Lesson 9 · from §74 of the 1910 text
Problems: Work and Motion
Work problems
If A can do a piece of work in a days, A does \frac1a of it in one day. When people work together, add the parts done in one day.
Example. A can do a piece of work in 2 days, B in 3 days, and C in 4 days. How long will it take them, working together?
Let x = the number of days needed by all three. Then they do \frac1x of the work in one day, and \frac12 + \frac13 + \frac14 = \frac1x. Solving, \frac{13}{12} = \frac1x, so x = \frac{12}{13} of a day.
Motion problems
Distance = rate × time, so time = \dfrac{d}{r} and rate = \dfrac{d}{t}. A table keeps the facts straight.
Example 1. Two bicyclists start at the same time to ride from A to B, 80 miles distant. One travels 4 miles an hour faster. The faster one reaches B and at once returns, meeting the slower one 64 miles from A. Find the rate of each.
Together they ride twice the distance, 160 miles. The slower rides 64, so the faster rides 96. The times are equal. With x for the slower rate:
| d (miles) | r (mi/h) | t = d/r (hours) | |
|---|---|---|---|
| Slower | 64 | x | \frac{64}{x} |
| Faster | 96 | x + 4 | \frac{96}{x + 4} |
\frac{64}{x} = \frac{96}{x + 4}, \quad 64x + 256 = 96x, \quad x = 8. The rates are 8 and 12 miles an hour. Check: \frac{64}{8} = 8 and \frac{96}{12} = 8. ✓
Example 2. A man travels at a uniform rate from A to B, 120 miles. He travels the first 70 miles without stopping. The remaining 50 miles, including a delay of 2 hours, take the same time as the first part. Find his rate.
Here the rate is the same. Let x = the hours for the first 70 miles; then he spends x - 2 hours actually moving on the last 50: \frac{70}{x} = \frac{50}{x - 2}, \quad 70x - 140 = 50x, \quad x = 7. His rate is 70 \div 7 = 10 miles an hour.
Streams
With a current of c miles an hour, a rower whose still-water rate is r goes r - c upstream and r + c downstream.
Some of these problems lead to quadratics. Throw out a negative root: it can't be a number of days or a speed. (Problem 53 is unusual: both of its answers work.)
Try one:
Quick check
A can do a piece of work in 6 days and B in 3 days. How long will it take them, working together?
From the history of algebra
Work problems are among the oldest in mathematics. The Chinese Nine Chapters on the Mathematical Art (about 200 BC) asks how long several canals filling a pond together would take, and the Greek Anthology (about 500 AD) has a fountain-and-pipes version: "This lion of bronze spouts water from his eyes, mouth, and foot…" The trick of adding the work done in one unit of time is the same one used here.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Give fractional answers as fractions or mixed numbers, like 12/13 or 4 5/12.
A can do a piece of work in 2 days, B in 3 days, and C in 4 days. How long will it take them, working together?
A can do a piece of work in 2\frac12 days and B in 3\frac34 days. How many days will they require, working together?
A can do a piece of work in 2 days, B in 2\frac45 days, and C in 3\frac12 days. How many days will they require, working together?
A can do a piece of work in 8 days, and A and B together can do it in 4\frac45 days. How long would it take B alone?
A can do a piece of work in 3\frac12 days, B in 2\frac45 days, and A, B, and C together can do it in 1\frac16 days. In how many days can C do the work alone?
A can do a piece of work in 12 days and B in 15 days. After A works 3 days alone, A and B finish the work. How long do they work together?
Two bicyclists start at the same time to ride from A to B, 80 miles distant. One travels 4 miles an hour more than the other. The faster bicyclist reaches B and at once returns, meeting the slower one 64 miles from A. Find the rate of each.
A man travels at a uniform rate from A to B, 120 miles distant. He travels the first 70 miles without stopping. The remainder of the journey, including a delay of 2 hours, requires the same time as the first part. Find his rate.
A bicyclist traveling 10 miles an hour was overtaken 5\frac23 hours after he started by an automobile which left the same starting point 1 hour and 40 minutes later. What was the rate of the automobile?
Two bicyclists, A and B, start at the same time to ride from X to Y, 63 miles distant. A travels 3 miles per hour less than B. The latter reaches Y and at once returns, meeting A 9 miles from Y. Find the rate of each.
A leaves a certain point and travels at the rate of 4\frac12 miles an hour. Two and one half hours later B leaves the same point and travels in the opposite direction at the rate of 10\frac12 miles an hour. How much time must elapse after A starts before they will be 40 miles apart?
A bicyclist traveled to a certain point and returned, making the trip in 18 hours. The total distance was 160 miles, and the rate going was 8 miles an hour. At what rate did he return?
A and B start at the same time from two towns 150 miles apart and travel toward each other. Their respective rates are 9 and 12 miles an hour. A rests 3 hours and B rests 4\frac12 hours before they meet. How far has each of them traveled when they meet?
A and B leave the same place at the same time for a point 63 miles distant. A travels 3\frac12 times as fast as B. The former reaches the point and returns immediately, meeting B 8 hours from the time of starting. Find the rates of A and B.
A man rows 4\frac14 miles per hour in still water. He finds that it requires 5 hours to row upstream a distance which it requires 3 hours to row down. Find the rate of the current.
A man who can row 4 miles per hour in still water rows up a stream the rate of whose current is 2 miles per hour. After rowing back he finds that the entire trip took 12 hours. How far does he row upstream?
A man who can row 4\frac13 miles an hour in still water rows downstream and returns. The rate of the current is 2\frac14 miles per hour and the time required for the trip is 13 hours. How many hours does he require to return?
A and B together can do a piece of work in 1\frac15 days. A alone can do the work in one day less than B. Find the time each requires alone.
A and B together can do a piece of work in 2\frac25 days. B alone can do the work in two days less time than A. Find the number of days required by each alone.
A man rows upstream and back, a total distance of 20 miles, in 6 hours. His rate upstream is 2\frac12 miles per hour. Find the rate of the current and his rate in still water.
A farmer pays $96 for a number of sheep. He sells all but 2 of them for $100, and gains $2 on each sheep sold. Find the number of sheep bought.
A piece of cloth is bought for $64. Four yards are cut off and the remainder is sold, at an advance of $2 per yard, for $72. Find the cost per yard.
A train runs 100 miles. On the return trip it increases its rate 5 miles per hour and makes the run in one hour less time. Find the rate going and returning.
An automobile makes a run of 120 miles. The chauffeur then increases the speed 4 miles per hour and returns over the same route in 5 hours less time. Find the rate going and returning.
Two automobiles travel 72 miles over the same route. One travels 2 miles per hour more than the other and makes the run in 30 minutes less time. Find the rate of each.
A and B start at the same time from two points 144 miles apart and travel toward each other. A's rate is 4 miles less than B's. The latter, having been delayed 3 hours on the way, has traveled the same distance as A when they meet. Find the rate of each.
A and B start from the same place at the same time and travel in opposite directions. B is delayed 2 hours on the way, and at the end of a certain time the two are 172 miles apart. If A has traveled 28 miles farther than B and one mile more per hour, find the rate of each.