2. A rectangular field is 14 yards longer than it is wide. If its length were increased by 10 yards, and its width decreased by 4 yards, the area would remain the same.

the dimensions of the field.

Find

3. A rectangular field is twice as long as it is wide. If it were 50 feet shorter and 20 feet wider, it would contain 2000 square feet less. Find the dimensions of the field.

4. A certain sum invested at 4%

a sum $300 larger invested at 3%.

brings the same interest as Find the first sum.

5. A sum invested at 5%, and a second sum, twice as large, invested at 4%, together bring $52 interest. What are the two sums?

6. An investment of $2500 brings a yearly interest of $114. A part of the capital is invested at 4%, and the remainder at 5%. How many dollars are invested at 4%?

7. A bought 12 oranges for a certain sum. If each orange had cost one cent more, he would have received 10 oranges for the same money. What was the price of each orange?

8. Six persons bought an automobile, but as two of them. were unable to pay their share, each of the others had to pay $40 more. Find the share of each, and the cost of the automobile.

9. Ten yards of silk and 20 yards of cloth cost together $35. If the silk cost three times as much per yard as the cloth, how much did each cost per yard?

10. Twenty yards of silk and 30 yards of cloth cost together $85. If the silk cost 50¢ more per yard than the cloth, what was the price of each per yard?

How

11. A man bought 7 lbs. of coffee for $1.79. For a part he paid 24 per lb. and for the rest he paid 35¢ per lb. many pounds of each kind did he buy?

12. Sixteen persons subscribed $138. Six of them paid equal amounts, and the remaining ones paid each one dollar Find the share of each man.

more.

13. Twenty men subscribed equal amounts to raise a certain sum of money, but four men failed to pay their shares, and in order to raise the required sum each of the remaining men had to pay one dollar more. How much did each man subscribe?

14. A cistern is filled in a certain time by a pipe which lets in 20 gallons per minute. Another pipe letting in 25 gallons per minute fills the cistern in one minute less. In how many minutes does the first pipe fill the cistern?

15. A cistern is filled in a certain time by a pipe letting in 21 gallons per minute. If another pipe, which lets in 14 gallons per minute, is opened 3 minutes longer than the first, 6 gallons less than in the first case will be poured in. In how many minutes does the first pipe fill the cistern?

16. A sets out walking at the rate of 3 miles per hour, and three hours later B follows on horseback traveling at the rate of 6 miles per hour. After how many hours will B overtake A, and how far will each then have traveled?

17. A and B set out walking at the same time in the same direction, but A has a start of 3 miles. If A walks at the rate of 2 miles per hour, and B at the rate of 3 miles per hour, how far must B walk before he overtakes A?

18. A sets out walking at the rate of 3 miles per hour, and one hour later B starts from the same point traveling by coach in the opposite direction at the rate of 6 miles per hour. After how many hours will they be 27 miles apart?

19. A and B start walking at the same hour from two towns 17 miles apart, and walk toward each other. If A walks at the rate of 3 miles per hour, and B at the rate of 4 miles per hour, after how many hours do they meet and how many miles does A walk?

20. The distance from New York to Albany is 142 miles. If a train starts at Albany and travels toward New York at the rate of 40 miles per hour without stopping, and another train

starts at the same time from New York traveling at the rate of 42 miles an hour, how many miles from New York will they meet?

21. Two men start at 12 o'clock from two towns 17 miles apart, and travel toward each other. One walks at the rate of 3 miles per hour, but rests one hour on the way; the other travels at the rate of 4 miles per hour and rests 3 hours. At what hour do they meet?

22. A and B start from two towns 20 miles apart and travel toward each other. A starts at 1 P.M., B starts at 2 P.M., and

they meet at 5 P.M.

If B travels one mile per hour faster than A, find the number of miles each travels per hour.

23. A picture which is 2 inches longer than wide is surrounded by a frame 1 inch wide. If the area of the frame is 40 square inches, what are the dimensions of the picture?

MISCELLANEOUS PROBLEMS

24. The formula which transforms Fahrenheit readings of a thermometer into Centigrade readings is C = (F — 32). If C = 40°, find the value of F.

25. Change the following readings to Fahrenheit readings: (a) 0° C, (b) 100° C, (c) 50° C, (d) — 12° C.

26. At what temperature do the Centigrade scale and Fahrenheit scale indicate equal numbers?

27. The formula for the distance which a falling body passes over in t seconds is S = 1 gt2. (Ex. 7, p. 16.)

If S = 240 ft. and t = 4 seconds, find the value of g.

28. The formula for compound interest is

(For the meaning of the letters see Ex. 4, p. 16.)

Find the principal that will bring $662 interest in two years

at 10% compound interest.

29. A number increased by 7 gives the same result as the number multiplied by 7. What is the number?

30. If a number be added to 3, the sum multiplied by 3, the product diminished by 20, the difference multiplied by 6, and the product diminished by 55, the result will be 5. Find the

number.

31. A has as many dollars as B has cents. If A should give B $6.93, B would have as many dollars as A has cents. How much money has each ?

32. A man made as much money as he had and $100. He made as much money as he then had and $200; again he made as much as he then had and $300, and found that he had finally $3100. How many dollars had he at first?

33. A man met some beggars, and after giving each 4¢ had 9 left. He found that he lacked 7 to be able to give each beggar 64. How many beggars were there?

34. A mason working 8 hours a day, in the course of a week, builds a number of cubic meters which exceeds 43 as much as 43 exceeds the number of cubic meters which he would build working 7 hours a day. How many cubic meters does he build per hour?

35. A has $6 more than B and gives to B as much as B. has. Then B gives to A as much as A then has, and once more A gives to B as much as B then has; and finds that A has now as much as B. How many dollars has each at first?

36. A boy has the same number of sisters as brothers, while his sister has 1 times as many brothers as sisters. How many sons and daughters are there in the family?

CHAPTER VI

FACTORING

103. An expression is rational with respect to a letter, if, after simplifying, it contains no indicated root of this letter; irrational, if it does contain some indicated root of this letter.

+ √b is rational with respect to a, and irrational with respect

104. An expression is integral with respect to a letter, if this letter does not occur in any denominator.

a2

to b.

+ab+b2 is integral with respect to a, but fractional with respect

105. An expression is integral and rational, if it is integral and rational with respect to all letters contained in it; as,

a2+2ab+ 4 c2.

106. The factors of an algebraic expression are the quantities which multiplied together will give the expression.

In the present chapter only integral and rational expressions are considered factors.

Although Va2 - b2 × √a2 - b2 = a2 - b2, we shall not, at this stage of the work, consider Va2 - b2 a factor of a2 — b2.

107. A factor is said to be prime, if it contains no other factors (except itself and unity); otherwise it is composite.

The prime factors of 10 a3b are 2, 5, a, a, a, b.