1 dt you can view as 5/5 dt.

297.91 -> So 5/5 minus 1/5, you have 4/5 distance by train is equal to

306.14 -> 30 plus negative 10, or is equal to 20.

309.99 -> Now to solve for distance by train, we can just multiply

313.27 -> both sides of this expression, both sides of this equation,

316.06 -> by the inverse of 4/5.

317.97 -> So we'll multiply both sides by 5/4.

322.63 -> The whole point of that is that this cancels out, and we

325.92 -> are left with the distance by train is equal to 20 times

331.53 -> 5/4, that's the same thing as 20/1 times 5/4.

334.84 -> 20 divided by 4 is 5, 4 divided by 4 is 1.

338.85 -> So it's just 5 times 5.

340.85 -> So the distance traveled by train is 25 miles.

345.21 -> And we're done.

345.83 -> And we could go back if we wanted to figure out any of

347.62 -> the times, or the distance by bicycle.

350.27 -> Actually, it's very easy.

351.06 -> The distance by train is 25, distance by bicycle has to be

354.28 -> 5 if they're going to add up to be 30 miles.

Source: https://www.youtube.com/watch?v=at4T4n4JYNc