PROBLEM III.

451. Given, the percentage and rate, to find the base.

1. 18 is 5% of what number?

Ans. 750.

2. 54 is 15 % of what number? 3. 17.5 is 2% of what number? 4. 2.28 is 5% of what number? 5. 414 is 120 % of what number? 6 6119 is 105% of what number? 7. .43 is 71% of what number? 8. The percentage is $18.75, and the rate is 23 %; what is the base?

Ans. 5800.

Ans. .6.

Ans. $750.

9. The percentage is 311, and the rate 31 %; what is the base?

10. I sold my house for $4578, which was 84 % of its cost; what was the cost? Ans. $5450. 11. A wool grower sold 3150 head of sheep, and had 30 % of his original flock left; how many sheep had he at first?

12. A man drew 40 % of his bank deposits, and expended 13 % of the money thus drawn in the purchase of a carriage worth $116; how much money had he in bank? Ans. $2175.

13. If $147.56 is 13 % of A's money, and 43 % of A's money is 8 % of B's, how much more money has A than B? Ans. $461.121.

14. In a battle 4 % of the army were slain upon the field; and 5 % of the remainder died of wounds, in the hospital. The difference between the killed and the mortally wounded was 168; how Ans. 21000.

many men were there in the army?

=

NOTE.-100 % ·4% 96 %, left after the battle; and 5 % of 96 % %, the part of the army that died of wounds.

=

15. A owns of a prize and B the remainder; after A has taken 40 % of his share, and B 20 % of his share, the remainder is equitably divided between them by giving A $1950 more than B; what is the value of the prize? Ans. $7800.

PROBLEM IV.

452. Given, the amount and rate, to find the base. 1. What number increased by 5 % of itself is equal to 378?

RULE.

ANALYSIS. If any number be increased by 5 % of itself

the amount will be 1.05 times the number. We therefore divide the given amount, 378, by 1.05, or 1⁄2, and obtain the base, 360, which is the number required. Hence the

Divide the amount by 1 plus the rate.

NOTE 1.—The amount is always a product, of which the base is one factor, nd 1 plus the rate the other factor.

EXAMPLES FOR PRACTICE.

1. What number increased by 15 % of itself is equal to 644?

Ans. 560.

2. A has $815.36, which is 4 % more than B has; how much oney has B?

3. Having increased my stock in trade by 12 nd that I have $3800; how much had I at first?

Ans. $784.

% of itself, I

4. In 1860 the population of a certain city was 39600, which as an increase of 10 % during the 10 years preceding; what as the population in 1850?

of

5. My crop of wheat this year is 8 % greater than my crop last year, and I have raised during the two years 5200 bushels; what was my last year's crop? Ans. 2500 bu.

NOTE 2.-1.00 +1.082.08. Hence, 5200 bu. = =2.08 % of last year's crop. 6. The net profits of a nursery in two years were $6970, and the profits the second year were 5 % greater than the profits the first year; what were the profits each year?

Ans. 1st year, $3400; 2d year, $3570.

7. If a number be increased 8 %, and the amount be increased 7 %, the result will be 86.67; required the number.

NOTE 3.-The whole amount will be 1.08 X 1.07 = 1.155€ times the original number.

8. A produce dealer bought grain by measure, and sold it by weight, thereby gaining 1% in the number of bushels. He sold at a price 5 % above his buying price, and received $4910.976 for the grain; required the cost. Ans. $4608.

9. B has 6 %, and C 4 % more money than A, and they all have $11160; how much money has A? Ans. $3600.

10. In the erection of a house I paid twice as much for material as for labor. Had I paid 6 % more for material, and 9 % more for labor, my house would have cost $1284; what was its cost? Ans. $1200.

PROBLEM V.

453. Given, the difference and rate, to find the base. 1. What number diminished by 5% of itself, is equal to 342!

OPERATION.

1.05.95

ANALYSIS. If any number be diminished by 5% of itself, the dif

342.95 360, Ans. ference will be .95 of the number.

Or,

=

1 — 20
= = 18

342 ÷ 18

=

We therefore divide the given difference, 342, by .95, or 18, and obtain

360, Ans. the base, 360, which is the required number. Hence the

RULE. Divide the difference by 1 minus the rate.

NOTE.-The difference is always a product, of which the base is one factor, and 1 minus the rate the other.

EXAMPLES FOR PRACTICE.

1. What number diminished by 10 % of itself is equal to 504? Ans. 560.

2. The rate is 8 %, and the difference $4.37; what is the base?

3. After taking away 15 % of a heap of grain, there remained 40 bu. 3 pk.; how many bushels were there at first?

Ans. 48 bu. 4. Having sold 36 % of my land, I have 224 acres left; how much land had I at first?

5. After paying 65 % of my debts, I find that $2590 will discharge the remainder; how much did I owe in all?

Ans. $7400.

6. A young man having received a fortune, deposited 80 % of it in a bank. He afterward drew 20 % of his deposit, and then had $5760 in bank; what was his entire fortune?

Ans. $9000. 7. A man owning of a ship, sold 12 % of his share to A, and the remainder to B, at the same rate, for $20020; what was the estimated value of the whole ship? Ans. $26000.

8. An army which has been twice decimated in battle, now contains only 6480 men; what was the original number in the army?

Ans. 8000.

horse to C.

9. Each of two men, A and B, desired to sell his A asked a certain price, and B asked 50 % more. A then reduced his price 20 %, and B his price 30 %, at which prices C took both horses, paying for them $148; what was each man's asking price?

Ans. A, $80.

{

B, $120.

10. A buyer expended equal sums of money in the purchase of wheat, corn, and oats. In the sales, he cleared 6 % on the wheat, and 3% on the corn, but lost 17 % on the oats; the whole amount received was $2336. What sum did he lay out in each kind of grain? Ans. $800.

APPLICATIONS OF PERCENTAGE.

454. The principal applications of Percentage, where time is not considered, are Commission, Stocks, Profit and Loss, Insurance, Taxes, and Duties. And since the five problems in Percentage involve all the essential relations of the parts or elements, we have for the above applications the following

GENERAL RULE. Note what elements of Percentage are given in the example, and what element is required; then apply the spe cial rule for the corresponding case..

COMMISSION.

455. An Agent, Factor, or Broker, is a person who transacts business for another.

456. A Commission Merchant is an agent who buys and sells goods for another.

457. Commission is the fee or compensation of an agent, factor, or commission merchant.

458. A Consignment is a quantity of goods sent to one person to be sold on commission for another person.

459. A Consignee is a person who receives goods to sell for another; and

460. A Consignor is a person who sends goods to another to be sold.

461. The Net Proceeds of a sale or collection is the sum left, after deducting the commission and other charges.

NOTE. A person who is employed in establishing mercantile relations between others living at a distance from each other, is called the Correspondent of the party in whose behalf he acts. A correspondent is the agent of those whose custom or patronage he secures to the party in whose interest he is employed.

462. Commission is usually reckoned at a certain per cent. of the money involved in the transaction; hence we have the following relations:

I. Commission is percentage, (445).

II. The sum received by the agent as the price of property sold, or the sum invested by the agent in the purchase or exchange of property, is the base of commission, (446).