DENOMINATE NUMBERS.

97. A SIMPLE NUMBER is a unit or a collection of units. The unit may be either abstract or denominate.

98. A DENOMINATE NUMBER is a denominate unit or a collection of units: thus, 3 yards is a denominate number, in which the unit is 1 yard.

99. Numbers which have the same unit, are of the same denomination: and numbers having different units, are of different denominations. If two or more denominate numbers, having different units, are connected together, forming a single number, such is called a compound denominate number.

100. There are eight different units in Arithmetic: 1st. The abstract unit: 2d. The unit of currency: 3d. The unit of length: 4th. The unit of surface: 5th. The cubic unit or unit of volume: 6th. The unit of weight: 7th. The unit of time: 8th. The unit of circular measure.

ENGLISH MONEY.

101. The units or denominations of English money are guineas, pounds, shillings, pence, and farthings.

NOTES.-1. The primary unit in English money is 1 farthing. The number of units in the scale, in passing from farthings to

97. What is a simple number?

98. What is a denominate number?

99. When are numbers of the same denomination? When of different denominations? If several numbers having different units are connected together, what is the number called?

100. How many units are there in Arithmetic? Name them.

pence, is 4; in passing from pence to shillings, 12; in passing from shillings to pounds, 20.

2. Farthings are generally expressed in fractions of a penny. Thus, 1 far.=‡d.; 2 far.=‡d. ; 3 far.=‡d.

3. By reading the second table from right to left, we can see the value of any unit expressed in each of the lower denominations. Thus, 1d. = 4 far. ; 18. 12d.=48far.; £1=208.= 240d. =960far.

REDUCTION OF DENOMINATE NUMBERS.

102. Reduction is changing the unit of a number, without altering its value.

1. How many pence are there in 2s. 6d.?

ANALYSIS.—Since there are 12 pence in 1 shilling, there are twice 12, or 24 pence in 2 shillings: add the 6 pence: therefore, in 28. 6d. there are 30 pence.

2. How many pence in 4 shillings? In 4s. 8d.? In 5s. 6d.? In 3s. 8d.? In 6s. 7d.?

3. How many shillings in £2? 4. How many pence in £1 ?

£2 88.? How many in £3 7s.?

In £3 8s., how many?
How many shillings in

5. How many shillings are there in 48 pence?

ANALYSIS.-Since there are 12 pence in 1 shilling, there are as many shillings in 48 pence, as 12 is contained times in 48, which is 4: therefore, there are 4 shillings in 48 pence.

6. How many pounds in 40 shillings? In 60? In 80?

103. From the above analyses we see, that reduction of denominate numbers is divided into two parts:

1st. To change the unit of a number from a higher denomination to a lower.

2d. To change the unit of a number from a lower denomination to a higher.

101. What are the denominations of English money?

Notes. 1-What is the primary unit in English money? Name the units of the scale.

2.-How are farthings generally expressed?

3.-How is the second table read? What does it show?

102. What is Reduction?

103. Into how many parts is reduction divided? What are they?

PRINCIPLES AND EXAMPLES.

104. To reduce from a higher to a lower unit.

1. Reduce £27 6s. 8d. to the denomination of farthings

ANALYSIS. Since there are 20 shillings in £1, in £27 there are 27 times 20 shillings, or 540 shillings, and 6 shillings added, make 5468. Since 12 pence make 1 shilling, we next multiply by 12, and then add 8d. to the product, giving 6560 pence. Since 4 farthings make 1 penny, we next multiply by 4, and add 2 farthings to the product, giving 26242 farthings for the answer.

OPERATION.

£27 6s. 8d. 2far.

20

5468.

12

6560d.

4

26242 Ans.

NOTE. The units of the scale, in passing from pounds to shillings, are 20; in passing from shillings to pence they are 12; and in passing from pence to farthings, 4.

Hence, to reduce from a higher to a lower unit, we have the following

RULE.-Multiply the highest denomination by the units of the scale which connect it with the next lower, and add to the product the units of that denomination; proceed in the same manner through all thə denominations, till the unit is brought to the required denomination.

105. To reduce from a lower unit to a higher. 1. Reduce 3138 farthings to pounds.

ANALYSIS. Since 4 farthings make a penny, we first divide by 4. Since 12 pence make a shilling, we next divide by 12. Since 20 shillings make a pound, we next divide by 20, and find that 3138 far.=£3 58. 4d. 2 far.

Hence, to reduce from a lower to a higher denomination, we have the following

RULE.-I. Divide the given number by the units of the scale

104. How do you reduce from a higher to a lower unit?

105. How do you reduce from a lower to a higher unit? What will be the unit of any remainder? How do you prove reduction ?

which connect it with the next higher denomination, and set down the remainder, if there be one.

II. Divide the quotient thus obtained by the units of the scale which connect it with the next higher denomination, and set down the remainder.

III. Proceed in the same way to the required denomination, and the last quotient, with the several remainders annexed, will be the answer.

NOTE.-Every remainder will be of the same denomination as its dividend.

PROOF. After a number has been reduced from a higher denomination to a lower, by the first rule, let it be reduced back by the second; and after a number has been reduced from a lower denomination to a higher, by the second rule, let it be reduced back by the first rule. If the work is right, the results will agree.

2. In £31 8s. 9d. 3 far., how many farthings? Also proof. 3. In £87 148. 84d., how many farthings? Also proof. 4. In £407 19s. 11 d., how many farthings? Also proof. 5. In 80 guineas, how many pounds?

6. In 1549 far., how many pounds, shillings and pence? 7. In 6169 pence, how many pounds?

LINEAR MEASURE.

106. This measure is used to measure distances, lengths, breadths, heights and depths, &c.

106. For what is Linear Measure used? Repeat the table. What is a fathom? the units of the scale?

What are its denominations? What is a hand? What are

NOTES.-1. A fathom is a length of six feet, and is generally used to measure the depth of water.

2. A hand is 4 inches, used to measure the height of horses. 3. The units of the scale, in passing from inches to feet, are 12; in passing from feet to yards, 3; from yards to rods, 5; from rods to furlongs, 40; and from furlongs to miles, 8.

In 10 feet? In 16 feet?
In 54 feet? In 96?
In 96 inches? In 48?

1. How many inches in 5 feet? 2. How many yards in 36 feet? 3. How many feet in 144 inches? 4. How many furlongs in 3 miles? In 6 miles? In 8?