116. 976.64 is 43.1 times what number?

117. 43308 is 1.97 times what number?

118. Divide mentally, 69 by 3; 74 by 2; 64 by 2; 84 by 4; 66 by 2; 82 by 2; 93 by 3; 77 by 7; 39 by 3; 40 by 2; 280 by 7; 55 by 5; 90 by 5; 84 by 7; 560 by 8; 91 by 7; 78 by 6; 42 by 3; 85 by 5; 96 by 2; 76 by 4.

119. Divide mentally, 126 by 7; 387 by 9; 264 by 8; 175 by 7; 534 by 6; 192 by 8; 644 by 7; 594 by 9; 324 by 6; 528 by 6; 792 by 8; 945 by 5; 483 by 7; 672 by 8; 883 by 9; 770 by 10; 770 by 7; 770 by 70; 440 by 11; 594 by 11; 516 by 12; 792 by 11; 660 by 12; 429 by 13; 299 by 13; 403 by 13.

THE TRUE QUOTIENT FIGURE

Cannot always be determined, without repeated trials. The number of trials may, however, be often diminished, as in the following example:

Divide 7106 by 187. 187) 7106 (38

561

1496
1496

The divisor is between 100 and 200, and will therefore be contained in the dividend not as many times as 100, but more times than 200. In dividing by 100 or 200, we need employ only the first figure as a divisor; and if we employ the first figure of each of these numbers as a trial divisor, we shall obtain the limits within which the true quotient figure is contained.

Now, 1 is contained in 7, 7 times; 2 in 7, 3 times;-7 and 3 are therefore the limits of the first quotient figure, which must be either 3, 4, 5, 6, or 7.

As the divisor is nearer 200 than 100, we first try the smallest of these figures, which is found to be the true one. In the same way, we find that 14 and 7 are the limits of the second quotient figure, which must be either 7, 8, or 9, as no single figure can be greater than 9. Trying 7, we find it too small; but 8 is the true figure sought.

Always employ the first divisor figure, and a number one larger than the first divisor figure, as trial divisors.

REVIEW.

ALL Arithmetical questions, depend on the proper application of one, or more, of the five fundamental operations that have been illustrated in the foregoing chapters, viz., Numeration, Addition, Subtraction, Multiplication, and Division.

In Numeration, numbers are either given in figures to be read in words, or given in words to be written in figures.

In Addition, two or more numbers are given, and their sum, or amount, is required.

In Subtraction, two numbers are given, and their difference required.

In Multiplication, two numbers are given, and their product is required. The product is equivalent to the sum of one of the numbers, repeated as many times or parts of a time, as there are units, or parts of units, in the other. Multiplication is, therefore, a quick way of performing many additions.

In Division, two numbers are given, and their quotient is required. The quotient shows how many times one of the numbers, or a part of it, can be subtracted from the other. Division is, therefore, a quick way of performing many subtractions.

The termination nd, signifies to be. Thus the subtrahend, is the number to be subtracted; the minuend, the number to be diminished; the multiplicand, the number to be multiplied; the dividend, the number to be divided.

The terminations er and or, are applied to the one that does something. Thus the multiplier, is the

number that multiplies; the divisor, the number that divides; the numerator of a fraction, the one that numbers the parts that are taken; the denominator, the one that denominates, or names the value of the parts.

QUESTIONS FOR THE PUPIL.

What is the object of Numeration? How many figures are employed to represent Numbers? What are they? What is the use of 0? What is a unit? Repeat the Numeration table. What is the Decimal Point, or Separatrix? What is the effect of removing a figure one place to the left?-one place to the right? What is the effect of Os at the right of decimals?

What is the object of Addition? How do you write the numbers that are to be added? Where do you begin to add? If the sum of any column is more than 9, what is done with it?

What is the object of Subtraction? What is the Minuend? What is the Subtrahend? What is the Remainder? How are the numbers to be written? Where do you begin to subtract? If any figure in the Subtrahend is larger than the one over it in the Minuend, what must be done? How may the truth of the answer be proved?

What is the object of Multiplication? What is it a quick way of performing? What is the Multiplicand? What is the Multiplier? What is the Product? Where do you place the units' figure of the Multiplier? If the Multiplier contains more than one figure, where do you place the first figure of each partial product? Where will the decimal point of each product fall? What is the product of any number by 0? What are Factors? When the Multiplier can be resolved into Fac tors, how may the product be obtained? How many decimals will there be in the product?

What is the object of Division? What is it a quick way of performing? What is the Dividend? What is the Divisor? What is the Quotient? What is the Remainder? Where do you place the Divisor? If the Dividend has fewer decimals than the Divisor, what must be done? How many decimals will there be in the Quotient? When there are Factors to the Divisor, how may the Quotient be obtained? When there are Os at the right-hand of the Divisor, what may be done? How may the Quotient be proved. How may the Product in Multiplication be proved?

MISCELLANEOUS EXAMPLES.

The following abbreviations may often shorten the labour required in solving a question.

When the multiplier consists of any number of 9s, increase it by 1, and subtract the multiplicand from the product.

If the multiplier is 5, divide the multiplicand by .2. If the multiplier is 25, divide the multiplicand by .04.

If the multiplier is 75, multiply by 100, and subtract of the product.

If the multiplier is 125, divide by .008.

If the multiplier is 375, divide by .008, and multiply the quotient by 3.

If the multiplier is 625, divide by .008, and multiply by 5.

If the multiplier is 875, multiply by 1000, and subtract of the product.

If the divisor is either of the above numbers, (except 75 or 875,) reverse the process; i. e., multiply instead of dividing, and divide instead of multiplying. If the divisor is 75, divide by 100, and add the quotient.

of

If the divisor is 875, divide by 1000, and add ÷ of the quotient.

[Let the teacher give examples, illustrating each of the above abbreviations, on the Board.]

1. A bankrupt owed to one man, five thousand and six dollars and sixteen cents; to another, three thousand two hundred and four dollars and nine cents; to another, one thousand nine hundred and seventy dollars and sixty-eight cents; to another, eight hundred and ninety-six dollars and three cents; and to all others, ten thousand and five dollars and eleven cents. What was the whole amount of his debts?

2. From nine hundred and three million two thousand and seventeen-and two hundred and one, hundred thousandths, take eight hundred million seventy thousand and four-and five, ten-millionths.

3. If a man takes two thousand four hundred and six—and five, tenths steps in walking one mile, how many will he take in six-and one hundred and seventy-three, ten-thousandths miles?

4. A banking institution made ninety-one thousand and seven dollars, in one thousand and sixteen days. How much were the profits per day?

5. William Johnson bought of John Williamson, 4 pieces of muslin, each 31.5 yards, at $0.125 per yard.

6

6

do.
do.

5

do.

7

do.

each 29.75 yards, at $0.115 per yard.
each 30.25 yards, at $0.11 per yard.
each 31 yards, at $0.1275 per yard.
each 30.5 yards, at $0.135 per yard.

How many yards of muslin did he buy, and what was the amount of his bill?

6. James Clark bought of Jeremiah Appleton,

6 chests of tea, each 44.5 pounds, at $0.625 per pound.

3

5

4

do.

do.

do.

each 47.2 pounds, at $0.75 per pound. each 51.6 pounds, at $0.875 per pound. each 49 pounds, at $0.93 per pound. How many pounds did he buy, and what was the amount of his bill?

7. What is the total population of the world, there being in North America, 30960000 inhabitants ; South America, 14040000; Europe, 230000000; Asia, 450000000; Africa, 57000000, and Oceanica, 18000000?

8. What is the number of inhabitants to a square mile, the whole land surface of the earth containing 50000000 square miles?

9. A merchant purchased flour for $943.25; he paid $1.375 for carting, and $6.00 for storage. How much must he sell it for, to gain $16.43?