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82. A Simple Number is a unit or a collection of units, either abstract, or concrete of a single denomination ; thus, 1 dollar, 9 apples, 12, are simple numbers.

A Compound Number is a collection of concrete units of several denominations, taken collectively; thus, 12£ 18s. 9d. is a compound number.

A Denominate Number is any concrete number which may changed to a different denomination.

Note. - A Scale denotes the relations between the different units of a number. Simple numbers have a uniform scale of 10, but compound numbers generally have a varying scale.

83. Reduction is the changing of a number into one of a different denomination, but of equal value.

It is of two kinds, Reduction Descending and Reduction Ascending.

Reduction Descending is the changing of a number of a higher denomination into one of a lower denomination ; es pounds to shillings, &c. It is performed by multiplication.

Reduction Ascending is the changing of a number of a lower denomination into one of a higher denomination; as farthings to pence, &c. It is the reverse of Reduction Descending, and is performed by division.


84. English or Sterling Money is the Currency of England.

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82. What is a simple number? A compound number? — 83. What is reduction? How many kinds of reduction ? What are they? What is reduction descending ? Reduction ascending ? — 84. What is English money? Repeat the table.

Note 1. - £ stands for the Latin word libra, signifying a pound; s. for solidus, a shilling; d. for denarius, a penny; gr. for quadrans, a quarter.

NOTE 2. — Farthings are sometimes expressed in a fraction of a penny; thus, 1 far. = $d. ; 2 far. = 5 d.; 3 far. 4 d.

Note 3. — The usual current value in United States money of the sovereign, the gold coin representing a pound sterling, is $ 4.84. The guinea, or 21 shilling gold piece, is no longer coined.

Note 4. The term sterling is probably from Easterling, the popular name of certain early German traders in England.


1. How many farthings in 3 pence? In 9 pence?
2. How many pence in 2 shillings? In 6 shillings?
3. How many shillings in 7 pounds? In 10 pounds ?
4. How many pence in 8 farthings? In 24 farthings?
5. How many shillings in 24 pence? In 60 pence?
6. How many pounds in 40 shillings? In 80 shillings?


85. To reduce from a higher to a lower denomination.

Ex. 1. How many farthings in 17£ 8s. 9d. 3far. ?

17£ 8s. 9d. 3far.


3 4 8 shillings.

1 2

We multiply the 17 by 20, because 20 shillings make 1 pound, and to this product we add the 8 shillings. We then multiply by 12, because 12 pence make 1 shilling, and to the product we add the 9d. Again, we multiply by 4, because 4 farthings make 1 penny, and to this product we add the 3 far., and obtain 16743 farthings.

4185 pence.


Ans. 1 6 7 4 3 farthings.


Multiply the highest denomination given by the number required of the next lower denomination to make one in the denomination multiplied. To this product add the corresponding denomination of the multiplicand, if there be any. Proceed in like manner, till the reduction is brought to the denomination required.

85. How do you reduce pounds to shillings? Why multiply by 20? How do you reduce shillings to pence? Why? Pence to farthings? Why? Sovereigns to shillings ? The general rule for reduction descending ?


86. To reduce from a lower to a higher denomination. Ex. 2. How many pounds in 16743 farthings?

We divide by 4, because 4 farthings 4) 1 6 7 4 3 far. make 1 penny, and the result is 4185

pence, and the remainder, 3, is far12) 418 5 d. 3far.

things. We divide by 12, because 12 20 ) 3 4 8 s. 9d.

pence make 1 shilling, and the result is

348 shillings, and the 9 remaining is 17 £ 8s.

pence. Lastly, we divide by 20, beAns. 17£ 8s. 9d. 3far.

cause 20 shillings make 1 pound, and

the result is 17£ 8s.; and by annexing all the remainders to the last quotient, we find the answer to be 172 8s. 9d. 3far.

RULE. - Divide the lowest given denomination by the number which it takes of that denomination to make one of the denomination next higher. The quotient thus obtained divide in like manner, and so proceed until it is brought to the denomination required. The last quotient, with the remainders connected, will be the answer.

3. In 9£ 18s. 7d. how many pence?
4. In 2383d. how many pounds, &c. ?
5. How many farthings in 14£ 11s. 5d. 2far. ?
6. How many pounds in 13990far.?


87. Troy Weight is the weight used in weighing gold, silver, and jewels.


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86. How do you reduce farthings to pence? Why divide by. 4 ? How do you reduce pence to shillings? Why? Shillings to pounds? Why? Shillings to guineas ? What is the general rule for reduction ascending ? What is Troy Weight used for? Repeat the table.

NOTE 1.- The oz. stands for onza, the Spanish for ounce.

NOTE 2. A grain or corn of wheat, gathered out of the middle of the ear, was the origin of all the weights used in England. Of these grains, 32, well dried, were to make one pennyweight. But in later times it was thought sufficient to divide the same pennyweight into 24 equal parts, still called grains.

Note 3. Diamonds and other precious stones are weighed by what is called Diamond Weight, of which 16 parts make 1 grain ; 4 grains, 1 carat. 1 grain Diamond Weight is equal to of a grain Troy.

NOTE 4. — The Troy pound is the standard unit of weight adopted by the United States Mint, and is the same as the Imperial Troy pound of Great Britain.


1. How many gr. in 2pwt.? In 10pwt. ?
2. How many pennyweights in 4oz. ? In 20oz. ?
3. How many ounces in 21b. ? In 5lb. ? In 10lb. ?
4. How many pennyweights in 48gr.? In 96gr.?
5. How many ounces in 40pwt.? In 120pwt. ?
6. How many pounds in 24oz.? In 60oz ? In 120oz. ?


1. How many grains in 721b. 10oz. 15pwt. 7gr. ?

2. In 419887 grains, how many pounds ?



7 2 lb. 10oz. 15pwt. 7gr. 24) 419 8 8 7 gr.

20) 17 4 9 5 pwt. 7gr. 87 4 ounces.

2 ) 8 7 4 oz. 15pwt. 20

7 2 lb. 10oz. 17 49 5 pennyweights. 24

Ans. 721b. 10oz. 15pwt. 7gr.

6 9 987 3 49 90

Ans. 41 9 8 8 7 grains.

87. What was the original of all weights in England ? How many of these grains did it take to make a pennyweight? How many grains in a pennyweight now? By what weight are diamonds weighed ? What is the standard at the mint? How do you reduce pounds to grains ? Give the reason of the operation. How do you reduce grains to pounds ?

3. How many grains in 76pwt. 12gr.? 4. How many pennyweights in 1836gr.? 5. In 76lb. 5oz. how many grains ? 6. In 440160 grains how many pounds ? 7. How many pennyweights in 1441b. 9oz. ? 8. How many pounds in 34740pwt. ? 9. How many pounds in 17895gr.? 10. In 3lb. loz. 5pwt. 15gr. how many grains ?

11. A valuable gem weighing 2oz. 18pwt. 12gr. was sold for $ 1.37 per grain ; what was the sum paid ? Ans. $ 1923.48.


88. Apothecaries' Weight is used in mixing medicines.

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5760 Note 1.- In this weight the pound, ounce, and grain are the same as in Troy Weight.

Note 2. — Medicines are usually bought and sold by Avoirdupois Weight.

Note 3. — Of fluids, 45 drops, or a common tea-spoonful, make about 1 fluid dram; 2 common table-spoonfuls, about 1 fluid ounce.

MENTAL EXERCISES. 1. In 40 grains how many scruples? In 60gr.? In 120gr.? 2. In 5 scruples how many grains ? In 10sc. ? In 40sc. ? 3. In 3 drams how many scruples? In 10dr. ? In 17dr. ? 4. How many pounds in 48 ounces? In 96oz. ? In 144oz. ? 5. How many ounces in 24 drams? In 64dr. ? In 96dr. ?

88. For what is Apothecaries' Weight used? What denominations of this weight are the same as those of Troy Weight? By what weight are mcdicines usually bought and sold ? Repeat the table.

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