2. Bought 215 bar. superfine flour, at $4.85 per bar.; 355 bar. extra flour at $5.15 per bar.; 132 bar. rye flour, at $4.90 per bar.; 210 bar. corn meal, at $3.70 per bar.; and 642 sacks graham flour, at 88¢ per sack. What was the total cost?

3. A retailer bought 35 overcoats, at $9.75 each; 160 black suits, at $17.25 each; 125 plaid suits, at $14.65 each; 84 jean suits, at $6.90 each; and 50 pairs trousers, at $3.15 each. Find the total cost.

4. An invoice of six pieces of gingham of 511, 493, 502, 541, 492, and 512 yd. respectively, was sold at $.093 per yd. What was the amount of the sale?

5. Six men worked 193 days each, at $1.90 per day; 24 days each, at $1.80, 11 days each, at $1.65; and 314 days each, at $1.25. How much was earned by all in the entire time?

6. A laborer received $184.55 as a balance due him for his season's work. He paid a debt of $19.25; bought 83 yd. cloth, at $1.25 per yd.; 2 suits of clothes, at $13.25 per suit; hosiery and gloves for $2.85; 4 tons coal, at $5.65 per ton; 2 cd. wood, at $3.90 per cd.; 3 bar. flour, at $4.75 per bar.; 628 pounds of pork, at 634 per lb.; and loaned the remainder of his money. How much did he loan ?

DIVISION OF UNITED STATES MONEY.

292. To Divide United States Money.

Rule.-Divide as in abstract decimals.

EXAMPLES FOR PRACTICE.

293. 1. If $11421.75 be divided equally among five persons, what will be the share of each ?

2. B sold 187 acres of land at $105.25 per acre, and divided the proceeds equally among fifteen persons. What sum did each receive?

3. A charitable farmer gave 15% bushels of apples worth $.50 per bu., 213 bushels of potatoes worth $.75 per bu., and 30 bushels of turnips worth $.62 per bu., in equal shares to six families. What was the value of each share?

4. A dealer bought wheat at $.95 per bu., oats at $.45 per bu., and corn at $.65 per bu. He paid $332.50 for the wheat, $191.25 for the oats, and $113.75 for the corn. How many bushels did he buy in all ?

5. C invested $9659.50 in coal, at $5.85 per ton; $2645.30 in sand, at $2.80 per cubic yd.; $658.40 in lime, at $1.60 per barrel. If he sold the coal at $6.05 per ton, the sand at $2.75 per cubic yd., and the lime at $1.75 per barrel, what was the gain or loss?

6. Having sold my mill for $17250, and 316 barrels of flour in stock at $5.15 per barrel, I invested of the proceeds, $1185.85 in furnishing a house, $1259.30 in utensils, $1582.25 in live stock, and with the remainder paid in full for a farm of 163 acres. What was the cost of the farm per acre?

REMARK.-In case exact quotients are not obtained in division of dollars, add two decimal ciphers and continue the quotient to cents; if not then exact add one cent if the mills be 5 or more, but if less than 5, reject the mills.

ANALYSIS.

294. Arithmetical Analysis is the process of solving problems independently of set rules, by deducing, from the terms stated, the conditions and relations required in their solution.

REMARK.-The general subject of Analysis will be treated only as auxiliary to the subject of Common Fractions, and the SPECIAL APPLICATIONS of the Fundamental Rules.

EXAMPLE 1. If 5 men earn $30 in 4 days, how many dollars will 7 men earn in 9 days?

FIRST EXPLANATION (extended).—If 5 men earn $30 in 4 days, 1 man, or of 5 men will earn in 4 days of $30, or $6; and if 1 man earns $6 in 4 days, in 1 day, which is of 4 days, he will earn of $6, or $13. Then, since 1 man in 1 day earns $13, in 9 days, which are 9 times 1 day, he will earn 9 times $1, or $131; and if 1 man in 9 days earns $131, 7 men, which are 7 times 1 man, will earn 7 times $131, or $941.

SECOND EXPLANATION (abbreviated).—If 5 men earn $30 in 4 days, they will earn $7 in 1 day; and if 5 men earn $73 in 1 day, 1 man will earn of $71, or $11; since 1 man in 1 day earns $11, 7 men in 1 day will earn 7 times $11, or $101; and if 7 men in 1 day earn $101, in 9 days they will earn 9 times $101, or $941, the same as before found.

THIRD EXPLANATION (more abbreviated).—If 5 men in 4 days, doing 20 days' work, earn $30, $1 would equal 1 day's work; 7 men in 9 days do 63 days' work, and since 1 day's work equals $11, 63 days' work will equal $941, as before found.

EXAMPLE 2. If 6 men can cut 45 cords of wood in 3 days, how many cords can 8 men cut in 9 days?

FIRST EXPLANATION (extended).—If 6 men cut 45 cd. in 3 days, in 1 day, which is of 3 days, they can cut of 45 cd., or 15 cd.; and if 6 men can in 1 day cut 15 cd., 1 man in 1 day can cut of 15 cd., or 2 cd.; since 1 man in 1 day can cut 2 cd., 8 men can in 1 day cut 8 times 2 cd., or 20 cd.; and if 8 men in 1 day can cut 20 cd., in 9 days they can cut 180 cd.

SECOND EXPLANATION (abbreviated).—6 men in 3 days, doing 18 days' work, cut 45 cd.; hence 23 cd. can be cut by 1 man in 1 day; then 8 men in 9 days, doing 72 days' work, can cut 72 times 2 cd., or 180 cd., as before found.

EXAMPLE 3. If a post 4 ft. high casts a shadow 13 ft. in length, what must be the height of a post that will cast a shadow 125 ft. in length?

EXPLANATION.—If a post 4 ft. high casts a shadow 13 ft., a post 1 ft. high would cast a shadow 31 ft.; since a shadow 31 ft. is cast by a post 1 ft. high, a post that will cast a shadow 125 ft. in length must be as many times 1 ft. in height as 31 ft. are contained times in 125 ft., or 38 ft.

EXAMPLE 4. If the hour and minute hands of a clock are together at noon, at what times after noon will they again be together? At what time between 4 and 5 o'clock ?

EXPLANATION.—Since the minute hand passes the hour hand 11 times in 12 hours, it will pass it the first time in of 12 hours; the second time in of 12 hours; the third time in of 12 hours; the fourth time in of 12 hours. of 12 hours equals 4 hours, 21 minutes, and 49 seconds; therefore the hands will be together between 4 and 5 o'clock at 21 minutes 49 seconds after 4 o'clock.

REMARK.-Apply the same reasoning to all examples of this class.

EXAMPLE 5. If Grace were older than she is, her age would equal of her grandmother's. What is the age of each, if the age of both is 87 years ?

EXPLANATION.-If Grace were older than she is, she would be g of her present age; and since if she were § her present age, she would be only as old as her grandmother, the age of grandmother must be 4 times g or 24 of the age of Grace, and the age of both must be +24 or 29 of the age of Grace; since the age of both is 87 years, 87 years must be 22 of the age of Grace, who must be 15 years old. If Grace's age be increased by of itself, or 3 years, she will be 18 years of age; and since her age would then be only of grandmother's age, the age of grandmother must be 4 times 18 years, or 72 years.

EXAMPLE 6. A man being asked his age, replied: "My father was born in 1805 and my mother in 1806; the sum of their ages at the time of my birth was two and one-third times my age in 1851." How old was the man in 1888?

EXPLANATION.-If the father was born in 1805 and the mother in 1806, the sum of their ages in 1851 was 91 years; and since the sum of their ages at the time of the birth of the son was 21 times his age in 1851, and the parents each increased in years after the son's birth as fast as he did, in 1851 the sum of their ages must have been 4 times the age of the son; hence the son, in 1851, was 91 years ÷ 41, or 21 years of age, and he must have been born in 1830, and in 1888 would be 58 years old.

7. The sum of two numbers is 65, and their difference is equal to of the greater number. Find the two numbers.

8. How long after noon will it be when the minute hand passes the hour hand the third time?

9. How long after noon will it be when the minute hand passes the hour hand the eleventh time?

10. A's age is 24 times the age of B, and the age of C is 21 times the age of both A and B. If the sum of their ages is 116 years, what is the age of each? 11. A man bought 15 bushels of barley, and 36 bushels of oats, for $28.80, and 25 bushels of barley, 18 bushels of oats, for $29.10. How much per bushel did he give for each kind of grain?

12. Charles, when asked his age, replied: "My father was born in 1843, and my mother in 1847. The sum of their ages at the time of my birth was 5 times my age in 1887." In what year will Charles be 25 years of age?

SPECIAL APPLICATIONS.

295. Special Applications, as here treated, embraces the use, in the solution of problems, of any or all explanations heretofore given, and the consideration of cost, price, and quantity, as being the elements of every business transaction; it also treats of such contracted methods as may be employed in dealing with aliquot parts of the powers of 10, or of other numbers.

General Rules.-1. If the price and quantity be given, the cost may be found by multiplying the price by the quantity.

2. If the cost and quantity be given, the price may be found by dividthe cost by the quantity.

3. If the cost and price be given, the quantity may be found by dividing the cost by the price.

ALIQUOT PARTS.

296. The Aliquot Parts of a number are the even parts of that number. 25, 331, 12, are aliquot, or even, parts of 100.

REMARK-The component factors of a number must be integral, while the aliquot parts of a number may be either integral or mixed.

297. The even parts of other even parts may be called parts of parts; as, = of ; or, since 33 is a part of 100, of 333, or 113, must be a part of the part 331.

REMARK-Full illustrations of the use of aliquot parts will follow. Those of $1, equal to 1004, being the most valuable for use, will be mainly considered.

SUGGESTION TO TEACHER.-Let each one of the following conditions be given to the class as a question, the required answer to which is the rule.

INSTRUCTIONS FOR PRACTICE WITH ALIQUOT PARTS.

298. 1. To find the cost of a quantity when the price of 1 is 50 cents. RULE.-Consider the quantity as dollars, and divide by 2.

2. To find the cost when the price of 1 is 3344. considered as dollars, by 3.

RULE.-Divide the quantity,

3. To find the cost when the price of 1 is 254. RULE.-Divide the quantity, considered as dollars, by 4.

4. To find the cost at 204. RULE.-Divide the quantity, considered as dollars, by 5.

5. To find the cost at 1634. RULE.-Divide the quantity, considered as dollars, by 6.

6. To find the cost at 1214. RULE.-Divide the quantity, considered as dollars, by 8.

7.

To find the cost at 834. RULE.-Divide the quantity, considered as dollars, by 12

8. To find the cost at 614. RULE.-Divide the quantity, considered as dollars, by 16.

9. To find the cost at 104. RULE.-Point off from the right one place in the quantity, and consider as dollars.

10. To find the cost at 54. RULE.-Point off one place in the quantity, consider as dollars, and divide by 2.

11. To find the cost at 319. RULE.-Point off one place in the quantity, consider as dollars, and divide by 3.

12. To find the cost at 214. RULE.-Point off one place in the quantity, consider as dollars, and divide by 4.

13. To find the cost at 14. RULE.-Point off one place in the quantity, consider as dollars, and divide by 6.

14. To find the cost at 144. RULE.-Point off one place in the quantity, consider as dollars, and divide by 8.

MISCELLANEOUS CONTRACTIONS.

299. 1. To find the cost when the price of 1 is 75 cents. RULE.-From the quantity, considered as dollars, take of itself.

2. To find the cost when the price of one is 804. RULE.-From the quantity, considered as dollars, take of itself.

3. To find the cost when the price of one is 6634. RULE.-From the quantity, considered as dollars, take of itself.

4. To find the cost when the price of one is $1.25. RULE.-To the quantity, considered as dollars, add of itself.

5. To find the cost when the price of one is $1.50. RULE.-To the quantity, considered as dollars, add of itself.

6. To find the cost when the price of one is $2.50. RULE.—Annex a cipher to the quantity, consider as dollars, and divide by 4.