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RULE.

Place the several terms of the multiplier under the corresponding ones of the multiplicand. Beginning at the right-hand, multiply the several terms of the multiplicand by the several terms of the multiplier successively, placing the right-hand term of each of the partial products under its multiplier; then add the partial products together, observing to carry one for every twelve, both in multiplying and adding. The sum of the partial products will be the answer.

Examples.

1. What is the product of 3f. 7′ 2′′ by 7ƒ. 6' 3"?

3f. 72"

7f. 6' 3"

10 9 6!!

lf. 9 7 0

25f. 2' 2"

Ans. 27f. 0 7 9 6

2. What is the product of 7f. 6' 4" by 2f. 3 5"?

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3. What is the product of 7f. 8' by 6f. 4' 3"?

Ans. 48f. 8' 7".

4. What is the product of 6f. 9' 7" by 4f. 2'?

Ans. 28f. 3′ 11′′ 2′′",

5. What is the area of a marble slab, whose length is 7ƒ.

3', and breadth 2f. 11'?

6. How many square feet are contained in the floor of a hall 37f. 3' long, by 10f. 7' wide?

7. How many square feet are contained in a garden 100f. 6' in length by 39f. 7' in width?

8. How many yards of carpeting, one yard in width, will it require to cover a room 16f. 5' by 13f. 7' ?

9. What will the plastering of a ceiling cost at 13cts. a square yard, its length being 30f. 7 inches, and the breadth 22f. 4 inches?

10. How many cubic feet are contained in a rectangular stone, 7f. 4' long, 2f. 11' wide, and 1f. 10' thick?

NEW METHOD OF MULTIPLYING DUODECI.

MALS.

84. Duodecimals may be multiplied together by a process similar to that employed for decimals, by the help of two new characters or figures.

For this purpose, we will represent 10 by the symbol x, and 11 by ..

Examples.

1. What is the product of 3f. 2' 3" by 2f. 1' 4"?

OPERATION.

3.23

2.14

10 90

32 3

646

Ans. 6.89 00 6ƒ. 8′ 9′′.

EXPLANATION.

In this example, we separated the feet from the parts of feet by a point, as in decimals, we then multiplied as in whole. numbers, observing to carry one for every 12, when multiply. ing as well as when adding.

The work of the same example, by the usual rule, is as follows:

3f. 2' 3"
2f. 1' 4"

1' 0" 9 0!!!

3' 2" 3""

6f. 4' 6"

Ans. 6f. 8' 9' 0"" - 0

2. What is the product of 38f. 3' 1" by 31f. 2' 2" ?

OPERATION.

32.31 27.22

64 62

646 2

1x397

6462

Ans. 834.x5 82=(8×144+3×12+4) f. 10′ 5′′ 8′′ 2′′'''

=1092ƒ. 10′ 5′′′ 8′′ 2′′,

EXPLANATION.

In this example, we converted 38f. and 31f., which are now in the decimal scale of notation, into the duodecimal scale of notation. In this way we found that 38-3×12+2; and 31=2×12+7, so that 38 and 31, when expressed duodeci. mally, become 32 and 27.

In the product the whole part, or feet, 834, is expressed

duodecimally; it is therefore equal to 8×144+3×12+4= 1192f., as given in the answer.

3. What is the product of 5f. 2' 3" 4"" by 2f. 1' 2" 5""?

5.234

OPERATION.

2.125

21 148

x4 68

523 4 x468

Ans. x.x95 x08—10f. 10′ 9′′ 5′′ 10 0 8

4. Multiply 47f. 3' 4" 3" by 80f. 2' 3"-5", and exhibit the work by both methods.

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Ans. 22311.442 1163=(2×1728+2×144+3×12+11) f. 4′ 4′′ 2′′" 11" 6' 3''''''=3791ƒ' 4′ 4′′ 2′′ 11′′ 6′′ 3′′ .

CHAPTER XIII.

ALLIGATION.

85. ALLIGATION teaches the method of finding the mean value of a mixture composed of several ingredients of different values. It is usually divided into two distinct parts, viz :— Alligation Medial, and Alligation Alternate.

ALLIGATION MEDIAL.

86. ALLIGATION MEDIAL, teaches the method of finding the mean value of a compound, when its several ingredients and their respective values are given.

Suppose a grocer mixes 140 pounds of tea, which is worth 8s. per pound; 200 pounds, worth 6s. per pound; and 160 pounds, worth 10s. per pound. What is a pound of the mixture worth?

140 pounds of tea, at 8s. per pound, is worth 1120s.; 200 pounds, at 6s., is worth 1200s.; 160 pounds, at 10s., is worth 1600s. Therefore, the mixture, which is 500 pounds, is worth 1120+1200+1600-3920s. Hence, one pound of the mixture must be worth 72s.

Hence, to find the mean value of a compound, composed of several ingredients of different values, we have this

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