Lesson 5 · from §86 of the 1910 text
More Problems in Two Unknowns: Work, Streams, and Trains
More problems with two unknowns
Reciprocals
If the unknowns appear as reciprocals, use \frac1x and \frac1y as the unknowns, as in the last lesson.
Work
If A can do a piece of work in a days, A does \frac1a of it in one day. If A and B together do it in 6 days, then \frac1a + \frac1b = \frac16.
Streams and trains
A boat or rower with rate x in still water on a stream with current y goes downstream at x + y and upstream at x - y.
Two trains running in opposite directions pass each other at the sum of their rates; in the same direction, at the difference. To pass completely, they must together cover the sum of their lengths. (A useful fact: 60 miles an hour is 88 feet a second.)
Example. A man rows 12 miles downstream in 2 hours and back in 3 hours. Let x = his rate in still water and y = the rate of the river, in miles an hour. Downstream, x + y = \frac{12}{2} = 6; upstream, x - y = \frac{12}{3} = 4. Adding, 2x = 10: he rows 5 miles an hour and the river runs 1 mile an hour.
Sometimes the facts of a problem lead to the same equation twice. Then the problem has an unlimited number of answers, and the answer to "find them" is that they cannot be found from the facts given.
Try one:
Quick check
A boat goes 24 miles downstream in 2 hours and back in 4 hours. Find its rate in still water and the rate of the current.
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.
State two equations in two unknowns and solve.
The sum of the reciprocals of two numbers is \frac{5}{24}, and the difference of their reciprocals is \frac{1}{24}. Find the numbers.
The difference of the reciprocals of two numbers is 1\frac16. The quotient of the greater number divided by the less is 1\frac78. Find the numbers.
If 15 grams are taken from one pan of a balance and placed in the other, the sum of the weights in the first will be \frac12 the sum of those in the second. But if 85 grams are taken from the second and placed in the first, the sums of the weights in each pan will then be the same. Find the weight in each pan at first.
A gives B $20; then B has twice as much money as A. B then gives A $75 and has left \frac15 as much as A. How many dollars had each at first?
The circumference of the fore wheel of a carriage is 2 feet less than that of the rear wheel. The fore wheel makes as many revolutions in going 155 feet as the rear wheel in going 186 feet. Find the circumference of each wheel.
If the length and the width of a rectangle are each increased one foot, the area will be increased 18 square feet. But if the length and the width are each decreased one foot, the area will be decreased by 16 square feet. Can the length and the breadth be found? What must their sum be?
A and B working together can do a piece of work in 2\frac25 days. A works 50% more rapidly than B. How many days would each require alone?
A and B together can do a piece of work in 7\frac15 days. They work together for 5 days, and A finishes the job by himself in 3\frac23 days. How many days would each require alone?
If the length of a rectangle is increased by 4 feet and the width decreased by 2 feet, the area is increased 8 square feet. But if the length is decreased by 1 foot and the width increased by 3 feet, the area is increased 33 square feet. Find the dimensions of the rectangle in feet and its area in square yards.
A rectangle has the same area as one 10 feet longer and 6 feet narrower. It also has the same area as one 4 feet longer and 3 feet narrower. Find the dimensions of the rectangle.
The products of three pairs of numbers are equal. One number in the second pair is 2 greater, and one in the third pair 3 greater, than the first number in the first pair. The other numbers in the second and third pairs are respectively 15 less and 18 less than the second number of the first pair. Find each pair of numbers.
If the number of men who together purchased a piece of land had been 3 more, each would have had to pay $200 less than he did; but if the number of men had been 4 less, each would have had to pay $500 more than he did. Find the number of men and the price of the land.
Streams and trains
A man rows 10 miles downstream in 2 hours and returns in 2 hours and 30 minutes. Find the rate of the river and his rate in still water.
A boat goes downstream 36 miles in 3 hours and upstream 24 miles in 3 hours. Find its rate in still water and the rate of the current.
The rate of a boat in still water is 8\frac12 miles an hour. It goes down the river from A to B in 14 hours. It returns one half the distance from B to A in 10 hours. Find the rate of the river and the distance from B to A.
A boat which runs 12 miles an hour in still water goes downstream from A to C in 7 hours. It returns upstream to B, 36 miles below A, in 5 hours. Find the distance from A to C and the rate of the stream.
A train leaves A one hour late and runs from A to B at 25% more than its usual rate, arriving on time. If it had run from A to B at 24 miles an hour, it would have been 10 minutes late. Find the distance from A to B and the usual rate of the train.
A train leaves A 40 minutes late. It then runs to B at a rate 20% greater than usual, and arrives 16 minutes late. Had it run 15 miles of the distance from A to B at the usual rate and the rest of the trip at the increased rate, it would have been 22 minutes late. Find the usual rate and the distance from A to B.
The rate of a passenger train is 66 feet a second and the rate of a freight train 44 feet a second. When they run on parallel tracks in opposite directions they pass each other in 15 seconds. The length of the freight train is twice the length of the passenger train. Find the length of each.
The rate of a passenger train is 45 miles an hour and that of a freight train is 30 miles an hour. The freight train is 350 feet longer than the passenger train. When the trains run on parallel tracks in the same direction they pass each other in 1 minute and 15 seconds. Find the length of each.
The length of a freight train is 1540 feet and the length of a passenger train 660 feet. When they run on parallel tracks in opposite directions they pass each other in 20 seconds, and when they run in the same direction they pass each other in 1 minute and 40 seconds. Find the rates of the trains in miles an hour.
Two bicyclists travel in opposite directions around a quarter-mile track and meet every 22 seconds. When they travel in the same direction, the faster passes the slower once every 3 minutes and 40 seconds. Find the rate of each rider in miles an hour.