27. During a certain year rents increased 20%. If a flat rented for $60 a month before the increase, for how much did it rent after the increase?

28. The metal of a watch case weighing 2 ounces was 50% gold. How much gold did it contain?

29. During a January sale, the price of $48 overcoats was reduced 33%. Find the selling price.

30. A farmer uses 40% of his 800-acre farm for wheat, 20% of the remainder for corn, 331% of the remainder for oats. How much of the farm is not cultivated?

31. If the expense of operating a factory is 40% of the sales, what is the operating expense for a year in which sales amounted to $680,370?

32. A family has an income of $1800 a year. If 20% is used for rent, 30% for food, 30% for clothing, and 10% for incidentals, how much is paid for each of these items? How much is saved?

33. A family with a $2000 income spent 25% of it for food, 20% for rent, 15% for clothing, and 25% for pleasure. They saved 10% and used the remainder for incidentals. How much was spent for each purpose?

34. A man owns property valued at $8250. What is the amount of his taxes if the tax rate is $1.52 per $100?

35. If a house is insured for $5000 at an insurance rate of .4%, what is the annual premium?

36. The value of food depends upon the energy it produces. Energy is measured in calories of heat. Thus a pound of fat produces 4082 calories of heat, and a pound of protein produces 1814 calories. How many calories of heat are furnished by a pound of butter if it contains 85% fat and 1% protein?

37. A real-estate dealer sold one house at $6500 and another at $4300. On the first he made a profit of 12% of the selling price, on the other he lost 10% of the selling price. Find his actual gain or loss.

38. Find the number which increased by 6% of itself gives 424.

Suggestion: Let x be the required number.

Show that x+.06x=424, and solve this equation.

39. A number increased by 10% of itself gives 44. Find the number.

40. Find the number which decreased by 4% of itself gives 192.

41. An article was sold for $128 at a gain of 10% of the cost. Find the cost.

42. An article was sold for $175 at a loss of 12% of the cost. What was the cost of the article?

43. A real-estate dealer can sell a certain lot for $1800. How much can he pay for the lot so as to make 15% on his investment?

10%

44. If fertilizer contains 4% of nitrogen, 10% of phosphoric acid, and 8% of potash, how many pounds of each are there in 1500 pounds of fertilizer?

148. Graphical representation of the percentage formula. If r=4, the percentage formula is p=.04b. Notice that this formula is similar to the formula c=πd, the relation between the circumference and diameter of the circle (§135). We may say that for a given rate the percentage varies directly as the base.

The percentage formula may be represented graphically as follows:

ь

Ρ

0

0

50

1. Let b take the values 0, 50, 100, 150, 200, etc. Tabulate the pairs of corresponding values of b and p.

2. Select convenient units and plot 100 the number pairs in the table.

3. Through the points thus obtained draw a line. This is the graph of the

equation p = .04b.

150

200

250

300

350

EXERCISES
Ρ

1. Show that the ratio remains constant as p and b

vary. Hence p varies directly as b.

2. Make a graph of the equation p=.05b.

3. Make a graph of the equation p=.06b.

149. Finding the rate by means of the percentage formula. Exercise 1, below, explains how to find what per cent one number is of another.

EXERCISES

1. What per cent of 33 is 14?

Solution: Let r be the required number of per cent.
By the percentage formula

This shows that 14 is 4233% of 33, or 42% of 33 approximately. Briefly, the problem may be solved by dividing 14 by 33.

2. Out of 25 games played by a ball team, 18 were won. What per cent was won? What per cent was lost?

3. A pupil solved correctly 9 out of 15 problems. What per cent did he solve correctly?

TELLER

4. An alloy consists of 18 parts of silver and 5 parts of copper. What per cent of it is silver? What per cent of it is copper?

5. A girl earned $63.80 during her summer vacation and deposited $25.00 of this sum in her savings account. What per cent of her earnings did she save?

6. Out of 240 eggs placed in an incubator, 185 chicks were hatched. What per cent of the eggs hatched?

7. On a rainy day 362 pupils were present in a school whose enrollment is 420. What was the per cent of attendance?

8. In a school of 430 pupils 250 are boys. What per cent of pupils are boys?

9. A farmer set out 225 trees; 58 of them died. What per cent died?

10. One out of every three inhabitants in a certain city is foreign-born. What per cent is foreign born?

11. A man pays $130 for taxes on property valued at $10,000. What is the tax rate?

12. A dealer gains a profit of $2260 on an investment of $18,000. What is his rate of profit?

13. A library circulated 256,394 books during a certain year. Of these books 162,342 were fiction. What per cent was this of the total number of books?

14. A salesman received $80 for making sales amounting to $3250. What was the rate?

15. The number of books in a library was increased from 930 volumes to 1210. What was the per cent of increase?

Solution: Let r be the number of per cent in the increase.

16. During two consecutive years the number of immigrants to the United States from Russia increased from 162,395 to 291,040. Find the approximate per cent of increase.

17. A farmer increased his oat crop from 37.5 bushels to 46 bushels per acre. Find the per cent of increase.

18. A man's wages were raised from $4.25 per day to $4.75 per day. Find the per cent of increase.

19. A dealer pays $14.50 for apples and sells them at a loss of $1.50. What is his per cent of loss?