(8) What is the value of A in the following proportions:

The smaller one runs at
What is the ratio of

a speed of 750 rpm. and the larger at 200 rpm. their speeds?

(10) An airplane travels 400 miles in 2 hours. Set up a proportion and determine how far the airplane will travel in 14 hours.

2,800 miles.

Answer. (11) If o inch on a map represents 49 miles, how many miles are represented by 3 inches on the map?

(12) If a boat drifts down stream 40 miles in 12 hours, how far will it drift in 15 hours? 50 miles. Answer.

(13) On June 12, 1939, a pilot flew a glider plane across Lake Michigan a total distance of 92 miles in 52 minutes. He cut loose from the tow plane at 13,000 feet and descended only 5,000 feet in crossing. At the same rate of descent, how much farther could he have glided? How many more minutes would he have been in the air?

(14) A roadbed rises 3% feet in a horizontal distance of 300 feet. How many feet will the roadbed rise in 720 feet? 8 ft. Answer. (15) If 16 gallons of gas will drive a car 288 miles, at the same rate of using gas how many gallons will it take to drive the same car from Chicago to Memphis, a distance of 564 miles?

h. Conversion exercises. Obtain conversion factors required in the following examples from the appendix.

(1) Change 210 miles per hour to knots.

(2) How many feet per second are 32 miles per hour?

(5) A tank containing 125 U. S. gallons of gas would contain how many British gallons?

(6) How many U. S. gallons are there in 78.5 British gallons? 94.2 U. S. gal.

Answer.

14. Positive and negative numbers.-There are many quantities which by their contrary or opposite nature are best described as negative quantities in contrast to positive quantities. For example, temperatures above 0° Fahrenheit are considered as positive, whereas those below 0° are considered as negative. As a consequence it becomes necessary to consider negative and positive numbers and how to deal with them.

a. A negative number is indicated by prefixing a minus sign (—) in front of the number. Thus -5, -7.04, -90.003 are all negative numbers.

100°C

POSITIVE
READINGS

0°c

NEGATIVE
READINGS

FIGURE 21.

b. A positive number is indicated by prefixing it with a plus sign (+), if necessary. When there is no possible ambiguity the plus sign is usually omitted. Thus +5, 7.04, +63.0, 98.4 are all positive numbers.

c. It is convenient to imagine the numbers as representing distances along a straight line as follows:

POSITIVE

-4 -3 -2 -1

NEGATIVE

FIGURE 22.

+2

+3 +4

Negative distances are measured to the left and positive distances are measured to the right.

d. The signs and now have additional meanings. They not only indicate addition and subtraction, but positive and negative numbers as well. To distinguish the sign of operation from the sign of quality (positive or negative), the quality sign is inclosed in parentheses: 25+(+5), 25−(+5), 25+(-5), or 25-(-5). For the sake of brevity, the first and second are generally written simply as 25+5, and 25-5.

15. Addition of positive and negative numbers. To add two numbers which have the same signs, add the numbers and prefix the common (or same) sign. If the numbers to be added have unlike signs, find the difference and use the sign of the larger number. a. Example: (+6)+(-3)=?

Solution: Since the signs are different, subtract the numbers to obtain a remainder of 3. Since the sign of the larger number is positive the sign of the remainder is also positive: 6+(-3)=3 Answer. b. Example: (−3)+(+2)=?

Solution: Referring to figure 22, begin at -3 and count 2 units in a positive direction. The result is 1 space to the left of zero. Therefore (-3)+(+2)=−1.

This problem may also be done by using the rule stated in the preceding paragraph. Since the signs are unlike, subtract 2 from 3 and prefix the remainder by a minus sign since the larger number is negative. (-3)+(+2)=-1 Answer.

16. Subtraction of positive and negative numbers.-a. To subtract two numbers (positive or negative), change the sign of the number being subtracted and add the numbers as in addition (par. 15). Example: (-3)— (—4)=?

Solution: Changing the sign of the number being subtracted, the problem then becomes

(11) 18+.4=?

(12) 20-17.4+9=+11.6

(13) 37.3+19.4+17.8=?

(14) (-175.03)+19-156.03

Answer.

Answer.

17. Multiplication and division of positive and negative numbers.-a. If the two numbers to be multiplied have the same signs, then the product is positive. If the two numbers to be multiplied have opposite signs, then the product is negative.

(1) Example: Multiply (+3.04) by (17.8).

Solution: Since the signs are the same, the product is

(2) Example: (+.00395)X(−345.9)=?

+54.112 Answer.

Solution: Since the signs are unlike, the product is negative, or -1.366305 Answer.

b. Exercises. Find the product in each of the following exercises. (1) (−1.6) (.9) *

c. In division the quotient is positive if the divisor and dividend have the same sign; if the divisor and dividend have opposite signs, the quotient is negative.

Example: Divide (—15.625) by (12.5).

Solution: Since the dividend and the divisor have opposite signs, the quotient is negative.

d. Exercises. Find the quotient in each of the following exercises:

(8) 390.59 -28.1 (9)-72.9

-13.9 Answer.

-0.81 (10) 0.6118

87.4

0.007 Answer.

18. Square root. The process of extracting square root finds many uses in solving problems. Following are the rules:

a. Begin at the decimal point and point off as many periods of two digits each as possible. Point off to the left if the number is an integer, to the right if it is a decimal. Point off to both the left and the right if there are digits both left and right of the decimal point. If the last period of the decimal has but one digit, add a cipher to complete the period.

b. Find the largest integer the square of which is equal to or less than the left hand group, and write this integer for the first digit of the root and directly over the first group of digits.

c. Square the first digit of the root; subtract its square from the first group and bring down the second group.

d. Obtain a trial divisor by doubling the partial root already found, divide it into the remainder (omitting from the latter the right hand digit), and write the integral part of the quotient as the next digit of the root and directly over the group of digits used in determining it.

e. Annex the root digit just found to the trial divisor to make the complete divisor; multiply the complete divisor by this root digit, subtract the result from the dividend; bring down the next group for a new dividend.

f. Obtain a new trial divisor by doubling the part of the root already found, and proceed as before until the desired number of digits of the root have been found.

g. After extracting the square root of a number involving decimals, point off one decimal place in the root for every decimal group in the number.

h. If the root is exact, square it. The result should be the original number. If the root is inexact, square it and add to this result the remainder.