Hot X: Algebra Exposed

Transcription

1 Hot X: Algebra Exposed Solution Guide for Chapter 14 Here are the solutions for the Doing the Math exercises in Hot X: Algebra Exposed! DTM from p Okay, our units are consistent: bracelets and hours. So, first we need to write their rates as jobs per time. Katie s rate is 3 bracelets per 4 hours, in other words, 3 bracelets 4 hours or 3 4 of a bracelet in one hour. Similarly, Megan s rate is 4 of a bracelet in one hour. We want to know their combined rate, which we can get from adding these two rates together. How many bracelets can they make in one hour, working together? That s just To add these, we use copycats to get a common denominator of 20, and we get: =! ! 4 = = Their combined rate is So that means they can make bracelets in an hour; in other words, 31 bracelets in 20 hours. The question asked how long it would take them to make 31 bracelets, and how about that! It would take 20 hours. Answer: 20 hours

2 3. Our units are dollhouses and hours; no converting necessary. Let s write Fia s rate as jobs per time, and that would be: 1 dollhouse per 6 hours, in other words, 1 6 of a dollhouse in an hour. Her dad s rate is 1 dollhouse per 4 hours, in other words, 1 4 of a dollhouse in an hour. What s their combined rate? How much of a dollhouse could they paint in one hour, working together? That s: We ll use copycats to get a common 4 denominator of 12, and we get: = 2 2! ! 1 4 = = 12 So their combined rate is, meaning they could paint of a dollhouse in an hour, in other words, they could paint dollhouses in 12 hours. But the problem wants to know how long it would take them to paint just one dollhouse. So we can use our formula: jobs/time * time = total jobs done And in this case, the jobs/time is their combined rate,, and the total jobs done is We want to solve for the time, and so: 12 t = 1 Multiplying both sides by 12 to isolate t, we get: 12 t = 1 à 12! 12 t = 12!1 à t = 12 à t = 12 hours And what is 12 hours? First we ll write it as a mixed number: 2 2 hours. Okay, so what s 2 of an hour? Let s convert it to minutes: 2 60 min hours! 1 hour = 120 min = 24 minutes So it will take Fia and her father 2 hours and 24 minutes to complete the job. Done! Answer: 2 hours, 24 minutes

3 4. Our units are not consistent; we have hours and minutes. Let s take the suggestion and convert the hours to minutes first. So Lily s mom can build a snowman in 2 hours, or 120 minutes. In jobs per time that s 1 snowman per 120 minutes, in other words: of a snowman in a minute. Let s call Lily s rate r. We know their combined rate (adding mom s rate and Lily s rate) is 1 snowman in 4 minutes, in other words: 1 4 of a snowman in a minute. So that means: r = 1. With me so far? 4 Isolating r, we get:. How can we subtract these? Let s find the LCM of 4 and 120 by doing a quick birthday cake method (see p.23 to review this method). It s always a good idea to stay sharp on our fraction/factoring skills and not ever be intimidated by big numbers! The birthday cake method makes it a little better, doesn t it? J So we ll use an LCD of 360 for our subtraction, and we can look at the factoring X 3 X 3 X 8 = 360 to get clues for what copycats we ll need to use to attain a denominator of 360.

4 So this is Lily s snowman building rate, in terms of jobs per time, which means Lily can build a snowman in 72 minutes by herself. And what s 72 minutes? Why that s 1 hour and 12 minutes, of course! We could also do this problem by converting everything to hours instead of minutes. In that case, mom s rate is 1 snowman in 2 hours, in other words: of a snowman in 1 hour. And you probably know that 4 minutes is of an hour but it s not hard to do the conversion:. But this is tricky: You see, that s not the combined rate; that s just the time it takes them. The Jobs per time combined rate would be 1 snowman per hour, which can be written as:, a complex fraction we can solve using the means & extremes method from p. 2: And this is their combined rate: of a snowman per hour. Remembering that mom s solo rate is, and if we call Lily s solo rate by r, then we can set up this equation to solve for r: à This means Lily s rate (by herself) is of a snowman in an hour. The problem actually asks for the time it takes for Lily to build a snowman by herself, which we can find with our formula: jobs/time * time = total jobs. Filling in for the jobs/time and 1 for total jobs, and we get:

5 Isolating t, we get: à à t = hours And what s hours? It s hours. So what s of an hour? Let s convert! = 12 minutes. So it would take Lily 1 hour, 12 minutes to build a snowman by herself, and now we ve solved this problem two ways. Navigating the tricky part of this problem was made possible by remembering the golden rule from this chapter: Always write the rates in terms of jobs per time! Answer: 1 hour, 12 minutes. Our units are invitations and hours, so no conversions are necessary. Hm, lots of unknown rates here; let s label em! We ll call Jen s rate j, Emily s rate e, Sarah s rate s, and Hillary s rate h. What information are we given? That their combined rate equals 20 invitations per hour (already in Jobs per time format, great!), and that translates into: j + e + s + h = 20 We re also told some relationships between these rates which we should also translate into math: Jen is twice as fast as Sarah means that j = 2s. Emily is three times as fast as Sarah means that e = 3s, and Hillary is four times as fast as Sarah means that h = 4s. This means we can rewrite the above rate sentence like this: j + e + s + h = 20 à 2s + 3s + s + 4s = 20 and this simplifies to: 10s = 20 à s = 2. Hey! That answers part a of this problem, which asked for Sarah s rate. And we now know that it is 2 invitations per hour.

6 For part b, it asks how long it would take the girls to make 18 invitations without Hillary. First we should find the combined rate of all the girls without Hillary. That would be: j + e + s in other words, 2s + 3s + s. But since we know the value of s, which is 2, we can actually find this rate! The combined rate without Hillary is: 2s + 3s + s à 2(2) + 3(2) + 2 = = 12 invitations per hour. So how long would it take to make 18 invitations? We just do: jobs/time * time = total jobs à 12t = 18 à t = hours, in other words, 1 hour and 30 minutes. Done! Answer: Part a: 2 invitations per hour Part b: 1 hour and 30 minutes