4)18.27584.56885We divide 4 into 18 (units) and have 4 (units) quotient and 3 units remainder. Since the 4 is the unit’s figure of the quotient, we write the decimal point immediately after it. Then, the 2 units remainder and the 2 tenths of the dividend make 22 tenths to be divided by 4, and so on. Having reached the 4 (ten-thousandths) of the dividend, we find 8 (ten-thousandths) quotient and 2 remainder. This remainder is 20 hundred-thousandths, which when divided by 4 gives 5 (hundred-thousandths) and no further remainder.
4)18.27584.56885
We divide 4 into 18 (units) and have 4 (units) quotient and 3 units remainder. Since the 4 is the unit’s figure of the quotient, we write the decimal point immediately after it. Then, the 2 units remainder and the 2 tenths of the dividend make 22 tenths to be divided by 4, and so on. Having reached the 4 (ten-thousandths) of the dividend, we find 8 (ten-thousandths) quotient and 2 remainder. This remainder is 20 hundred-thousandths, which when divided by 4 gives 5 (hundred-thousandths) and no further remainder.
Example2: Divide 18.2758 by 11.
11)18.27581.66143636Here we find the digits 3, 6 repeated indefinitely in the quotient. Decimals of this sort will be fully consideredlater.
11)18.27581.66143636
Here we find the digits 3, 6 repeated indefinitely in the quotient. Decimals of this sort will be fully consideredlater.
Example3: Divide 354.43 by 184.
184)354.43(1.92625Ans.17044831150[16]460920Here we find the first figure of the quotient is obtained by dividing 184 into 354 units. Having now reached the decimal point in the dividend we also put the decimal point in the answer, and go on as before.
184)354.43(1.92625Ans.17044831150[16]460920
Here we find the first figure of the quotient is obtained by dividing 184 into 354 units. Having now reached the decimal point in the dividend we also put the decimal point in the answer, and go on as before.
[16]At this stage there is a remainder 115 hundredths. We bring down 0 from the dividend, and obtain 1150 thousandths, etc.
[16]At this stage there is a remainder 115 hundredths. We bring down 0 from the dividend, and obtain 1150 thousandths, etc.
(b) Division of a decimal.
Example4: Divide 10.6603 by 7.85.
Thus:
785)1066.03(1.358Ans.281045536280Here 7.85 is 785 hundredths, and 10.6603 is 1066.03 hundredths; so that the required quotient is obtained by dividing 1066.03 by 785.Therefore, to divide by a decimal, move the point as many places to the right as will make the divisor a whole number; move the point in the dividend the same number of places to the right. Then proceed as in Example 3.
785)1066.03(1.358Ans.281045536280
Here 7.85 is 785 hundredths, and 10.6603 is 1066.03 hundredths; so that the required quotient is obtained by dividing 1066.03 by 785.
Therefore, to divide by a decimal, move the point as many places to the right as will make the divisor a whole number; move the point in the dividend the same number of places to the right. Then proceed as in Example 3.
Example5: Divide 176.4 by .00012.
12)17640000Ans.1470000Here, to make the divisor a whole number, we have to move the point 5 places. Therefore we also move the point 5 places to the right in the dividend, first writing enough 0’s after the 176.4 to enable us to do so.
12)17640000Ans.1470000
Here, to make the divisor a whole number, we have to move the point 5 places. Therefore we also move the point 5 places to the right in the dividend, first writing enough 0’s after the 176.4 to enable us to do so.
To divide a decimal by 10, 100, etc.
Rule.—Remove the (.) as many places to the left as there are ciphers in the divisor.
Expression of decimal fractions as common fractions.
Example: Express 5.375 as a common fraction.
Example:.375 = 375 thousandths.
Therefore5.375 = 53751000= 538Ans.
Rule.—Take the digits of the decimal for numerator; for the denominator put down 1 followed by as many ciphers as there are digits in the decimal. Reduce this fraction to its lowest terms.
Expression of common fractions as decimals.
We have seen that a common fraction represents the quotient of the numerator divided by the denominator. Therefore, to convert a common fraction to a decimal fraction, we divide the numerator by the denominator.
Example:Express332as a decimal.
4)3.08).75.09375Ans.
It will be found in many cases that there is always a remainder, so that the quotient can be continued indefinitely.
The learner has already discovered that some common fractions cannot be changed to exact decimal fractions, as—
1⁄3=.33333 on to infinity.2⁄3=.66666 on to infinity.7⁄33=.212121, etc.
These decimals are known asCirculates,RecurringorCirculatingdecimals.
The part which recurs is called theRepetend.
This is marked by putting a dot over the first and last figures of it. For instance, if we write the 21 in the last case above, this way: 2̊1̊, it indicates that, if written out, the result would be 21212121, etc., on to infinity.
Where a circulating decimal occurs in work, it is best to reduce it to a common fraction. If need be, it may be expressed in the result as a circulate to any number of decimal places.
To change a pure circulate to a common fraction.
Rule.—Omit the (.) and write the figures of the repetend for the numerator, and as many 9’s for the denominator as there are places in the repetend.
Examples: Change the pure circulates .3̊, .2̊7̊, .1̊42857̊, to common fractions.
.3̊,(39=13)Ans.1⁄3.
.2̊7̊,(2799=311)Ans.3⁄11.
.1̊42857̊,(142857999999=17)Ans.1⁄7.
To change a mixed circulate to a common fraction.
Rule.—From the whole decimal subtract the finite part, and make the remainder the numerator. For the denominator, write as many 9’s as there are figures in the repetend, and annex as many 0’s as there are finite places.
Example: Change the mixed circulates .16̊ and .416̊ to common fractions.
16 - 1 = 15,1590=16.Ans.1⁄6.
416 - 41 = 375,375900=512.Ans.5⁄12.
To add, subtract, multiply and divide circulates, reduce them to common fractions, then apply the respective rules.
When one of the numbers is analiquot partof 100, the process of multiplication and division can often be very much shortened, as shown below.
Find cost of 27 yards of goods at 162⁄3c ($1⁄6) per yard. At $1 per yard, 27 yards cost $27; at $1⁄6, (27 ÷ 6), $41⁄2.Ans.$41⁄2.
Find cost of a bale of cotton, 528 pounds at 81⁄3c ($1⁄12) per pound. At $1 per pound, 528 pounds cost $528; at $1⁄12(528 ÷ 12) $44.Ans.$44.
Find cost of 1845 pounds of iron, at 31⁄3c ($1⁄30) per pound. Take1⁄30of 1845, since 31⁄3c is1⁄30of $1. (1845 ÷ 30 = 611⁄2).Ans.$611⁄2.
Find cost of 16 pounds of butter at 371⁄2c ($3⁄8) per pound. Here we take3⁄8of 16. Say1⁄8of 16 is 2, and3⁄8is (2 × 3) 6. Or say 3 times 16 is 48, and1⁄8of 48 is 6.Ans.$6.
Find cost of 171⁄2bushels of apples at 75c ($3⁄4) per bushel. The shortest way to find3⁄4of $17.50 is to diminish it by1⁄4of itself.
4)17.50at $13.371⁄2at $1⁄413.121⁄2at $3⁄4
Ans.$13.121⁄2.
At 61⁄4c per pound how much sugar will $5 buy? As 61⁄4c is1⁄16of $1, evidently each dollar will buy 16 pounds.Ans.80 pounds.
In multiplying by a fraction, write the quantity in a line with the numerator and cancel common factors.
Find cost of 72 yards of carpet, at 871⁄2c ($7⁄8) a yard. Cancel 8, also 72 and write 9 instead.Ans.$63.
78×729 = 63
Of 28 pounds of coffee, at 183⁄4c ($3⁄16) per pound. Cancel 28 and 16, write 7 and 4.Ans.$51⁄4.
316×4287=214or 51⁄4
At 662⁄3c ($2⁄3) per bushel, how many bushel of wheat will $34 buy?Ans.51 bushel.
32×3417 = 51
In division, invert terms of fraction.
How much syrup, at 412⁄3c ($5⁄12) per gallon can be bought for $15?Ans.36 gallons.
125×153 = 36
Table of Aliquot Parts of 100
This table embodies all the aliquot parts of 100 and their equivalent fractions which are generally used in practical calculations.
To find the value of articles sold by the unit, hundred or thousand.
Rule.—Multiply the quantity by the price, or vice versa, and point off the proper number of decimal places in the result.
Find the cost of a bale (518 pounds) of cotton at 73⁄8c per pound.
518×.07=36.26„×.003⁄8=1.941⁄4Ans.$38.201⁄4At 7c (.07) per pound, 518 pounds cost $36.26; at3⁄8c, $1.941⁄4. For3⁄8of 518, multiply by 3, and divide product by 8.
518×.07=36.26„×.003⁄8=1.941⁄4Ans.$38.201⁄4
At 7c (.07) per pound, 518 pounds cost $36.26; at3⁄8c, $1.941⁄4. For3⁄8of 518, multiply by 3, and divide product by 8.
Find cost of a lot of hogs, weighing 8740 pounds, at $4.35 per hundredweight.
87.404.35380.1900The price being $4.35 per 100 pounds and as in 8740 pounds there are 87.40 hundredweight, four decimal places are pointed off.Ans.$380.19.
87.404.35380.1900
The price being $4.35 per 100 pounds and as in 8740 pounds there are 87.40 hundredweight, four decimal places are pointed off.Ans.$380.19.
Find the cost of 2864 feet of lumber, at $171⁄4per 1000 feet.
Price being dollars per 1000, point off three places. (2.864 × 171⁄4= 49.404.)Ans.$49.40.
To find the value of articles sold by the ton (2000 pounds).
Rule.—Multiply the weight by the price and take half of the product.
Find the cost of 2680 pounds of hay, at $111⁄2per ton.
Point off three places, when price is dollars; five if dollars and cents. (2680 × 111⁄2= 30820; 30820 ÷ 2 = 15.410.)Ans.$15.41.
When the long ton of 2240 pounds is used.
Rule.—Multiply the weight by the price and divide the product by 2.240.
Find the cost of 4800 pounds coal, at $63⁄4per long ton. (4800 × 63⁄4) ÷ 2.24 = $14.46,Ans.
To find the cost of grain, when the price per bushel and weight is given.
Rule.—Reduce the weight to bushels, and multiply by the price.
Find the cost of 3570 pounds of shelled corn, at 36c per bushel.
56)3570(63.75bu..36Ans.$22.9500To reduce pounds of shelled corn to bushels, divide by 56. At 36c per bushel, 63.75 bushels come to $22.95.
56)3570(63.75bu..36Ans.$22.9500
To reduce pounds of shelled corn to bushels, divide by 56. At 36c per bushel, 63.75 bushels come to $22.95.
Find cost of 2900 pounds of wheat, at 57c per bushel.
To reduce pounds of wheat to bushels divide by 60. 2900 ÷ 60 = 481⁄3bushels; 481⁄3× .57 = $27.55,Ans.
In computing the value of grain, the operation can often be abbreviated by cancellation.
Rule.—Write the weight and price per bushel, on the right of a vertical line, and the number of pounds to the bushel on the left. Then cancel common factors, as explained above.
Find the cost of 3230 bushels of wheat, at 72c per bushel.
6032307212323 × 12=38.76Here we cancel the 0’s on both sides; then, 6 and 72, which leaves 323 and 12. Their product being the answer.
6032307212323 × 12=38.76
Here we cancel the 0’s on both sides; then, 6 and 72, which leaves 323 and 12. Their product being the answer.
At 28c per bushel, what will 4080 pounds of oats cost?
3240805104287Ans.$35.70Oats, 32 pounds to the bushel. Seetable,page 861. Cancel 32 and 4080, then, 4 and 28, leaving the factors 510 and 7.
3240805104287Ans.$35.70
Oats, 32 pounds to the bushel. Seetable,page 861. Cancel 32 and 4080, then, 4 and 28, leaving the factors 510 and 7.
Other short cuts for computing cost of merchandise, produce, etc.
Find cost of 261⁄2dozen eggs, at 181⁄2c a dozen.
26 × 18=4.681⁄2of 44=.221⁄2×1⁄2=1⁄44.901⁄4Ans.$4.90.When both fractions are1⁄2. To product of the whole numbers, add1⁄2of their sum, and annex1⁄4to answer.
26 × 18=4.681⁄2of 44=.221⁄2×1⁄2=1⁄44.901⁄4Ans.$4.90.
When both fractions are1⁄2. To product of the whole numbers, add1⁄2of their sum, and annex1⁄4to answer.
Of 533⁄4pounds of butter, at 283⁄4c per pound.
53 × .28=14.843⁄4of 81=.603⁄43⁄4×3⁄4=9⁄1615.45Ans.$15.459⁄16.To the product of the whole numbers, add3⁄4of their sum, plus the square of3⁄4.
53 × .28=14.843⁄4of 81=.603⁄43⁄4×3⁄4=9⁄1615.45Ans.$15.459⁄16.
To the product of the whole numbers, add3⁄4of their sum, plus the square of3⁄4.
Of 131⁄4yards of flannel, at 311⁄4c per yard.
13 × .31 = 4.03 + .11 = 4.14,Ans.
To 4.03 add .11,1⁄4of 44 (13 + 31). The1⁄16(1⁄4×1⁄4) is disregarded.
Simple denominate numbers.—When we speak of measures, whether they are of money, extension, time, or weight, we use terms like 5 dollars, 4 yards, 3 hours, or 10 pounds to express the quantity we are talking about.
Sometimes we use two or more terms or names to express the measure, as 3 hours, 15 minutes, 10 seconds; 4 gallons, 3 quarts, 1 pint.These are compound denominate numbers.
The chief differences between compound numbers and simple numbers is, that with the exceptions of United States money, and the metric system of weights and measures, the denominations of compound numbers do not increase or decrease by the scale of ten.
Reduction.—Reduction of Compound Numbers is the process of changing them from one denomination to another without altering their value.
Reduction Descendingis changing the denomination of a number to another that is lower, as: 2 hours = 120 minutes; 2 feet = 24 inches.
Reduction Ascendingis changing the denomination of a number to another that is higher, as: 120 minutes = 2 hours; 24 inches = 2 feet.
First.—Write the names of the different units to be used in addition, placing them in a horizontal row, the largest to the left.
Next.—Write the numbers of each unit to be added, below the names of the units, each in its proper place.
Then.—Add and place each sum below the column added.
Example: Add 7 hours 15 minutes 30 seconds, 9 hours 30 minutes 40 seconds, and 11 hours 40 minutes 32 seconds.
Work:
Explanation: 32 seconds + 40 seconds + 30 seconds = 102 seconds. But, 102 seconds = 1 minute 42 seconds. Write the 42 below and carry the 1 minute. 1 minute (carried) + 15 minutes + 30 minutes + 40 minutes = 86 minutes. But, 86 minutes = 1 hour 26 minutes. Write the 26 and carry the 1 hour. 1 hour + 11 hours + 9 hours + 7 hours = 28 hours. Result = 28 hours 26 minutes 42 seconds.
Example: Subtract 6 tons 12 cwt. 9 pounds 10 ounces from 15 tons 7 cwt. 13 pounds 9 ounces.
Work:
Explanation: (1) Place as in addition of denominate quantities. 10 ounces cannot be taken from 9 ounces, so we must take 1 pound from the 13 pounds and add it to the nine ounces. 16 ounces + 9 ounces = 25 ounces. 25 - 10 = 15. Write the 15 below.
(2) Now there are only 12 pounds left to take the 9 from. 12 - 9 = 3. Write the 3 below.
(3) 12 is larger than 7. 1 ton + 7 cwt. = 27 cwt. 27 - 12 = 15. Write the 15 below.
(4) 14 - 6 = 8. Write the 8 below.
(5) Result = 8 tons 15 cwt. 3 pounds 15 ounces.
Example: Multiply 21 yards 2 feet 11 inches by 6.
Work:
Explanation: (1) 6 × 11 inches = 66 inches = 5 feet 6 inches. Write the 6 below and carry the 5.
(2) 6 × 2 feet = 12 feet. 12 feet + 5 feet (carried) = 17 feet, or 5 yards 2 feet. Write the 2 below and carry the 5.
(3) 6 × 21 yards = 126 yards. 126 yards + 5 yards = 131 yards.
(4) Result = 131 yards 2 feet 6 inches.
Problem: Divide 3 years 9 months 4 days by 12.
Work
Explanation: (1) We cannot divide 3 by 12, so we reduce 3 years to months. 3 years = 36 months. 36 months + 9 months = 45 months. 45 ÷ 12 = 3, and a remainder 9. Write the 3 and carry the remainder 9.
(2) 9 months (carried) = 270 days. 270 days + 4 days = 274 days. 274 ÷ 12 = 22, and a remainder 10. Write the 22 and carry the 10.
(3) 10 days = 240 hours. 240 ÷ 12 = 20. Write the 20.
(4) Result = 3 months 22 days 20 hours.
Rules: 1. Divide the given denomination by the number which will reduce it to the next higher denomination. Divide the quotient in the same manner, and continue the operation until the entire quantity is reduced.
2. To the last quotient annex the several remainders in their proper order. The result will be the answer.
Example: Reduce 201458 inches to higher denominations.
201458 inches = 3 miles 57 rods 2 yards 1 foot 8 inches.
Rules: 1. Write the given quantity in the order of its denominations, beginning with the highest, and supply vacant denominations with ciphers.
2. Multiply the highest denomination by the number which will reduce it to the next lower denomination, and add to the product the units of the lower denomination, if there be any.
3. Proceed in the same manner until the entire quantity is reduced to the required denomination.
Example:Reduce 10 yards 8 feet 10 inches to inches.