16. A 14-foot piece of timber must be cut into two parts one of which is the length of the other. Find the lengths of the two parts.

17. A teacher asked a class to divide one half a certain number by 4 and the other half by 12. Instead of doing so, one member of the class divided the entire number by half the sum of the two divisors. His answer was too small by 4. What was the number?

18. A man started on a journey with a certain sum of money. He spent for car fare, of the remainder for hotel bills. When he returned home he found that he had $25.00. How much did he start with?

19. A man lost of his money in speculation, lent of the 10 remainder to a friend, and spent the rest for a home, paying $5460 for it. What was his original capital?

20. The difference between two numbers is 22 and of the greater exceeds the smaller by 19. Find the numbers.

21. A tank can be filled by one pipe in 10 hours, and by another pipe in 15 hours. How long will it take to fill the tank if both pipes are open together?

SOLUTION.

Then

Let x= the number of hours.

1

x

=

=

the part of the tank both can fill in one hour.

the part of the tank the first pipe can fill in

one hour.

the part of the tank the second pipe can fill in one hour.

22. How long will it take two pipes to fill a tank if one can fill it in 5 hours and the other can fill it in 12 hours?

23. Two pipes are connected with a tank. The large one can fill it in 7 hours; the small one can empty it in 9 hours. How long will it take to fill the tank if both pipes are open?

24. A plows a field of corn in 4 days, and B plows it in 5 days. How long will it take to plow the field if they work together?

[

HINT. If x=

= the number of hours it will take both, we get the

+ Compare Ex. 21.]

25. A does a piece of work in 4 days, B in 6 days, and C in 8 days. How long will it take them working together?

26. A can do a piece of work in 16 hours, and B can do it in 20 hours. If A works for 10 hours, how many hours must B work to finish?

27. A brick mason can build a wall in 10 days; if another mason helps him, they can build it in 3 days. How long will it take the second mason alone?

28. It takes 5 hours for a railroad tank to be filled with water. Locomotives draw out and use a tank full every 8 hours. How long will it take to pump the tank full?

29. A's age is two fifths of B's; ten years ago one fourth of B's age equaled A's. What is the age of each?

30. A son's age is one third that of his father's. 12 years ago he was as old as his father. How old is each now? 31. Five gallons of 80% alcohol are to be made 50% alcohol. How many gallons of water must be added?

[HINT. The equation is .50(x+5)=5×.80. Explain.]

32. How much water must be added to 65 lb. of a 10% solution of salt to make it an 8% solution?

33. 18% of the weight of wheat is lost in grinding it into flour. How many bushels of wheat, 60 lb. each, must be used to make 984 lb. of flour?

34. Separate 169 into two parts such that if one part is divided by 21 and the other by 14, the sum of the quotients is 11.

35. If a certain number is subtracted from the numerator and added to the denominator of the fraction, the fraction is diminished by 12. What is the number?

36. The fare from St. Louis to Chicago and back is $15.00. A special rate was made in connection with a convention so that I made the trip, paid my expenses which were equal to 2 times my fare, and spent all together only $2.50 more than the regular fare. What was the special rate?

37. Democritus lived of his life as a boy, as a man, and spent 13 years in his dotage. he when he died? (From a collection of Metrodorius.)

as a youth, How old was problems by

38. An aviator flew 75 miles and back in 4 hours. His rate when returning was 40 miles per hour. What was his rate going?

[HINT. If one travels s miles at the rate of r miles an hour, the time t occupied is given by the formula t=s/r hours.]

39. An aviator made a trip of 95 miles. After flying 40 miles he increased his speed 15 miles an hour and made the remaining distance in the same time as it took him to fly the first 40 miles. What was his rate during the first 40 miles of the trip?

For further exercises on this topic, see Appendix, p. 303.

106. Literal Equations. Equations in which some, or all, of the known numbers are represented by letters are called literal equations.

b

Thus, x-a=b, −+a=c, 2 ax−b=3, are literal equations.

Here the first letters of the alphabet, a, b, c, etc., represent (as usual) the known numbers, while x represents the unknown number.

Literal equations are solved by the same methods as the equations in § 105.

EXAMPLE 1. Solve for x in the equation

known letters; that is, it contains only the a and the b. This must

be true of the solution of every literal equation.

EXAMPLE 2. Solve for x in the equation ax=bx+7 c.

SOLUTION. Transposing, we find

Dividing by a-b,

x=

7c. a-b

Ans. (See Note on p. 174.)

CHECK. Substituting the answer for x in the given equation

Solve for x in each of the following equations.

In the following equations the first letters of the alphabet are the known letters. Solve for x. Remember that in each case it is necessary to get the value of x in terms of the known letters. (See Note, § 106.)

1. 3x+b=x -3 b.

2. 3(a-x)= 12 d.

3. (a+x)2=a2+x2.

[HINT. Use Formula V, § 59, to simplify the first member.]