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Find the price of the following: 2. If 42 tons of hay cost $546; what is the price per ton ?

3. Bought 536 yards cloth for $1608; how much did I pay per yard ?

Ans. $3. 4. Sold 196 acres land for $10192; how much did I receive an acre ?

. Ans. $52. 5. Paid $1029 for 147 barrels flour; what did I pay per barrel ?

Ans. $7. 6. Received $980 for 28 weeks' work; what was my wages per week?

Ans. $35. 7. The total cost for conducting a certain school for 14 years was $252000; what was the yearly expense ? Ans. $18000.

8. A merchant pays his clerks annually $3744. How much is this per week?

Ans. $72. 159. PROB. III.To find the cost when the number of units and the price of two or more units are given.

1. At $15 for 3 cords of wood, what is the cost of 39 cords?

SOLUTION 1.-Since 3 cords cost $15, 39 cords must cost as many times $15 as 3 is contained times in 39. Hence, first step, 39 = 3= 13; second step, $15 x 13 = $195, the cost of 39 cords.

SOLUTION 2.-Since 3 cords cost $15, each cord cost as many dollars as 3 is contained times in 15 ; hence each cord cost $5, and 39 cords cost 39 times $5, or $195. Hence, first step, 15 + 3 = 5; second step, $5 x 29 = $195, the cost of 39 cords.

Observe carefully the difference between these two solutions. Let both be used in practice. Take for each example the one by which the division can be most readily performed.

Solve and explain the following:

2. Paid $28 for 4 barrels of flour; how much, at the same rate, will I pay for 164 barrels ?

Ans. $1148. 3. If 9 stoves cost $135, what is the cost at the same rate, of 84 stoves ?

4. A farmer paid for 32 sheep $128; what is the cost at the same rate, of 793 ? ·

Ans. $3172.

5. A man traveled by railroad 1728 miles in 3 days; how many miles, at the same rate, will he travel in 54 days?

6. A merchant bought 150 yards of cotton for $18; how much will he pay, at the same rate, for 1350 yards ?

7. A book-keeper receives for his service at the rate of $624 for 13 weeks; what is his yearly salary? Ans. $2496.

160. PROB. IV.-To find the number of units when the cost and the price per unit are given.

1. At $7 a ton, how many tons of coal can be bought for $658 ?

SOLUTION.—Since 1 ton can be bought for $7, there can be as many tons bought for $658 as $7 is contained times in $658. Hence, $658+$7 =94, the number of tons that can be bought for $658.

Solve and explain the following:

2. For $765, how many barrels of pears can be bought at $9 a barrel ?

Ans. 85. 3. How many horses can be bought for $9928, at $136 per horse ?

Ans. 73. 4. A man paid for a farm $6134, at $38 per acre; how many acres does the farm contain ?

5. A mechanic received at one time from his employer $357. He was paid at the rate of $21 a week; how many weeks had he worked ?

Ans. 17. 6. A lumber merchant paid for walnut lumber $3528, at $98 a thousand feet; how many thousand feet did he buy?

161. PROB. V.-To find the number of units that can be purchased for a given sum when the cost of two or more units is given.

1. When 8 bushels of wheat can be bought for $12, how many bushels can be bought for $6348 ?

SOLUTION.—Since 8 bushels can be bought for $12, there can be as many times 8 bushels bought for $6348 as $12 is contained times in $6348. Hence, first step, $6348 = $12 = 529; second step, 529 x 8 = 4232, the number of bushels that can be bought for $6348.

Solve and explain the following:

2. If 28 pounds of sugar cost $4, how many pounds can be bought for $348 ?

Ans. 2436 pounds. 3. The cost of 4 boxes of oranges is $12. How many boxes, at the same rate, can be bought for $552 ?

4. When peaches are sold at $6 for 8 baskets, how many baskets must a man sell to receive $582 ? ·Ans. 776 baskets.

5. A farmer sold a quantity of butter at $35 a hundred pounds, and received $1715; how many pounds did he sell ?

6. A carpenter was paid at the rate of $42 for 12 days, and received $588; how many days was he employed ?

7. At $69 for 12 cords of wood, how many cords can be bought for $966 ?

Ans. 168 cords.

REVIEW EXAMPLES.

162. 1. I sold 60 pounds of cheese at 12 cents a pound, and laid out the proceeds in coffee at 20 cents a pound; how many pounds of coffee did I buy? Ans. 36 pounds.

2. Bought a quantity of wood for $3959, and sold it for $6095, thus gaining $3 on each cord sold; how much wood did I buy?

Ans. 712 cords. 3. I sold a farm of 244 acres at $48 an acre, and another farm of 160 acres at $72 an acre; how much more did the first farm bring than the second ?

Ans. $192. 4. The expenses of a young lady at school were $75 for tuition, $20 for books, $68 for clothes, $17 for railroad fare, $5 a week for board for 42 weeks, and $36 for other expenses ; what was the total expense ?

Ans. $426. 5. I paid $8960 for 8 city lots, and sold them at a loss of $12 on each lot; how much did I receive for 3 of them ?

6. I bought 27 acres of land at $41 an acre, and 26 acres at $27 an acre, and sold the whole at $43 an acre; how much did I gain or lose ?

Ans. $470 gain. 7. What is the total cost of 45 acres of land at $17 an

acre, two horses at $132 each, a yoke of oxen for $130, a horse-rake for $65, and a plow for $17? Ans. $1241.

8. A grocer bought 14 barrels of apples for $42; how much will he pay at the same rate for 168 barrels ? Ans. $504.

9. The sum of two numbers is 73, and their difference 47; what are the numbers ?

SOLUTION.—The sum 73 is equal to the greater number plus the less, and the less number plus the difference 47, are equal to the greater ; hence, if 47 be added to the sum 73, we have twice the greater number. Hence, first step, 73 + 47 = 120; second step, 120 ; 2= 60, the greater number; third step, 60 — 47 = 13, the less number.

10. The sum of two numbers is 8976, and the difference 452; what are the numbers ?

Ans. 4714 and 4262. 11. Two men owed together $3957, one of them owed $235 *more than the other; what was each man's debt?

12. A house and lot are worth $7394. The house is valued at $2462 more than the lot; what is each worth?

13. If 15 cows can subsist on a certain quantity of hay for 10 days, how long will the same suffice for 3 cows?

14. A fruit merchant bought 576 barrels of apples at $36 for every 8 barrels, and sold them at $33 for every 6 barrels; how much did he gain on the transaction ? Ans. $576.

15. A drover bought cattle at $47 a head to the amount of $2961, and sold them for $3528; what was the selling price per head ?

Ans. $56. 16. A farmer sold 62 bushels of wheat for $50, also 14 cords of wood at $5 a cord, 4 tons of hay at $15 a ton, and 2 cows at $30 apiece; he took in payment $145 in money, a coat worth $50, a horse-rake worth $21, and the balance in cloverseed at $4 a bushel; how many bushels of seed did he receive ?

Ans. 6 bushels. 17. A farmer paid $22541 for two farms, and the difference in the cost of the farms was $3471. The price of the farm for which he paid the smaller sum was $64 an acre, and of the other $87 an acre. How many acres in cach farm?

PROPERTIES OF NUMBERS.

DEFINITIONS.

163. An Integer is a number that expresses how many there are in a collection of whole things.

Thus, 8 yards, 12 houses, 32 dollars.

164. An Exact Divisor is a number that will divide another number without a remainder.

Thus, 3 or 5 is an exact divisor of 15. All numbers with reference to exact divisors are either prime or composite.

165. A Prime Number is a number that has no exact divisor besides 1 and itself.

Thus, 1, 3, 5, 7, 11, 13, etc., are prime numbers.

166. A Composite Number is a number that has other exact divisors besides 1 and itself.

Thus, 6 is divisible by either 2 or 3 ; hence is composite.

167. A Prime Divisor is a prime number used as a divisor.

Thus, in 35 • 7, 7 is a prime divisor.

168. A Composite Divisor is a composite number used as a divisor.

Thus, in 18 ; 6, 6 is a composite divisor.

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