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error, the uncertainty may be indicated by the double sign, #, read, plus, or minus, and placed after the product.

4. When the number of decimal places in the multiplicand is less than the number to be retained in the product, supply the deficiency by annexing ciphers.

EXAMPLES FOR PRACTICE.' 1. Multiply 236.45 by 32.46357, retaining 2 decimal places, and 2.563789 by .0347263, retaining 6 decimal places in the product.

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7676.02 + 2. Multiply 36.275 by 4.3678, retaining 1 decimal place in the product.

Ans. 158.4 +. 3. Multiply .24367 by 36.75, retaining 2 decimal places in the product.

4. Multiply 4256.785 by .00564, rejecting all beyond the third decimal place in the product.

Ans. 24.008 +. 5. Multiply 357.84327 by 1.007806, retaining 4 decimal places in the product.

6. Multiply 400.756 by 1.367583, retaining 2 decimal places in the product.

Ans. 548.07 +. 7. Multiply 432.5672 by 1.0666666, retaining 3 decimal places in the product.

8. Multiply 48.4367 by 27, extending the product to three decimal places.

Ans. 103,418 +. 9. Multiply 715z by 3339, extending the product to three decimal places.

10. The first satellite of Uranus moves in its orbit 142.8373 +

degrees in 1 day; find how many degrees it will move in 2.52035 days, carrying the answer to two decimal places.

Ans. 360.00 degrees. 11. A gallon of distilled water weighs 8.33888 pounds; how many pounds in 35.8756 gallons ? Ans. 299.16 + pounds.

12. One French metre is equal to 1.09356959 English yards; how many yards in 478.7862 metres. Ans. 523.58 #yards.

13. The polar radius of the earth is 6356078.96 metres, and the equatorial radius, 6377397.6 metres; find the two radii, and their difference, to the nearest hundredth of a mile, 1 metre being equal to 0.000621346 of a mile.

DIVISION.

223. In division of decimals the location of the decimal point in the quotient depends upon the following principles :

I. If one decimal number in the fractional form be divided by another also in the fractional form, the denominator of the quotient. must contain as many ciphers as the number of ciphers in the denominator of the dividend exceeds the number in the denominator of the divisor. Therefore,

II. The quotient of one number divided by another in the decimal form must contain as many decimal places as the number of decimal places in the dividend exceed the number in the divisor. 1. Divide 31.368 by 5.37.

ANALYSIS. We first divide as OPERATION.

in whole numbers; then, since the 5.37 ) 34.368 ( 6.4 32 22

dividend has 3 decimal places and the divisor 2, we point off 3

2 2 148

=1 decimal place in the quotient, 2 148

(II). The correctness of the work is shown in the proof, where the dividend and divisor are written as

common fractions. For, when we Х

PROOF.

have canceled the denominator of

the divisor from the denominator of the dividend, the denominator of the quotient must contain as

34368 1000

100 537

if = 6.4

many ciphers as the number in the dividend exceeds those in the divisor. 224. Hence the following

Divide as in whole numbers, and from the right hand of the quotient point off as many places for decimals as the decimal places in the dividend exceed those in the divisor,

Notes.—1. If the number of figures in the quotient be less than the excess of the decimal places in the dividend over those in the divisor, the deficiency must be supplied by prefixing cipbers.

2. If there be a remainder after dividing the dividend, annex ciphers, and continue the division: the ciphers annexed are decimals of the dividend.

3. The dividend should always contain at least as many decimal places as the divisor, before commencing the division ; the quotient figures will then be integers till all the decimals of the dividend have been used in the partial dividends.

4. To divide a decimal by 10, 100, 1000, etc., remove the point as many places to the left as there are ciphers on the right of the divisor.

EXAMPLES FOR PRACTICE; 1. Divide 9.6188 by 3.46.

Ans. 2.78. 2. Divide 46.1975 by 54.35.

Ans. .85. 3. Divide .014274 by .061.

Ans. .234. 4. Divide .952 by 4.76. 5. Divide 345.15 by .075.

Ans. 4602. 6. Divide by 476.3.

Ans. .001679+. 7. Divide .0026 by .003. 8. Divide 3.6 by .00006,

Ans. 60000. 9. Divide 3 by 450. 10. Divide 75 by 10000. 11. Divide 4.36 by 100000. 12. Divide .1 by .12. 13. Divide 645.5 by 1000.

14. If 25 men build 154.125 rods of fence in a day, how much does each man build ?

15. How many coats can be made from 16.2 yards of cloth, allowing 2.7 yards for each coat ?

16. If a man travel 36.34 miles a day, how long will it take him to travel 674 miles ?

Ans. 18.547+ days. 17. How many revolutions will a wheel 14.25 feet in circumference make in going a distance of 1 mile or 5280 feet?

CONTRACTED DIVISION 225. To obtain a given number of decimal places in the quotient.

In division, the products of the divisor by the several quotient figures may be contracted, as in multiplication, by rejecting at each step the unnecessary figures of the divisor, (220).

1. Divide 790.755197 by 32.4687, extending the quotient to two decimal places.

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SECOND CONTRACTED METHOD.

ANALYSIS. In the first method 32.4687790.755197

of contraction, we first compare the 53.42 141 3

3 tens of the divisor with the 79 114

tens of the dividend, and ascertain

that there will be 2 integral places 17

in the quotient; and as 2 decimal 1

places are required, the quotient

must contain 4 places in all. Then assuming the four left hand figures of the divisor, we say 3246 is contained in 7907, 2 times; multiplying the assumed part of the divisor by 2. and carrying 2 units from the rejected part, as in Contracted. Multiplication of Decimals, we have 6494 for the product, which subtracted from the dividend, leaves 1413 for a new dividend. Now, since the next quotient figure will be of an order next below the former, we reject one more place in the divisor, and divide by 324, obtaining 4 for a quotient, 1299 for a product, and 114 for a new dividend. Continuing this process till all the figures of the divisor are

rejected, we have, after pointing off 2 decimals as required, 24.35 for a quotient. Comparing the contracted with the common method, we see the extent of the abbreviation, and the agreement of the corresponding intermediate results.

In the second method of contraction, the quotient is written with its first figure under the lowest order of the assumed divisor, and the other figures at the left in the reverse order. By this arrangement, the several products are conveniently formed, by multiplying each quotient figure by the figures above and to the left of it in the divisor, by the rule for contracted multiplication, (222), and the remainders only are written as in (112).

226. From these illustrations we derive the following

RULE. I. Compare the highest or left hand figure of the divisor with the units of like order in the dividend, and determine how many figures will be required in the quotient.

II. For the first contracted divisor, take as many significant figures from the left of the given divisor, as there are places required in the quotient; and at each subsequent division reject one place from the right of the last preceding divisor.

III. In multiplying by the several quotient figures, carry from the rejected figures of the divisor as in contracted multiplication.

NOTES. — 1. Supply ciphers, at the right of either divisor or dividend, when necessary, before commencing the work.

2. If the first figure of the quotient is written under the lowest assumed figure of the divisor, and the other figures at the left in the inverted order, the several products will be formed with the greatest convenience, by simply multiplying each quotient figure by the figures above and to the left of it in the divisor.

EXAMPLES FOR PRACTICE.

1. Divide 27.3782 by 4.3267, extending the quotient to 3 decimal places.

Ans. 6.328 +. 2. Divide 487.24 by 1.003675, extending the quotient to 2 decimal places.

3. Divide 8.47326 by 75.43, extending the quotient to 5 decimal places.

4. Divide .8487564 by .075637, extending the quotient to 3 decimal places.

Ans. 11.221 E.

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