KEY TO CARD No. 25.

Third. Shillings.

Two in one I cannot; but 1s. makes 12 pence and 5 are 17-two in 17-8 times and one over-1 penny makes 4 farthings-2 in 4-2 times.

Thus we find the answer to be 8d. 2q.

LESSON 14.

A merchant bought 5 hundred weight of lead for £14 11 8, what was that a pound?

7

7 3

2

2

1

6

21

First. Pounds.

Five in 14-2, and £4 over.

Shillings.

£ 4 over make 80s. and 11 are 91. Five in 91-18 times and 1 over. Pence.

One shilling over makes 12 pence and 8 are 20-5 in 20 -4 times.

Second. Pounds.

Eight in 2-no times, but 2 pounds make 40s. and 18s. make 58s,

Shillings.

Eight in 58-7 times and 2 over.

Pence.

Two shillings over make 24 pence and 4 are 28--8 in 28— 8 times and 4 over.

Four

Farthings.

pence over make 16 farthings-8 in 16-2 times. Third, Shillings.

Seven in 7-1.

Pence.

Seven in 3-0, but 3d. remain undivided.

Farthings.

Three pence remaining undivided, make 12 farthings and 2 are 14-7 in 14 twice.

Fourth. Shillings and Pence.

Two in 1-0, but 1s. makes 12d.-2 in 12-6 times; set 6 for pence.

KEY TO CARD No. 26.

Farthings.

Two in 2 farthings-1; set 1 for farthings. The answer

is 6d. 1q. a pound.

Compound Division is proved by Compound Multiplication. EXAMPLE.

Reverse the last lesson or question and say, "If 1 pound of Lead cost 6d. 1q., what will five hundred weight cost?"

£. s. d. q.

6

1

7 3 2

= Two parallel lines denote Equality; as, 3s.=36 pence; that is, three shillings equal thirty-six pence.

+

perpendicular and horizontal line forming a cross, signify Addition; as, 4+2=6. 3+2+4-9. Read thus, 4 more 2 equal 6. 3 more 2 more 4 equal 9.

*A single dash denotes Subtraction; as, 7-2-5. 7-2 -1-4. Read, 7 less 2 equal 5. 7 less 2 less 1 equal 4. Two lines forming a cross diagonally, signify Multiplication; as, 4x2=8. That is, 4 multiplied by 2 equals 8. A dash with two periods, one above, and one below the centre, signify Division; as, 12÷2-6. That is, 12 divided by 2 equals 6. An inverted parenthesis also denotes division. In this book, the dash is not used arithmetically till after division.

KEY TO CARD No. 26.

:: Four points set in the middle of four numbers, denote the numbers to be proportional to one another, in the Rule of Three; as 3:6:: 12:24. That is, as 3 are: to 6:: so are 12: to 24.

A Square signifies that the square of any number is required; as, 14-16; or the square of 4 equals 16. ✔ This reversed figure seven, signifies that the Square Root of a number is required, as 16-4; that is to say,

the Square Root of 16 equals 4, or is 4.

These characters are used to save written words, and to show when to add, subtract, multiply, or divide, without a tedious series of writing. The learners may copy them several times on their slates. See Card No. 26.

DECIMALS.

Having given a hint of decimals in Multiplication, I will now insert a few lessons to aid the learner in common business; but as the whole subject of fractions is tedious and little in use, shall give a view of the most necessary cases only.

What is a fraction?

It is some part of an integer; that is, some part of a whole number; as,

One fourth. One half. Three fourths. Seven eighths.

These are called vulgar fractions; and for ease, plainness, and despatch of business, decimals have been invented for a substitute. Therefore our first business will be to form fractions into decimals.

In order to understand ourselves, let us have a name for ch part of the aforegoing fractions.

KEY TO CARD No. 26.

The upper number is called Numerator; and is made by a remainder after division.

The lower number is called, Denominator; and is the divisor in Division. Thus, divide 17 cents equally between 4 boys, each share is 4; that is, four and one fourth.

*

To turn fractions into decimals, observe the following RULES.

1. Annex, that is, place on the right side of the Numetator, one or more ciphers, and call it a Dividend.

2. Take the lower number, the Denominator, for a divisor : The quotient will be the decimal required.

In decimal operations, Long Division is preferable to Short Division, on account of knowing where to place the dividing point. This point is the only difficulty to be met with in decimals. It is a period called the separatrix, to separate decimals from integers, and integers from decimals.*

When we shall readily know where to place this point, decimals will become as familiar as integers; and greatly expedite business in frequent instances.

To know how, and where to place this point, let us attend. to a few particular rules.

DIVISION.

1. Count the decimal places in the dividend. Suppose them to be two as in lesson first.

2. Count the number of decimals in the divisor, if any, then begin at the right hand figure in the quotient, and

Integer signifies a whole number; as, a pound, an ounce, a bushel, a gallon, &c. "The whole of any thing."

KEY TO CARD No. 26.

continue counting towards the left, till you have a number equal to the decimals in the dividend: there place the sepa ratrix or point.

3. But if the number of figures in the quotient fall short, then prefix ciphers on the left till your counting is completed: there place the point.

4. When the dividend is less than the divisor, annex-'ci-, phers on the right according to discretion.

NOTE. In forming fractions into décimals, we consider. the Numerator, and Denominator, as two distinct whole.. numbers.

For instance, observe lesson first. We place I foi divă dend, and 4 for a divisor. We then see that 4 cannot be contained in 1; therefore place a point and annex ciphers to the right of the point.

These ciphers show us how many decimal parts are contained in the dividend; and that the digit, 1, is considered as an integer.

In the next place we proceed with Division; when one, begin to count and see where the point must be placed in the quotient. As there is no decimal place in the divisor of lesson 1, begin at the right hand of the quotient and count towards the left, till having a number equal to the decimal places in the dividend: then place our point as on the left side of .25 hundredths. This counting proves its being placed right at first.

only; consequently there is but one decimal place in the dividend. And as there is but one figure in the quotient, the point must be placed on the left of that figure. Thus, .5 tenths.

*Digit, signifies any number under ten.