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17. What per cent. of MMMDLXX. is 25 per cent. of MMDCCCLVI?

18. Gas is reduced from $1.50 to $1.00 per M. What per cent. of the original cost is saved ?

19. If į of a ton of coal is sold for what 1000 lb. cost, what is the gain per cent. ?

20. The cost was $3486, the selling price was $4161. Find the gain and the gain per cent. Also point out the base, the amount, the percentage, and the difference.

To Find the Base.

Since Percentage is the product of Base and Rate, obviously

Percentage
Base

Rate

EXERCISES. 1. $82 is 121 per cent. of what base ? Process.

By Analysis. 121% = .124 = .125

121% of B. = $82.

1% of B. B.

$656.

100% of B.

82 122

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25

P.
R.

1964

$656.

82 .1 25

= 16400

25

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Explanation.
The percentage is $82.00; the rate is 121 %; the base is required.

P.

and obtain Since the base is required, we use the formula, B.

R.' B. = $656. 2. Find of what number: 1. 385 is 121%.

7. 168 men is 8%. 2. 396 is 11%.

8. 462 oxen is 7%. 3. 250 is 15%.

9. 12 is 16 % 4. 8.25 is 331%.

10. 70 is 66%. 5. $64.36 is 10%.

11. 300 is 331%. 6. 38.6 bu. is 13%.

12. 100 is 621%.

13. 72 is 444 %.

20. 5 cwt. is 40%. 14. 48 is 371%.

21. 75 yd. is 181%. 15. 84 is 871%.

22. $.50 is 311%. 16. 126 is 90%

23. 160 is 1064% 17. 24 is 83%.

24. 75 lb. is 64%. 18. 10 is 61%.

25. 837 gal. is 6%. 19. 14 is 14%.

26. 9006 is .06%. 3. $281.25 is 371% of what number? 4. If 28% of a number $71.68, what is the number?

5. If 25% of a number = $324, what is 40% of that number?

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PROBLEMS.

1. A man sold a horse at a gain of $15, which was 15% of the cost. Find the cost and selling price.

2. A farmer sold 384 barrels of apples, which was 96% of all he had. How

many

had he? 3. A farm was sold for $536 less than cost, which was at a loss of 20%. What was the cost of the farm ?

4. The immigrants of a population number 143,000 persons, or 11% of the whole. Find the total population.

5. A farmer sold 110A. 43 sq. rd. of land, which was 20% of his land. How much land had he at first?

6. A man pays $500 rent a year; 85% of this sum is 33} per cent. of į of his income. Find his income. 7. On the sale of a patent $1600 was lost.

What was the value, if the rate of loss was 16% ?

8. A sale resulted in a loss of $38.46, which was 1% of the cost. Find the cost.

9. By selling an article for 664% of its cost, $23.25 was realized. What was the cost ?

10. At a gain of 4%, a profit of $36 was realized. What was the cost ?

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11. 30% of B.'s money is in a bank, and 50% in a farm; the remainder, $2000, is in P. R. R. stock. How much money does B. own?

To Find the Base when the Rate and the Amount or

Difference are Given.

EXERCISES. 1. What number increased by 67% of itself equals 510? Process.

Explanation. 64% = 1 18 + 16 = 17 16. 16

1, the number, + to of the = 510. 16 = 30,16 = 480.

number equals 17 of the number

510; and since 1% of the numOr, 1 +.064 = 1.061

ber 510, la of the number 510. 510 : 1.0625 480.

ity of 510 = 30; and since te

30, 16 16 times 30 or 480. Or, since once the number and .064 times the number 510, that number must equal 510 = 1.061, or 480.

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num.

3

400

3 40

2. What number diminished by 3% of itself equals 794 ? Process.

Explanation.

400, the number, % 487.

47 of the

794 ; 488 =

387 of the num. = 794.

and since 37 of the num. 794, ao = 2. 400 2

do
357 of 794

2; and since 400 800. Or,

do 2, 408 : 400 times 2 or 800. 1 .0075 .9925

Or, since once the num. - .0075

times the num. = .9925 times the 794 ; .9925 800.

num., and since .9925 times the 794, that num. must equal 794 = .9925, or 800.

400

num. =

Hence we have the formula :

Difference Base

1- Rate

; or, B.

D. 1- R.

:

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3. What number increased by :

1. 15% of itself = 4830 ? 9. 2% of itself = 516 ?
2. 27% of itself=508? 10. 163% of itself = 11?
3. 333% of itself = 984? 11. 16%% of itself=1400 ?
4. 163% of itself=658?

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12. 67% of itself = 901 ?

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5. 62% of itself=1820 ? 13. 16% of itself=261 ?
6. 10% of itself=15,400? 14. 3% of itself = 28?
7. 15% of itself = 690 ? 15. 140% of itself = 630 ?

8. 87% of itself=4140.15? 16. % of itself = 1897? 4. What number diminished by:

1. 25% of itself = 1200 ? 6. 12% of itself = $616? 2. 55% of itself=1240 ?

7. 144% of itself = 96 ? 3. 331% of itself=1260 ? 8. 36% of itself = 336 ? 4. 11% of itself=4539? 9. 28% of itself = 1? 5. 3% of itself=2667.50 ? 10. 64% of itself = 675 ?

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PROBLEMS.

1. A man owes $14,300, which is 8% more than his property is worth. Find the value of his property ?

2. In 5 years the population of a town has increased 25%. The population is now 7675. What was it 5 yr, ago?

3. A regiment lost 64% of its men, and then had left 750. Find the original number.

4. Having sold 30% of my land, I had then 34 acres remaining. How many acres had I at first ?

5. Two farms contain 630 acres, and one farm is 25% larger than the other. Find the size of each farm ?

6. What was the cost of hats sold at $1.00 apiece, which was at a loss of 163% ?

7. A horse was sold for $112, and the loss was 331%. What did he cost ?

8. Silk sold at $5.50 a yd. brings in a profit of 10%. What did it cost?

9. A wheat speculator lost at one time 10% of his money, and at another time 121% of the remainder, and had then $50,400 left. How much had he previous to those losses ?

10. A farmer sold two cows for $50 apiece. On the one he gained 25%; on the other he lost 25%. Did he gain or lose by the sales, and how much ?

11. An army lost 25% of its men in battle, and 25 per cent. of the remainder were discharged. 2223 men still remained. Find the original number of men in the army.

12. A farmer having sold 110 A. 43 sq. rd. of land had 80% left. How much had he at first?

13. A patent was sold for $8000. The seller lost 84% of the original value. Find the original value ?

14. A flock of sheep was increased by 250% of itself, and then numbered 1400. Find the original number.

15. How many lb. of tallow must be mixed with 91 lb. of rosin that the mixture may contain 24% of tallow?

16. By selling goods at 60 cents a pound, 8% is lost. What advance must be made in the price to gain 15% ?

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1. Show how formulæ 2 and 3 are derived from formula 1. 2. Show how formulæ 4 and 5 are derived.

3. Invent five problems to which above formulæ may respectively apply.

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