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225. Terms used in Percentage. The number of which some per cent is to be taken is called the base.

The number of hundredths of the base is called the rate. For example, in 25% of $300, $300 is the base and 25% is the rate.

Sometimes 25 is called the rate per cent, 25% being called the rate, but these two terms are commonly used to mean the same thing.

The result found by taking a certain per cent of the base is called the percentage.

Therefore, the percentage is the product of the base and the rate.

226. To find Some Per Cent of a Number. Required to find 231% of 275. 231% = 0.234.

275 Multiplying by 0.23}, we have 65.314.

.233 If we wish, we may write 239% as 0.2375 and then multiply. 2061 We may also, if we choose, write the product 65.3125.

825

550 Therefore, to find a required per cent of a number,

65.314 multiply the number by the given rate.

EXERCISE 130 Find the following: 1. 10% of 80. 6. 6% of $75. 11. 337% of 66. 2. 25% of 48. 7. 4% of $50. 12. 664% of 99. 3. 75% of 96. 8. 5% of $80. 13. 374% of 48. 4. 15% of 60. 9. 3% of $90. 14. 624% of 168. 5. 40% of 40. 10. 41% of $80. 15. 16%% of 360. 16. In a school of 300 pupils 11% are in the sixth grade. How many pupils are in the sixth grade ?

17. In a box of 192 oranges 67% have spoiled. How many have spoiled ?

18. From a barrel of flour weighing 196 lb., 25% has been sold. How many pounds have been sold ?

19. A contractor agreed to build a stone wall 72 ft. long. He has already built 20% of it. How many feet has he still to build ?

20. A trolley car has a run of 56 mi. a day. When it has run 15% of this distance, how many miles has it run ?

21. How much will 48% of a cubic foot of steel weigh if a cubic foot weighs 490 lb. ?

22. If a steel car when full carries 96,000 lb. of coal, how much does it carry when it is 621% full ?

23. If out of 240 problems in arithmetic you solved 931% without any errors, how many did you solve correctly?

24. If a locomotive weighing 116 tons can exert a pull equal to 22% of its weight, how many tons of pull can it exert?

25. A ball team has played 32 games this season and has lost 374% of them. How many games has it won ?

26. A cubic foot of water weighs 62} lb. Ice is 92% as heavy as water. What does 25. cu. ft. of ice weigh?

27. The diameter of a circle is about 313% as long as the circumference. What is the diameter of a circle whose circumference is 15 ft. ?

28. A meter is 91%% longer than a yard. How many inches in' a meter ?

29. A knot, used at sea, is 151% longer than a mile. How many feet are there in a knot ?

30. A sidewalk 112 ft. long and 4 ft. 2 in. wide is to be made 121% longer and 20% wider. How many feet will it be extended ? How much wider will it be? By how many square feet will its area be increased ?

227. The Per Cent One Number is of Another. Because the percentage is the product of the base and the rate ($ 225), therefore

The rate equals the percentage divided by the base.
For example, what per cent of 5 is 4 ?

Since 4 is the product of 5 by some per cent, therefore 4 = 5, or 80%, equals the required per cent.

EXERCISE 131

Find what per cent the second number is of the first : 1. 5, 3.

6. 100, 6. 11. $125, $6.25. 2. 10, 5. 7. 150, 25.

12. $150, $9.00. 3. 12, 4.

8. 175, 25. 13. $180, $9.00. 4. 25, 5.

9. 225, 75. 14. $250, $7.50. 5. 75, 15. 10. 250, 50. 15. $475, $23.75. 16. A man bought a horse for $150 and sold it for $175. How much did he gain ? What per cent of the cost did he gain?

17. A man bought a farm for $6500 and sold it for $7150. How much did he gain? What per cent of the cost did he gain ?

18. A boy weighed 75 lb. a year ago and he now weighs 82} lb. What per cent of 75 lb. has he gained ?

19. A village had a population of 1200 five years ago. It now has a population of 1332. What has been the per cent of increase ?

20. A poultry raiser has 231 chickens this year. Last year he had 220 chickens. What is the per cent of gain this year ? The year before he had 200 chickens. What was the per cent of gain last year ?

21. A man saved $750 more this

year than last year. Last year he saved $2000. What is the per cent of increase ?

22. A man weighed 160 lb. in September and 10 lb. more the next April. What per cent had he gained on his weight in September ?

23. A farm of 300 acres has 48 acres of woodland. What per cent of the farm is woodland ?

24. In a certain school 22 of the pupils are in the sixth grade. There are 275 pupils in the school. What per cent are in the sixth grade ?

25. A contractor has agreed to erect a building for $48,000. He has been paid $18,000 on account. What per cent of the contract price has he been paid ?

26. In an arithmetic class it is planned to solve 550 problems in a certain number of days. All but 44 have been solved. What per cent of the problems remain to be solved ? What per cent of them have been solved ?

27. A baseball team has won 32 games and lost 28 games. What per cent of the total number of games has it won ? What

per

cent has it lost? 28. A cubic foot of ice weighs 574 lb. A block of ice containing 4 cu. ft. loses 283 lb. by melting. What per cent does it lose ?

29. A cubic foot of water weighs 624 lb. From a tank containing 16 cu. ft. there is drawn off an amount of water that would weigh 125 lb. What per cent of the water is drawn off ?

30. To a class of 32 pupils 8 more are admitted. What per cent of the original number are admitted ? What per cent of the present number are admitted ? If 10 pupils are now taken out, what per cent are taken out ?

228. One Number is a Given Per Cent of what Number ? Because the percentage is the product of the base and the rate ($ 225), therefore

The base equals the percentage divided by the rate.

For example, if 15% of a number is 390, what is the number?

390 - 15% = 39,000 + 15 = 2600. Ans.

The simple equation may be used in cases of this kind, it being explained that 2 x means 2 times some number that is to be found. For example,

0.15 x = 390. Therefore

2600, by dividing these equals by 0.15

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EXERCISE 132

Of what numbers are the following the given per cents ? 1. 720, 12% 4. 175, 25%

7. $84, 6% 2. 300, 15% 5. 360, 75%

8. $75, 5% 3. 690, 23% 6. 480, 124% 9. $88, 4% 10. What is the number of which $75 is 125%? That is, what number increased by 25% equals $75 ?

11. What number increased by 331% equals $480 ? 12. What number increased by 50% equals $600 ? 13. $260 is 4% more than (or 104% of) what number? 14. $420 is 5% more than what number?

15. $375 is 75% of what number? That is, what number decreased by 25% equals $375 ?

16. What number decreased by 25% equals $900 ?
17. $435 is 6% of what sum of money ?
18. What number increased by 331% equals. $500 ?

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