Lesson 8 · from §87 of the 1910 text
More Problems with Letters: Money, Work, and Motion
More problems with letters
These are the work, money, and motion problems of earlier lessons with letters in place of some numbers.
- Work: if A needs x days, A does \frac1x of the work in a day.
- Motion: time =\dfrac{\text{distance}}{\text{rate}}, distance = rate \times time.
- Interest: a\% of x dollars is \dfrac{ax}{100} dollars.
Example. A man walks m miles at 5 miles an hour and rides back at 6 miles an hour. Going takes \frac m5 hours and returning \frac m6 hours, so the whole trip takes \frac m5 + \frac m6 = \dfrac{11m}{30} hours.
Try one:
Quick check
A works twice as fast as B. Together they can do a piece of work in d days. How many days would each require alone?
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.
Solve each problem. The answers are expressions in the letters.
A and B have together $40. A gives h dollars to B, after which B gives k dollars to A. Then they have equal sums. How many dollars had each at first?
A gives r dollars to B and then has \frac12 as much money as B. Then B gives $8 to A and has left \frac34 as much money as A. How many dollars had each at first?
A part of $1000 is invested at a% and the remainder at b%. The yearly income from both investments is c dollars. How many dollars are there in each investment?
A portion of x dollars is invested at 5% and the remainder at 4%. The yearly income is y dollars. How many dollars are there in each investment?
A works three times as fast as B. Together they can do a piece of work in c days. How many days would each require alone?
A works h times as fast as B. Together they can do a piece of work in 4 days. How many days would each require alone?
A and B together can do a piece of work in h days. A can do \frac23 of the work in 6 days. How many days does each require alone?
A and B together can do a piece of work in 5 days. A can do \frac25 of it in k days. How many days does each require alone?
B requires twice as much time as A to do a piece of work which they can do together in n days. How many days does each require alone?
A and B together can do a piece of work in p days. A works q times as fast as B. How many days does each require alone?
A man travels n miles and then returns to his starting point. Going, his rate is 3 miles an hour; returning, it is 4 miles an hour. How many hours did the entire journey take?
A and B start at the same time from two towns k miles apart and travel toward each other until they meet. A travels 3 miles an hour and B travels 5 miles an hour. In how many hours do they meet? How far does each travel?
In Problem 34, what would the required time have been, if the rate going had been p miles an hour and the rate returning q miles an hour?
In Problem 35, what would have been the respective distances if A had rested h hours on the way before he met B?
A and B start at the same time from two points c miles apart and travel toward each other until they meet. A travels p miles an hour and B travels q miles an hour. In how many hours do they meet?
In Problem 38, how many miles does each travel?
A man rides in a carriage d miles and returns on foot at the rate of 3 miles an hour. The time of riding is h hours less than the time of walking. Find the rate of the carriage.
In the preceding problem, if the rate of walking had been c miles an hour, what would have been the rate of the carriage?
A man rides a distance of p miles and walks back at the rate of q miles an hour. The entire trip took t hours. Find his rate of riding.
A and B start from the same point at the same time and travel in opposite directions for n hours. They are then 50 miles apart. A travels 2 miles an hour more than B. Find the rate of each.
In Problem 43, what would have been the respective rates if A had traveled k miles an hour more than B, and at the end of n hours they were h miles apart?
A man has just t hours at his disposal. How far can he ride in a carriage which travels p miles an hour, and yet have time to walk back at the rate of q miles an hour?