21. What number, increased by 62% of itself, equals 39424 ?

22. What number, increased by 114% of itself, equals 875 ?

23. 1215 is some number plus 35 per cent of itself. What
is the number?

24. The Mississippi River is 4200 miles long, and is 5%
longer than the Nile, which is 63% longer than the
Amazon. How long is the Nile? The Amazon?
25. The standing army of Russia is 975,000 men, whicn
is 95% greater than the standing army of France.
How many men in the latter?

26. The area of Delaware is 2050 square miles, which is 64% greater than the area of Rhode Island. What is the area of Rhode Island ?

27. The diameter of the planet Jupiter, 88000 miles, is 142% greater than the diameter of the planet Saturn. What is Saturn's diameter ?

28. A tract of 8360 acres of land was divided between two purchasers, so that one had 9% more than the other. How many acres had each?

29. The territory of Dakota contained 152,000 square miles, and was divided into the two States of North Dakota and South Dakota, so that South Dakota has 23% greater area than North Dakota. What is the area of each State?

369. 1. 40 is 80% of what number?

2. 40 is 20% less than what number?

SOLUTION. I. This is another form of the preceding example. The number sought is the base, or 100%, of which number 40 is 80%, that is, 100%-20%. 40÷.80=50.

SOLUTION. II. Since 80% = 40 is of some number; 40 is of 40X =50.

In such examples, the number given is called the Difference, since it is the difference of the base and the rate. The number sought is the base. Such examples give rise to

CASE V.

370. Difference and Rate per cent given, to find the Base. Divide the Difference by 100% minus the Rate per cent expressed as a decimal.

371. Cases III, IV, and V are essentially one. In each of them a product and a multiplier are given, and the number to be found is a multiplicand.

In Case III, the given product is the percentage.

In Case IV, the given product is a percentage, which includes 100%, representing the base itself.

In Case V, the given product is also a percentage, the remainder of 100% after a given per cent has been subtracted.

In all of the cases named, the Base is sought by dividing a product (Percentage, Amount or Difference) by the number expressing the rate per cent which the product is of the number sought.

ORAL.

1. 12 is 75% of what number?

2. 12 is 25% less than what number?
3 36 is 10% less than what number?

4. 63 is 12% less than what number?

5. A woman's age is 30 years, which is 142% less than her husband's age. How old is her husband ?

6. A man sold a buggy for $80, which was 60% less than it cost him. What did it cost him?

7. What number decreased by 20% of itself equals 28 ?

8. What number decreased by 90% of itself equals 32 ? 9. A boy gave to his playmates 75% of his apples, and had 6 left. How many had he at first?

10. When 80% of a journey had been finished, 50 miles remained. How long was the journey?

11. After 44% of a cargo of potatoes had been sold, 560 bushels remained. How many bushels were in the cargo?

12. The sum of the ages of two boys is 14 years. The age of one of them is 25% less than the age of the other. How old is each?

SUGGESTION. The sum of the ages is 175%, or . of the age of the older boy.

13. Two horses are valued at $1500, one being worth 121% less than the other. What is the value of each? 14. Two casks contain together 48 gallons, one of them holding 40% less than the other. How many gallons

does each hold?

WRITTEN.

15. 7360 is 92% of what number?

16. 1320 is 12% less than what number? 17. 1445 is 15% less than what number?

18. What number, decreased by 23% of itself equals 1155? 19. What number, decreased by 46% of itself equals 1152? 20. A drover sold 34% of his cattle, and had 990 left; how many did he sell?

21. A man worth $1,220,000 is unhappy because his fortune is 39% less than his neighbor's. What is his neighbor's fortune?

22. Mt. St. Elias is 17900 feet high, and is 38.4% less than Mt. Everest. What is the height of Mt. Everest? 23. The Rio Grande and Yukon rivers are, together, 3100

miles long? The Rio Grande is 6.25% less in length than the Yukon. How long is each river?

24. A cubic foot of anthracite and a cubic foot of bitu

minous coal weighed together 171 lbs. The weight of the latter was 20% less than the weight of the former. What did each weigh?

SUMMARY.

372. Per Cent means hundredths.

373. Rate per Cent means a certain number of hundredths. The rate is the numerator of a fraction of which the denominator is always 100.

374. The Base is the number of which the per cent is computed. The Base is always considered as 100 per cent. 375. The Percentage is that part of the base which is indicated by the rate per cent.

The word Percentage also denotes that part of Arithmetic in which the standard of comparison is 100 per

cent.

376. The Amount is the sum of the base and the percentage. 377. The Difference is the base less the percentage. 378. 1. The Base is the Multiplicand.

2. The Rate per cent is the Multiplier.

3. The Percentage is the Product of which the Base and the Rate per cent are the factors.

Percentage Base x Rate Per Cent.

4. The Amount is the Product of the Base and 100 per cent plus the Rate.

Amount Base × ( 100%+Rate.)

5. The Difference is the Product of the Base and 100% minus the Rate.

Difference=Base × ( 100%-Rate.)

PROFIT AND LOSS.

379. Cost is the sum paid for an article.

380 Price is the sum asked or received for an article.

381. Profit, or Gain, is the excess of the price above the cost. 382, Loss is the excess of the cost above the price.

383. The cost is always considered the base. The gain or loss is a certain percentage of the base.

The price is either amount or difference, according as it is either greater or less than the cost.

ORAL.

1. Potatoes that cost 80 cents a bushel are sold at a gain of 25%. What is the gain? What is the price? SOLUTION. The gain is 25% of 80 cents, which is 20 cents.

is the cost, 80 cents,+the gain, 20 cents, =$1.

The price

2. How much was the profit on a cow which was bought for $60, and sold at a gain of 10%? What was the price?

3. When oats that cost 40 cents a bushel are sold for 50
cents, what is the profit on a bushel? What is the
gain per cent ?

SOLUTION. The profit is the excess of the price over the cost, 10 cents.
The gain per cent is the profit, divided by the cost, 10÷40=25%.
4. Cattle were purchased at an average of $30 a head,
and sold at an average of $40 a head. What was the

profit ?

What was the gain per cent?

5. Cloth that cost 60 cents was sold for 50 cents. What

was the loss per cent ?

6. For how much must I sell a horse that cost me $200 in order to gain 20% ?

7. Coal costing $3 a ton was sold for $5 a ton. What was the gain per cent ?