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VIII. ABBREVIATED METHOD OP DIVISION OF DECIMALS, WHEN ONLY A CERTAIN NUMBER OF DECIMAL PLACES ARE REQUIRED IN THE QUOTIENT.

Rule-1. Proceed to find the figures of the quotient by the ordinary rule, till the number that remain to be found is less than the number of figures in the divisor. If this be so at first, find one figure of the quotient by the ordinary rule.

2. Instead of bringing down the next figure to the remainder, cut off the last figure of the divisor, and divide the remainder by the remaining figures to obtain the next figure of the quotient.

3. In forming the subtrahend, take care to carry the nearest ten from the product of the quotient-figure, and the figure cut off.

4. Cut off another figure of the divisor, and repeat the process, and so proceed till no figures of the divisor are left.

5. If the divisor at first contain more figures than the quotient, the excess may be rejected from the beginning, attention however being paid to (3).

EXAMPLE. Divide 2.71828180 by 3.1415927 to 8 places of decimals. 3.1,4,1,5,9,2,7)2.71828180(.86525596

2 51327416

20500764
18849556

1651208 1570796

80412
62832

17580
15708

1872
1571

301
283

18
19

Ans. .86525596.

IX. TO REDUCE A VULGAR FRACTION TO A DECIMAL.

Rule. Write the numerator as a decimal by annexing ciphers preceded by a decimal point, and divide by the denominator, by the preceding Rule. If the denominator be composed by the product of 2's and 5's only, the division will terminate, and the decimal is then a terminating, or nonrecurring decimal. If the denominator have any other prime factor than

! or 5, the division will not terminate, and must be carried on till the emainder, and therefore the quotients begin to recur.

Obs. A non-terminating decimal, if the figures recur, is called a reurring, or circulating decimal. A dot is placed over the first and last f the recurring figures.

EXAMPLES. 1. Express as decimal fractions /, 73, 516, ; 23.

7 7.000 (1)

=.875.
8 8

3 31 31.00
(2) 7-=-= = 7.75.

4 4

5 1 85 X 98 5 5.0 ( 5-2

=--= 2.5.
16 8 16 x 17 2 2
2. Express as decimal fractions 13, 14, 414.
12)11.0000

9)11.0000000
Ans. .916.
.9166

6) 1.2222222..
.2037037.

Ans. .2037. 814)417.00.51228501

407 0

(3)

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X.

TO CONVERT A RECURRING DECIMAL INTO A VULGAR FRACTION,

Rule. For the numerator take the whole decimal diminished by the jon-recurring part, and for denominator as many nines as there are re-urring figures, followed by as many ciphers as there are non-recurring figures.

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2. Convert .25515 into a vulgar fraction.

25515 — 25 25490 2549 .25515 =

99900 99900 9990 3. Convert 7.013 into a vulgar fraction.

13-1 12 1 7.013 = 7

= 7-- =7900 900 75

X.

RATIO, PROPORTION, AND VARIATION.

Def. 1. Ratio is the relation which exists between two magnitudes of the same kind, it being considered what multiple, part or parts, the 1st is of the 2nd.

A ratio is written thus :- 3:4 (= the ratio of 3 to 4.) Def. 2. The former of the two magnitudes composing a ratio is called the antecedent, the latter the consequent.

Def. 3. A ratio of greater inequality is one in which the antecedent is greater than the consequent. A ratio of less inequality is one in which the antecedent is less than the consequent.

Def. 4. If, by multiplying together the antecedents of two or more ratios for a new antecedent, and the consequents for a new consequent, a new ratio be formed, this ratio is said to be compounded of the former ratios. Def. 5. If two ratios are equal, the four magnitudes are proportional.

A proportion is written thus 3 : 4= 6:8 or 3 : 4 ::6:8. Def. 6. Magnitudes are said to be continually proportional, when the 1st: 2nd : : 2nd : 3rd :: 3rd : 4th : : &c.

Def. 7. The 1st and 4th terms of a proportion are called the extremes, the 2nd and 3rd the means.

Def. 8. The antecedents of the ratios are called homologous terms of a proportion, as also the consequents.

Def. 9. If by multiplying together the corresponding terms of two or more proportions, a new proportion be formed, this proportion is said to be compounded of the others.

Def. 10. One quantity is said to vary directly as another, when, if the former be changed, the latter is changed in the same ratio. Thus if A vary directly, or vary, as B, and A?, B?, be other corresponding values of A and B, A: A1=B:B1. This is expressed thus:-AO B.

Def. 11. One quantity is said to vary inversely as another, when if the former be changed, the reciprocal of the other is changed in the same ratio.

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B

B

Thus if A vary inversely as B, and A?, B1 be other corresponding values of A and B, A : A=! This is expressed thus :- A o 1.

B1 Def. 12. One quantity is said to vary as two others jointly when, if the first be changed, the product of the others is changed in the same ratio.

Def. 13. One quantity is said to vary directly as one quantity and inversely as another, when, if the first be changed, the product of the second and the reciprocal of the third is changed in the same ratio.

I. TO FIND THE RATIO OF ONE NUMBER TO ANOTHER.

Rule. Divide the 1st by the 2nd, the quotient, whether integral or fractional, will be the ratio required.

EXAMPLES. Find the ratio of 16 to 48, and of 21 to 1$.

16 1 (1) 16 : 48 =

48 3

1 1 e 15 7
(2) 2:1-=
2 3

8 8

II. TO SIMPLIFY THE TERMS OF A RATIO, WHEN FRACTIONAL. Rule. Multiply both terms by the L. C. M. of the denominators of the fractions.

EXAMPLE Simplify the terms of the ratio 914 : 104.

Here L. C. M.= 42.

3 5 129 215 ...9-:10—=

= 387 : 430. 14 21 14 21

Ill. TO COMPOUND TWO OR MORE RATIOS.

Rule. Multiply the antecedents together for a new antecedent, and the consequents for a new consequent; and divide the terms by their G.C.M.

EXAMPLE.
Find the ratio compounded of the ratios 8 : 15; 12 : 32.
Compound ratio = 8 X 12 : 15 X 32

=1:5.

IV. TO COMPARE TWO RATIOS.

Rule. Multiply the antecedent of the one by the consequent of the other, that ratio will be the greater, the product of whose antecedent with the other consequent, is the greater.

EXAMPLE Compare the ratios 6:7, and 4 : 5.

6:7 > or < 4:5 As 6 X 5 > or 4X7 As 30 >or28

.: 6:7 4:5.

V.

TO DIVIDE A NUMBER INTO PARTS WHICH SHALL BEAR TO

EACH OTHER GIVEN RATIOS.

Rule. Divide the number by the sum of the numbers composing the given ratios, and multiply the quotient by each of the numbers. The products will be the parts required.

EXAMPLE.
Divide 286 into 3 parts in the ratio of 1, 3, 9.

1+3 +9=13 : 286 · 13 = 22

22 X 3 = 66: 22 X 9= 198. .. The parts required are 22, 66, 198.

VI.

TO DETERMINE WHETHER FOUR NUMBERS BE PROPORTIONALS

IN A GIVEN ORDER.

Rule. Convert the first two and last two numbers into ratios, and compare them. If they be equal, the numbers are proportional; if unequal, they are not.

Or—Multiply the two extremes together and the two means, if these products are equal, the numbers are proportional; if unequal, they are not.

EXAMPLE.

Determine whether the numbers 3, 9, 6, 12, be proportionals. 3 1

6 1 3:9=-=

6:12 9 3

12 2
1

1
Now – is less than
3

2
..3:9 is less than 6 : 12
and the numbers are not proportionals.
Or thus: 3 X 12= 36 : 9 X 6 = 54

Now 36 is less than 54.
.. the numbers are not proportional.

VII. HAVING GIVEN THREE NUMBERS, TO FIND A FOURTH SUCH, THAT

ALL SHALL BE PROPORTIONAL.

Rule. Multiply the 2nd and 3rd together, and divide the product by the 1st, the quotient is the number reqnired.

EXAMPLES.
1. Find a fourth proportional to the numbers 8, 6, 16.

6 X 16 96
No. required = =-= 12.

8
8

E

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