him turn to the exercise for the slate to which reference is made, and let him apply it in illustration of the answer he gives. Thus

Ans. Subtract the subtrahend from the minuend, and the difference will be the remainder, as Ex. 6, (slate,) where the minuend and subtrahend are given to find the remainder, we subtract the subtrahend 3481 from the minuend 7842, and the difference, 4361, is the remainder.

2. When the minuend and remainder are given, how do you find the subtrahend? Ex. 7.

do

3. When the subtrahend and the remainder are given, how you find the minuend? Ex. 8.

4. When you have the sum of two numbers, and one of them given, how do you find the other?

Ex. 9.

5. When you have the greater of two numbers, and their difference given, how do you find the less number? Ex. 10. 6. When you have the less of two numbers, and their difference given, how do you find the greater number? Ex. 11. 7. When the sum and difference of two numbers are given, how do you find the two numbers? Ex. 12.

EXERCISES FOR THE SLATE.

T45. 1. If the multiplicand (squares in a row) be 754, and the multiplier (rows of squares) be 25, what will be the product (no. of squares)?

2. If the product (no. of squares) be 18850, and the multiplicand (squares in a row) be 754, what must have been the multiplier (rows of squares)?

3. If the product (no. of squares) be 18850, and the multiplier (rows of squares) be 25, what must have been the multiplicand (squares in a row) ?

4. If the dividend (no. of squares) be 144, and the divisor (squares in a row) be 8, what is the quotient (no. of rows)?

5. If the dividend (no. of squares) be 144, and the quotient (no. of rows) be 18, what must have been the divisor (squares in a row)?

6. If the divisor (squares in a row) be 8, and the quotient (rows of squares) be 18, what must have been the dividend (no. of squares) ?

7. The product of three numbers is 525, and two of the numbers are 5 and 7, what is the other number? Ans. 15.

MENTAL EXERCISES.

When the factors are given, how do you find the product? Ex. 1.

When the product and one factor are given, how do you find the other? Ex. 2 and 3.

When the dividend and quotient are given, how do you find the divisor? Ex. 5.

When the divisor and quotient are given, how do you find the dividend? Ex. 6.

When the product of three numbers and two of them are given, how do you find the other? Ex. 7.

EXERCISES FOR THE SLATE.

46. 1. What will be the cost of 15 pounds of butter, at 13 cents a pound?

2. A man bought 15 pounds of butter for 195 cents; what was that a pound?

3. A man buying butter, at 15 cents a pound, paid out 195 cents; how many pounds did he buy?

4. When rye is 75 cents a bushel, what will be the cost of 984 bushels? how many dollars will it be?

5. If 984 bushels of rye cost 738 dollars, (73800 cents,) what is the price of 1 bushel?

6. A man bought rye to the amount of 738 dollars, (73800 cents,) at 75 cents a bushel; how many bushels did he buy?

7. If 648 pounds of tea cost 284 dollars, (28400 cents,) what is the price of 1 pound? 28400648 how many?

MENTAL EXERCISES.

1. When the price of one pound, one bushel, &c., of any commodity is given, how do you find the cost of any number of pounds, or bushels, &c., of that commodity? Ex. 1 and 4. If the price of the 1 pound, &c., be in cents, in what will the whole cost be? if in dollars, what? if in shillings?

if in pence? &c. 2. When the cost of any given number of pounds, or bushels, &c., is given, how do you find the price of one pound, or bushel, &c.? Ex. 2, 5, and 7. In what kind of money will

the answer be?

3. When the cost of a number of pounds, &c., is given, and also the price of one pound, &c., how do you find the number of pounds, &c.? Ex. 3 and 6.

EXERCISES FOR THE SLATE.

T47. 1. A boy bought a number of apples; he gave away ten of them to his companions, and afterwards bought thirty-four more, and divided one half of what he then had among four companions, who received 8 apples each; how many apples did the boy first buy?

Let the pupil take the last number of apples, 8, and reverse the process. Ans. 40 apples. 2. There is a certain number, to which if 4 be added, and from the sum 7 be subtracted, and the difference be multiplied by 8, and the product divided by 3, the quotient will be 64; what is that number? Ans. 27.

3. If a man save six cents a day, how many cents would he save in a year, (365 days?) - how many in 45 years? how many dollars would it be? how many cows could he buy with the money, at 12 dollars each?

Ans. to the last, 82 cows, and 1 dollar 50 cents remainder. 4. A man bought a farm for 22464 dollars; he sold one half of it for 12480 dollars, at the rate of 20 dollars per acre; how many acres did he buy? and what did it cost him per acre? Ans. to the last, 18 dollars. 5. How many pounds of pork, worth 6 cents a pound, can be bought for 144 cents?

6. How many pounds of butter, at 15 cents per pound, must be paid for 25 pounds of tea, at 42 cents per pound?

7. A man married at the age of 23; he lived with his wife 14 years; she then died, leaving him a daughter 12 years of age; 8 years after, the daughter was married to a man 5 years older than herself, who was 40 years of age when the father died; how old was the father at his death?

Ans. 60 years.

8. The earth, in moving round the sun, travels at the rate of 68000 miles an hour; how many miles does it travel in one day, (24 hours?) how many miles in one year, (365 days?) and how many days would it take a man to travel this last distance, at the rate of 40 miles a day? how many years? Ans. to the last, 40800 years.

Problems in the Measurement of Rectangles and Solids.

NOTE.A rectangle is a figure having four sides, and each of the four corners a square corner.

PROBLEM I.

T48. The length and breadth of a rectangle given, to find the square contents.

1. How many square rods in a plat of ground 5 rods long and 3 rods wide?

D

C

B

SOLUTION. A square rod is a square measuring 1 rod on each side, like one of those in the annexed diagram. We see from the diagram that there are as many squares in a row as there are rods on one side, and as many rows as there are rods on the other side; that is, 5 rows of 3 squares in a row, or 3 rows of 5 squares in a row. We multiply the number of squares in

one row by the number of rows; 5X315 square rods, Ans. Hence the

RULE.

Multiply the length by the breadth, and the product will be the square contents.

NOTE. Three times a line 5 rods long is a line 15 rods long. Hence the pupil must not fail to notice, that we multiply the number of square rods in a piece of ground 1 rod wide and of the given length by the number of rods in the width.

EXAMPLES.

2. How many square rods in a piece of ground 160 rods long (squares in a row) and 8 rods wide (rows of squares)? Ans. 1280 square rods. 3. How many square feet in a floor 32 feet long and 23 feet wide? Ans. 736.

4. How many yards of carpeting, 1 yard wide, will it take

Questions. 48. Describe a rectangle; a square rod. How do you determine the number of squares in a row, and the number of rows? Give the rule. What is the quantity really multiplied? What absurdity in considering it otherwise?

to cover the floors of two rooms, one 8 yards long and 7 yards wide, and the other 6 yards long and 5 yards wide? Ans. 86 yards.

5. How many square feet of boards will it take for the floor of a room 16 feet long and 15 feet wide, if we allow 12 Ans. 252. square feet for waste?

6. There is a room 6 yards long and 5 yards wide; how many yards of carpeting, a yard wide, will be sufficient to cover the floor, if the hearth and fireplace occupy 3 square Ans. 27. yards?

PROBLEM II.

T49. The square contents and width given, to find the length.

1. What is the length of a piece of ground 3 rods wide, and containing 15 square rods?

SOLUTION. -In this example we have 15, the number of squares in several rows, (see the diagram, problem I.,) and 3 the number of squares in 1 row, to find the number of rows. We divide the squares in the number of rows by the squares in 1 row. Hence,

RULE.

Divide the square contents by the width, and the quotient will be the length. Or really, since the divisor and dividend must be of the same denomination, we divide the whole number of square rods by the square rods in a piece of land 3 rods long by 1 rod wide; thus, 15÷35 rods in length, Ans.

EXAMPLES.

2. A piece of ground containing 1280 square rods, is 8 rods in width; what is its length? Ans. 160 rods. 3. A floor containing 736 square feet, is 23 feet wide; what is its length? Ans. 32 feet.

PROBLEM III.

¶ 50. The square contents and length given, to find the width.

1. What is the width of a piece of ground, 5 rods long, and containing 15 square rods?

Questions.¶49. Repeat the 2d problem; the example. What two things are given in the example, and what required? Give the rule. What is really the divisor. and why?