undivided numbers below; proceed with these in the same manner, and repeat the process till there remain not any two numbers commensurable: the continued product of the divisors, quotients, and undivided numbers, will be the least common multiple.

(1) Required the least common multiple of 2, 3, 4, 5, 6, 7, 8, 9, and 10.*

8.

(2) Find the least number divisible by 3, 4, 5, 6,7, and

Ans. 840.

(3) What is the least common multiple of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12?

Ans. 27720.

Case 10. To reduce fractions to a common denominator. RULE 1. Multiply each numerator into all the denominators, except its own, for a numerator; and all the denominators for a.common denominator.

Or,

RULE 2. Find the least common multiple of the denominators, which will be the least common denominator. Divide this by each denominator, and multiply the several quotients by the respective numerators for the required numerators.

(1) Reduce

and to a common denominator.† (2) Reduce,, and ğ, to a common denominator.

Ans.,, and f§; or 1, §, and §. (3) Reduce, 4, 1%, and 9, to a common denominator. Ans. 16, 18, 188, and 728.

8409

(4) Reduce,, and, to a common denominator.

Ans. 48, 5, and 48.

of 2, to a common denominator.
Ans. $185, 18%, and
660 539

1133

of 14, to a common denomiAns. 75, 13, and 5.

2 and 4, being factors of 8, 3 a factor of 9, and 5 a factor of 10, may be omitted. Thus,

2)6, 7, 8, 9, 10

Then 2 X 3X 7 X4 X3 X5 = 42 X 60

2520, the least number divisible by all the given numbers.

numerators. Ans. 11 and §§.

28 the denominator.

ADDITION.

RULE. Reduce the given fractions to a common denominator, over which place the sum of the numerators.

(1) Add and together. 3+1+1=?=1. Ans.

(2) Add 3,, and .
(3) Add 1, 4, and .*
(4) Add 78 and together.
(5) Add, and of 2.

Fractional quantities may

(6) Add 53, 63, and 4.

(7) Add 14, 34, and 1⁄2 of 7. (8) Add of 6%, and 4 of 74. (9) Add of 9%, and 3 of 4. }

10

be reduced to their proper

values, and the sum found by Compound Addition.

(10) Add

of a pound to g of a shilling. Ans. 8s. 4d.

(11) Add d. gs. and £3. (12) Add 3 lb. troy,

Ans. 14s.

oz. and ğ oz. Ans. 7 oz. 19 drts.

of a cwt.

Ans. 12 cwt. 1 qr. 1 lb.

(14) What is the sum of 3 of £17..7..6d.,

of a crown?

of £13. and

Ans. £13..0..24.

(15) Add of 3 a. 1 r. 20 p., of an acre, and of 3 roods, 15 perches. Ans. 3 a. 2 r. 331 p.

SUBTRACTION.

RULE. Reduce the given fractions to a common denominator, over which place the difference of the numerators.. When the numerator of the fractional part in the subtrahend is greater than the other numerator, borrow a fraction equal to unity, having the common denominator; then subtract, and carry one to the integer of the subtrahend. (1) From take • 2—4=8}=}}=2'5. Ans. (2) From take 3. (6) From 644 take of 2. (7) From 15 take 12.

(3) From 53 take

(4) From

(5) From

of §.

take

of.

take

of .

Fractional quantities may be

values, as directed in Addition.

* When there are integers among the given numbers, first find the sum of the fractions, to which add the integers.

Thus in Ex. 3, }+}=}; then 1+1=2+R={}; anl 4+11= 413. Ans.

(10) From of a pound take of a shilling. Ans. 7s. 14d. (11) From 13s. take of 74d.

(12) What is the difference between

of £1.?

5

Ans. 1s. 3d.
of £1. ; and
Ans. 2d. 3 qrs.

(13) Subtract & cwt. from 4 ton. Ans. 10 cwt. 2 qrs. 103 lb. lb. troy subtract § of 34 oz.

(14) From

of 5

(15) Subtract 71

9 yds.

Ans. 3 lb. 2 oz. 1 dwt. 23 gr. furlongs from 15 mile. Ans. 4 fur.

MULTIPLICATION.

RULE. Prepare the given numbers (if they require it) by the rules of Reduction: then multiply all the numerators together for the numerator of the product, and all the denominators for the denominator.

(12) Required the product, in square feet, of 14 ft. 7 in. by 8 ft. 9 in. Ans. 127 sq. fl.

DIVISION.

RULE. Prepare the given numbers (if they require it) by the rules of Reduction; then invert the divisor, and proceed as in Multiplication.

(1) Divide

by 3.t

(2) Divide by .

(3) Divide 672 by 13.

(4) Divide 7935 by 189. (5) Divide 16 by 24.

(6) Divide by 44.

A number inverted becomes the reciprocal of that number; which is the quotient arising from dividing unity by the given number: thus (1÷7=4, the reciprocal of 7; 1÷=, the reciprocal of §.

!

(11) Divide £1181 by 3 of 13. (12) Divide 1s. 44d. by of §. (13) Divide 3 qrs. 241 lb. by 1 cwt.; and value the quotient.

qrs.

Ans. £3..17..10.
Ans. 6d. 3

of 12, in the fraction of a Ans. 1 cwt. 1 qr. 15ğ lb. (14) What must £7..14..6. be multiplied by, to produce £21..17..9? Ans. 25.

THE RULE OF THREE.

RULE. Prepare the terms, previous to stating, so that no subsequent Reduction will be necessary: then, having stated the question, as previously directed, invert the dividing term, and the continued product of the three will be the answer. (1) If 3 of a yard cost £. what will of a yard cost ?* (2) If yd. cost £3. what will yd. cost? Ans. 14s. 8d. (3) If of a yard of lawn cost 7s. 3d. what will 100 yards Ans. £4.19..10.

cost? (4) If lb. costs. how much will §s. buy? Ans. 1 lb. 5) If 48 men can build a wall in 244 days, how many men can do the same in 192 days? Ans. 6 men. (6) If of a yard of Holland cost £. what will 123 ells cost at the same rate? Ans. £7..0..8 5. yard wide, be sufficient of a yard wide, will Ans. 4 yards.

(7) If 31 yards of cloth, that is 13 to make a cloak, how much that is make another of the same size?

(8) If 12 yards of cloth cost 15s. 9d. what will 48 yards cost at the same rate?

Ans. £3..0..9

(9) If 25%s. will pay for the carriage of 1 cwt., 145 miles, how far may 6 cwt. be carried for the same money?

(10) If of a cwt. cost £14. 4s. what

cwt.?

Ans. 22 miles. is the value of 7 Ans. £118..6..8. what will 36,7 lb.

Ans. £61..3..4. inches broad, will Ans. 2011 inches. broad cloth, each 27% Ans. £85..14..31 §.

3

1

3 1

: 10

yd. $ 16:4

9

4

£.

==15s. Ans.

3

(14) If a penny white loaf weigh 7 oz. when a bushel of wheat costs 5s. 6d. what is the bushel worth when a penny white loaf weighs but 24 oz.? Ans. 15s. 4d. 3} qrs.

(15) What quantity of shalloon that is of a yard wide will line 74 yards of cloth, that is 14 yard wide? Ans. 15 yards. (16) Bought 3 pieces of silk, each containing 248 ells, at 6s. Od. per ell. How must I sell it per yard, to gain £5. by the bargain? Ans. 5s. 91d. 19.

THE DOUBLE RULE OF THREE.

(1) If a carrier receive £21. for the carriage of 3 cwt. 150 miles, how much ought he to receive for the carriage of 7 cwt. 34 grs., 50 miles? Ans. £1..16..9.

(2) If £100. in 12 months gain £51. interest, what principal will gain £33. in 9 months? Ans. £85..14..31 4.

(3) If 9 students spend £103. in 18 days, how much will

20 students spend in 30 days?

(4) Two persons earned 4ğs. for much would 5 persons earn in 10

Ans. £39..18..40. one day's labour: how days, at the same rate? Ans. £6..1..4.

(5) If £50. in 5 months gain £24. what time will £13. require to gain £1.? Ans. 9 months. (6) If the carriage of 60 cwt., 20 miles, cost £144. what weight can I have carried 30 miles for £5? Ans. 15 cwt.

DECIMAL FRACTIONS.

IN Decimal Fractions the unit is supposed to be divided into tenths, hundredths, thousandth parts, &c. consequently the denominator is always 10, or 100, or 1000, &c.

In our system of Notation, the figures of a whole number follow each other in a decimal (or tenfold) proportion. Hence, the numerator of a decimal Fraction is written as a whole number, only distinguished by a separating point prefixed to it. Thus 5 for 1o, 25 for 1, 123 for 1.

The denominator is, therefore, not expressed; being always understood to be 1, with as many ciphers affixed, as there are places in the numerator.

The different values of figures will be evident in the annexed Table.