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add 10 units to 4 units, and the sum is 14 units; and 6 units from 14 units leave 8 units. We now add 1 (ten) to the 4 tens, and the sum is 5 tens, and 5 tens from 9 tens leave 4 tens; therefore 48 dollars remain unpaid. The reason of this method of performing subtraction depends upon a self-evident truth, viz., that if two unequal numbers be equally increased, their difference remains the same.
From the above examples and illustrations, we derive the following
RULE. - Write down the greater number; then write the less number under it, placing units under units, tens under tens, hundreds under hundreds, and draw a line underneath.
Begin with the units, and subtract each figure of the less number in succession from the figure over it, and write the remainder underneath.
Whenever a figure of the less number is greater than the figure over it, add ten to the upper figure, sub tract the lower figure from the amount, then add one to the next lower figure before it is subtracted.
Subtraction is indicated by a short horizontal line, thus, 12—8=4; which is read, 12 minus 8 equals 4, or 12 less 8 equals 4.
EXERCISES TO BE PERFORMED ON THE SLATE.
LESSON CXXII. 1. A merchant borrowed 1,275 dollars, and has paid
750 dollars ; how many dollars remain unpaid ? 2. A trader purchased 2,500 yards of cloth, and has
sold 1,745 yards of it; how many yards has he
remaining unsold ? 3. If a farm and the buildings on it be valued at 12,000
dollars, and the buildings be valued at 4,575 dol
lars, what is the value of the land ? 4. A man purchased an estate, for which he paid 13,750
dollars, and has sold it for 15,225 dollars; what
number of dollars did he gain by trading ? 5. The Rocky Mountains are 12,500 feet above the
level of the ocean, and the Andes are 21,440 feet; how many feet higher are the Andes than the Rocky
Mountains ? 6. A merchant purchased a quantity of goods for which
he paid 2,750 dollars; the goods being damaged, he sold them for 395 dollars less than cost. What
did he get for the goods ? 7. A farmer raised 1,275 bushels of wheat, and has sold
786 bushels of it ; what number of bushels has he
remaining unsold? 8. A farmer sold a load of cheese for 195 dollars, and
took goods in payment to the amount of 49 dollars, and the balance in money ; how much money
did he receive ? 9. A man paid 125 dollars for a horse, and 75 dollars
for a wagon; how much did the cost of the horse
exceed the cost of the wagon ? 10. A tree 95 feet in height was broken off by the wind;
the top part which fell was 39 feet in length. How
high was the part which was left ? 11. A merchant purchased a piece of broadcloth, for which
he paid 240 dollars ; but being damaged, he sold it
for 185 dollars. How many dollars did he lose ? 12. A cargo of fruit cost 1,075 dollars, but it was so much
damaged by a storm that the owner sold it for 890
LESSON CXXIII. 1. A gentleman purchased a gold watch for 65 dollars,
a chain for 12 dollars, and a key for 5 dollars ; he afterwards sold the whole for 100 dollars. Did he
gain or lose, and how much? 2. Charles had 175 marbles ; how many had he left
after giving 25 to William, 19 to George, 38 to
Samuel, 49 to Edward, 41 to John, and losing 2 ? 3. Purchased a farm for 2,125 dollars, paid for build
ings and stock 1,250 dollars ; for how much must
I sell the whole to gain 500 dollars ? 4. A trader sold from a piece of silk containing 45 yards,
at one time 15 yards, at another time 10 yards, and at another time 12 yards ; how many yards
were left in the piece ? 5. “A boy bought a sled for 28 cents, and gave 14 cents
to have it repaired; he sold it for 40 cents. Did
he gain or lose by the bargain, and how much ?” 6. A man owing 350 dollars, paid at one time 47 dollars,
at another time 84 dollars, at another time 35 dollars, and at another time 120 dollars; how much
did he then owe ? 7. A lady gave 6,000 dollars to humane institutions ;
viz., 1,500 dollars to the Orphan Asylum, 2,500 dollars to the Institution for the Blind, and the remainder for the education of the Deaf and Dumb. What was the sum given to the last-named institu
tion ? 8. A market-man having 575 pounds of butter, sold
125 pounds to one man, 95 pounds to another, and 45 pounds to another ; how many pounds had he
remaining unsold ? 9. 915 + 624 + 450 + 375 — 1,125 how many
? 10. 15,475 + 25,750 + 145,725 85,000 how 11. 75,425 + 84,676 + 95,500 — 150,000 = how 12. 94,645 + 15,106 + 44,350 — 105,666 = what number?
MULTIPLICATION. MULTIPLICATION is the method of taking or repeating one of two given numbers as many times as there are units in the other. Multiplication is also a short method of performing addition, when all the numbers to be added are alike.
One of the two given numbers is called the multiplicand, and is the number to be multiplied.
The other is called the multiplier, and it is the number to multiply by, and shows the number of times the multiplicand is to be taken or repeated.
The number produced by multiplying one of the two given numbers by the other is called the product, and it contains either of the two given numbers as many times as there are units in the other.
The multiplicand and multiplier are called factors of the product.
The sign of multiplication is an inclined cross; thus, 8 X5=10, which is read, 8 multiplied by 5 is equal to 40.
Illustration First. Suppose we wish to find the cost of 3 yards of cloth, at 12 cents a yard. It is plain that 3 yards will cost three times as many cents as one yard. First Method.
Second Method. Multiplicand, 12 cents, the price of 1 yd.
12 cents. Multiplier, 3, the number of yards.
12 cents. Product, 36 cents, the cost of 3 yds.
The cost of 3 yards, 36 cents. We
may write down 12 cents, the price of one yard, and multiply them by 3, the number of yards, as is done in the first method ; or, we may write down 12 cents, the price of one yard, 3 times, and add them, as is done in the second method : the result is the same in each of the methods.
Illustration Second. -If an acre of land is worth 125 dollars, how many dollars are 9 acres worth ?
If one acre is worth 125 dollars, 9 acres are worth 9
times 125 dollars; and nine times 125 dollars can be found by multiplying 125 by 9, in the following manner :
We write down the multi
plicand, 125 ; and write the Multiplicand, 125 dols. multiplier, 9, under the units Multiplier,
of the multiplicand. We then 9
say, 9 times 5 units are 45 Product, 1125 dols. units, or 4 tens and 5 units ;
we write down the 5 units, and reserve the 4 tens: then, 9 times 2 tens are 18 tens, and 4 tens which we reserved are 22 tens, or 2 hundred, and 2 tens; we write down the 2 tens, and reserve the 2 hundred: then, 9 times 1 hundred are 9 hundred, and 2 hundred which we reserved are 11 hundred, or 1 thousand, and 1 hundred ; both of which we write down, and the whole product is 1,125, the number of dollars that 9 acres are worth.
From the preceding questions and illustrations, we derive the following rule for multiplication, when the multiplier does not exceed 12.
RULE. — Write down the multiplicand; then write the multiplier under the multiplicand, placing units under units, tens under tens, and draw a line underneath.
Multiply each figure of the multiplicand, beginning with units, by the multiplier : when the product of any figure does not exceed nine, write it under the figure multiplied; when the product exceeds nine, write down the right-hand figure of it, and add the left-hand figure, or figures, to the product of the next figure, and write down the whole product of the last figure.