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(2) Example: Divide 6.646250 by 10.634. Solution:

16725 ano. 1 0 6 34464646125

6 3 8 0 41

265 85
2 1 2 6 8

53 / 70
5 3 170

FIGURE 13.

g. If decimal points are involved in a division, as a rule the remainder is not indicated when the division does not come out even. In lieu thereof extra zeros are added to the dividend, and the division is continued until the quotient has as many figures as desired. (1) Example: Find 46.47/0.6. Solution:

77.45 ano.

6.146,470

'EXTRA ZERO

FIGURE 14.

(2) Example: Find 6.646250: 10.637.

Solution:

16248 2* ano.
1 0,63.7.16,644.25020
6 38 2'2

EXTRA
2 6 4 o 5 ZEROS
21 274

51310
4 2 5 4 8

8 7 6 20
85096
2 5240
2 1 274
3 966 REMAINDER

FIGURE 15.

3 h. Mixed numbers.-In e above it was stated that «,4647

=774+ 6

6 When the plus (+) sign is omitted, then 774%6 is called a mixed number. A mixed number is simply the sum of a whole number and a fraction written without the plus sign. To convert a mixed number to a pure fraction, multiply the whole number by the denominator of the fractional part and add the numerator. This is the new numerator; the denominator does not change.

Example: Change 78% to a pure fraction.
Solution:

78X5+4=394

394

ano. 5

78*

FIGURE 16.

i. Checks.-Any division may be checked by multiplying the divisor by the quotient and adding the remainder. The result is always the dividend.

Example: Check the answer to example in g (2) above.
Solution: .62482 X 10.637=6.64621034
6.64621034+remainder=6.64621034

.00003966
6.64625000

Check, j. Units.-As in multiplication, any two different quantities may be divided, even though the units are not the same. The quotient is expressed in a unit which is itself the quotient of the units of the dividend and the divisor. (1) Example: Divide 175 miles by 10 hours.

175 miles miles
Solution:
= 17.5

Answer. 10 hours hours In units of this type it is customary to write the denominator in the singular and to use the stroke (/) to separate the numerator from the denominator: 17.5 miles/hour, or 17.5 miles/hr. Although "miles/ hour” really means miles divided by hours it is usual to substitute the word "per" for "divided." Hence "miles/hour” is read as miles per hour, the standard abbreviation for which is mph. (2) Example: Divide 500 pounds by 50 square inches.

500 pounds Solution:

=10 pounds/square inch=10 lb/sq. in. 50 square inches

Answer. (10 pounds per square inch) k. Exercises.-Perform the indicated divisions and express the quotient as a mixed number.

(1) 894/16 (2) 755/24

314 Answer.

31
Solution: 24/755

72
35
24

11 (3) 1,025/314

638 Answer

(4) 215/72

Answer.

72 (5) 4,723/353

10,099 (6) 6,754,000/11,411

591

Answer.

11,411 (7) 9,001/30

2 (8) 11,415/45

253- Answer.

3 (9) 673/37

3 (10) 1,371/38

36

. 1. Exercises.- In the following exercises, express the quotient as a decimal. Round off the decimal part of the quotient to two places.

(1) 73.01/3.4 (2) .345/.36

.958
Solution: 36/34.500

32 4
2 10
1 80

300
288

Since only two decimal places are to be obtained, the quotient is rounded off to .96.

Answer. If the figure to be thrown away is greater than or equal to 5, increase the figure on the left by 1. If the figure is less than 5, do not change the preceding figure. Thus: .953=.95; 1.057=1.06; 1.053= 1.05, etc.

(3) 13.37/.834
(4) 14.705/8.64

1.70 Answer. (5) (157 miles)/(17.3 hours) (6) 1,942.4/.0035

554971.43 Answer. (7) 9.63/145.4 (8) (198 miles)/(59 minutes)

3.19 miles/min. Answer. (9) (5,280 feet)/60 seconds) (10) 19.437/38.6

.50 Answer.

8. Conversion of decimal fractions to common fractions.-a. A number which consists of a decimal point followed by a sequence of figures is called a decimal fraction. Thus, .33, .9899, .00467, and .00335 are all decimal fractions. Since 33 divided by 100 is .33, then 33 9,899

467 .33= Similarly, .9899=

and .00467

There100 10,000'

100,000 fore, to express any decimal fraction in fractional form, write the number without the decimal point and divided it by 1 followed by as many zeros as there are figures after the decimal point in the given number. (1) Example: Express .023678 in fractional form.

Solution:

[blocks in formation]

(2) Example: Express 4.0785 in fractional form. Solution: First express .0785 as a fraction:

785
.0785=

10,000
785

785
4.0785=4+

Answer.. 10,000=410,000

54 6. As stated in paragraph 7e, a fraction such as 54/16 or is really

16 just an indicated division. Very frequently in calculations it is much easier to carry a fraction along as a fraction than it is to “divide it out.” Later on, this problem will be considered in detail. At present, however, there is one very important rule of operation on fractions which should be mastered.

(1) This rule is that both the dividend (numerator) and divisor (denominator) of any fraction may be divided or multiplied by any number (except zero), without changing the value of the fraction. For example, if the numerator (54) and the denominator (16) of the 54

54 27 fraction

16
are both divided by 2, then according to this rule

16 8 (2) This rule allows zeros to be added to any given decimal number.

27 since .27

100'
for example, by the rule, both numerator and denomi-

270 nator can be multiplied by 10. Then

-.270 and because the

1,000 fractional value is unchanged, .27=.270.

C. A fraction is said to be in its lowest terms or simplest form if there is no number which will divide both the numerator and denominator evenly. The operation of finding the simplest form of a fraction is called reduction to lowest terms or simplification.

632 Example: Simplify

32 Solution: Divide numerator and denominator by 4;

632 158

32 8 This can be simplified still more by dividing by 2: 158 97

Answer. 8 4 d. Exercises. In the following exercises express the given decimal fractions in fractional form and then simplify. When possible, express the simplified fraction as a mixed number. (1) 1.875=?

15 (2) .9375= ?

Answer.

16 (3) 2.109375=?

1 (4) .125= ?

Answer. (5) .890625=?

53 (6) 3.828125=?

3 Answer.

64 (7). .625=?

3 (8) 4.375=?

4

Answer.

8 (9) 1.6875= ?

5 (10) 2.3125= ?

2- Answer.

16 e. Percentage.--Percent means a number with an understood de

50 nominator of 100. For example, 50 percent (%) means

or .50.

100 (1) To change from percent to a decimal, divide the number of percent by 100, which is equivalent to moving the decimal point two places to the left, and omit "percent." (a) Example: Change 42 percent to a decimal. Solution: 42 percent=42/100=.42

Answer. (b) Example: Change .9 percent to a decimal. Solution: .9 percent=,9/100=.009

Answer. (c) Example: Change -% to a decimal.

2 1

Answer. 2

:
Solution: 5%=.5%=.5/100=.005

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