FIRST: Express the percent as a decimal. THEN: Perform the indicated mathematical manipulations and reduce to lowest 52% = .52 terms.
NEXT: Divide the known number by the (800 × 7.2) 5760 decimal expression. R = 10,080 - 10,080 =→ 0 4:7 80 ÷ .52 = 153.8 ohms total resistance. What is the ratio in gallons?
FIRST: Express the ratio in fractional form.
RATIO 800 gal.
An important application of the common frac- R = 10,080 lb.
parison of one number to another number. Com- tion is that of ratio. A ratio represents the com- NEXT: Express both quantities in the same parison by the use of ratios has widespread appli- unit (gallons).
cation in the field of aviation. A ratio is used to R = 800 gal.
express the comparison of the volume of a cylinder 10,080 when the piston is at bottom center to the volume 7.2 of a cylinder when the piston is at top center.
This is referred to as the compression ratio. The THEN: Perform the indicated mathematical aspect ratio of an aircraft wing is a comparison manipulations and reduce to lowest of the wing span to the wing chord. The relation- terms.
ship of maximum speed, wing area, wing span, 800 800 loaded weight, and horsepower of different makes R = 10,080 - 1,400 = → or +7 and models of aircraft may be compared through the use of ratios. 7.2 A ratio is the quotient of one number divided by another number, expressed in like terms. A ratio, therefore, is the fractional part that one number is of another. A ratio may be expressed as a fraction, or it may be written using the colon (:) as the symbol for expressing ratio; thus the ratio ½ can be written 7:8.
2. If the cruising speed of an airplane is 200 knots and its maximum speed is 250 knots, what is the ratio of cruising speed to maximum speed?
FIRST: Express the cruising speed as the numerator of a fraction whose denominator is the maximum speed.
Finding the Ratio of Two Quantitites To find a ratio, the first term is divided by the second term. Both quantities of both terms must be expressed in the same units, and reduced to their lowest terms.
EXAMPLES
1. What is the weight ratio of a fuel load of 800 gallons to one of 10,080 pounds? Assume that the fuel weighs 7.2 pounds per gallon.
FIRST: Express the fuel load in gallons as the numerator of a fraction whose denominator is the fuel load in pounds.
R = 10,080 lb. 800 gal.
NEXT: Express both quantities in the same unit (pounds).
R = (800 × 7.2) 1b. 10,080 lb.
R = 250 200 NEXT: Reduce the resulting fraction to its lowest terms.
R = 250 200 THEN: Express the result as a ratio of one.
4 R = 3 or 8:1 (Read 8/10ths to one) Finding the Quantity of the First Term Now consider the situation when the ratio and the quantity that corresponds to the second term are given, and it is required to find the quantity that corresponds to the first term. To solve this type problem, multiply the term that corresponds to the second term by the fraction that represents 10
EXAMPLE The given ratio is 5/7 and the quantity that corresponds to the second term is 35. Find the quantity that corresponds to the first term.
FIRST: Express the problem as the product of the second term times the ratio.
35 × 5/7 = NEXT: Perform the indicated operation.
38 × 5/7 - 25 The first term is 25. The proof of this can be demonstrated by showing that the ratio of 25 to 35 is 5:7, reduced to lowest terms.
25/35 = 5/7 Finding the Quantity of the Second Term To solve a problem of this type, the ratio of the two quantities and the quantity that corresponds to the first term must be known. The solution is obtained by dividing the known number by the fraction that represents the ratio.
EXAMPLE The ratio of two quantities is 2/3; the quantity that corresponds to the first term is 100. Find the quantity that corresponds to the second term.
FIRST: Express the problem as the quotient of the first term divided by the ratio.
100 ÷ 2/3 = NEXT: Perform the indicated operation.
100 ÷ 2/3 = 50 100 × 3/2 = 150 The second term is 150. Again, this can be proved by expressing 100 as a ratio of 150.
100/150 = 2/3
PROPORTION A proportion is a statement of equality between two or more ratios. Thus, = g: or 3:4 = 6:8.
This is read 3 is to 4 as 6 is to 8. The first and last terms of the proportion are called the extremes.
The second and third terms are called the means.
In any proportion, the product of the extremes is equal to the product of the means. In the proportion 2:3 = 4:6 the product of the extremes, 2 X 6, is 12; the product of the means, 3 X 4, also is 12. An inspection of any proportion will show this to be true. This rule simplifies the solution of many practical problems.
An airplane flying a distance of 300 miles used 24 gallons of gasoline. How many gallons will it need to travel 750 miles?
300:750 = 24:x
(300) (x) = (750) (24) 300x = 18,000 × = 60 Sixty gallons of gasoline will be required to travel a distance of 750 miles.
POSITIVE AND NEGATIVE NUMBERS Positive and negative numbers are numbers that have directional value from a given starting point or from zero. Numbers above or to one side, usually right, of zero are designated as positive (+); those below or to the opposite side, usually left, of zero are designated as negative (-).
Figure 1-3 is representative of signed numbers on a horizontal scale.
-5 3-21 +1+2+3+4+5 FIGURE 1-3. A scale of signed numbers.
The sum of positive numbers is positive.
The sum of negative numbers is negative.
Addition To add a positive and a negative number, find the difference in their actual values and give this difference the sign (tor-) of the larger number.
EXAMPLE The weight of an aircraft is 2,000 pounds. A radio rack weighing 3 pounds and a transceiver weighing 10 pounds are removed from the aircraft.
What is the new weight? For weight and balance purposes, all weight removed from an aircraft is given a minus sign, and all weight added is given a plus sign.
FIRST: Add the values for the removed weights.
10+3=13 NEXT: Prefix the sign for removed weights.
- 13 FIRST: Change the sign of the number to be subtracted. +20 becomes - 20.
NEXT: Combine the two terms, following the rule for adding like signs.
(-6) + (-20) = -26 degrees.
Multiplication The product of two positive numbers is positive (+). The product of two negative numbers is positive (+). The product of a positive and a negative number is negative (-).
EXAMPLES 3X6=18 - 3X6= - 18 -3X -6=18 3X -6= -18 THEN: Add the sum of the removed weights to the total weight, following the rule for unlike signs.
+ 2000-13 = + 1987 pounds.
Division The quotient of two positive numbers is positive.
The quotient of two negative numbers is positive.
The quotient of a positive and negative number is negative.
EXAMPLES Subtraction To subtract positive and negative numbers, change the sign of the subtrahend (the number to be subtracted from another) and proceed as in addition.
EXAMPLE What is the temperature difference between a temperature reading of +20 at 5,000 feet and a reading of -6 at 25,000 feet? Follow the rule, "a change in temperature is equal to the first reading, subtracted from the second reading."
6÷3=2 -6÷3=-2 -6÷-3=2 6÷-3= -2
POWERS AND ROOTS Power When one number, the base, is used as a factor two or more times, the result is a power of the base. A positive integral exponent, written as a small number just to the right and slightly above the base number, indicates the number of times the base is used as a factor. Thus, 4 squared, or 12
4' means 4 X 4, which is 16. The 4 is the base, the 2 is the exponent, and the 16 is the power.
Roots A root of a number is one of two or more equal numbers that, when multiplied together, will produce the number. Such a number is called an equal factor. Thus, two equal factors that will produce 9 when multiplied together are 3 and 3.
Therefore, the square root of 9 equals 3. This may be written V9 = 3. The symbol V" is called a radical sign. Another method of indicating the square root of a number is to use a fractional exponent such as 912 = 3. If the root to be taken is other than a square root, it may be shown in a similar manner; that is, the cube root of 9 may be written 91. For example, the cube root of 8 equals 2 and may be written 18 = 2, or 81/3 = 2;
the fourth root of 256 equals 4 and may be written V256 = 4, or 256%4 = 4.
Computation of Square Root It is comparatively easy to determine the square root of such numbers as 4, 9, 16, and 144. The numbers are the perfect squares of small numbers.
Unfortunately, all numbers are not perfect squares; neither are they small. The square of a number is the product of that number multiplied by itself. Extracting the square root of a number is the reverse process of squaring a number, and is essentially a special division process. A description of this process follows and is presented in example form.
EXAMPLE Find the square root of 213.16 FIRST: Starting at the decimal point, and marking off in both directions from the decimal point, separate the number into periods of two figures each.
The last period at the left end need not have two figures; all others must have two figures. A zero may be added to the right end so that the period will have two figures.
V213.16 NEXT: Select the largest number that can be squared in the first period. Place the selected number above the radical sign, and place the square of this 13 number under the first period and subtract.
V213.16 THEN: Bring down the next pair.
(1) Multiply the root by 2 and place the product to the left of the remainder as the trial divisor.
(2) Determine the number of times the trial divisor will go into that portion of the remainder that is one digit more than the trial divisor. Write this number to the right of the digit in the trial divisor to form the final divisor and also to the right of the digit in the root.
(3) Multiply this number times the completed divisor. If the resulting product is larger than the remainder, reduce the number by one, both in the root and in the final divisor, and repeat the multiplication process.
(4) Subtract the product formed from the remainder and bring down the next pair to form a new remainder.
(5) To complete the solution of extracting the square root, simply repeat the procedure set forth in this step for each period of numbers remaining. It is unnecessary to carry the root beyond the number of digits possessed by the original number.
14. 6 V213.16 24 113 96 286 17 16 17 16
Two will divide into 11, 5 times.
However, 5 X 25 is greater than 113, so the 5 must be reduced to a 4.
The decimal is placed in the root so that the number of digits in the whole number portion of the root is equal to the sum of the periods, or pairs, in the whole number portion of the number from which the root was extracted.
(3) The value of a number does not change when it is both multiplied and divided by the same factor (5 X 10 ÷ 10 = 5).
Moving the decimal point of a number to the left is the same as dividing the number by 10 for each place the decimal point moves. Conversely, moving the decimal point to the right is the same as multiplying the number by 10 for each place the decimal point moves.
Powers of Ten
POWER OF
EXPANSION VALUE The difficulty of performing mathematical problems with very large (or very small) numbers and Positive Exponent 10° 10 × 10 × 10 × 10 × 10 x10 1,000,000 the counting and writing of many decimal places 105 10 × 10 × 10 × 10 × 10 100,000 10° 10 × 10 × 10 × 10 10,000 are both an annoyance and a source of error. 10° 10 × 20 × 10 1,000 The problems of representation and calculation 10ª 10 × 10 100 are simplified by the use of "powers of ten." (See 10' 10° 10 10 figure 1-1.). This system, sometimes referred to as The velocity of light, 30, 000, 000, 000 centimeters per second, simplifies to 3 × 10" centi- "Engineer's Shorthand, requires an understand« meters per second.
ing of the principles of the exponent. These are Negative Exponent summarized as follows: 10~ = 1a ł0=01 10°=10 10 × 10 ł00 = 00:
10 = 10° 10 × 10 × 10 1,000 = 0.001
(1) The positive exponent (or power) of a number is a shorthand method of indi- 10% = To 10 × 10 × 10 × 10 10.000 = 0.0001 cating how many times the number is 10→ = 10 10 × 10 × 10 x 10 × 10 100.000 = 0.00001 multiplied by itself. For example, 23 (read as 2-cubed or 2 to the third 10 ° = 10° 10 × 10 × 10 × 10 × 10 x 10 1,000,000 = 0,000001 power) means 2 is to be multiplied by The mass of an clectron, 0.000, 000, 000, 000, 000, 000, 000, 000, 000, 911 gram, become itself 3 times: 2 X 2 X 2 = 8. A number with a negative exponent may be defined as its inverse or reciprocal (1 divided by the number) with the same exponent made positive. For example, 2-3 (read as 2 to the minus 3 power) is the same as FIGURE 1-4. Powers of ten and their equivalents.
The procedure for use of powers of ten may be summarized as follows:
(2) 2x2X2-8
(2) Auy number, except zero, to the zero power is equal to 1. When a number is written without an exponent, the value of the exponent is 1. When an exponent has no sign (+ or -) preceding it, the exponent is positive.
(1). Move the decimal point to the place desired. Count the number of places the decimal point is moved.
(2) Multiply the altered number by 10 to a power equal to the number of places the decimal point was moved.
(3) The exponent of 10 is negative if the decimal point is moved to the right, and it is positive if the decimal point is moved to the left. An aid for remembering the sign to be used is: L, 14
A, R, D. When the decimal point moves Left, you Add; and when the decimal point moves Right, you Deduct.
In most instances, you will find it convenient to reduce the numbers used to numbers between 1 and 10 times 10 to the proper power. Unless otherwise specified, all answers to problems using powers of ten will conform to that requirement.
Powers of Ten Added and Subtracted Before using powers of ten in mathematical operations, it will be beneficial to review a few more principles governing exponents:
If two or more numbers are written with the same base, their product is equal to the base raised to a power equal to the algebraic sum of their exponents.
31 × 35×33 = 31+5+8 = 312 If two numbers are written with the same base, their quotient is equal to the base raised to a power equal to the algebraic difference of their exponents (numerator's exponent minus denominator's exponent).
= 45-3 = 42 A factor may be moved from numerator to denominator or from denominator to numerator by changing the sign of its exponent. Thus we have +==33X13=32 The bases must be the same before numbers can be multiplied or divided by the addition or subtraction of their exponents. Thus, d' X b' cannot be combined because the bases (a and b) are not the same.
Note particularly that the rules specify algebraic addition and algebraic subtraction of the powers. Here are some summarizing examples:
37 X 3-11 = 37+ (-11) = 37-11 = 3-4 = 15X15 = 8113 = 17 = 7 58 56 = 58(6) = 58+6 = 514.
612-6013 = 68=51 Multiplication and division employing powers of ten may be performed in three simple steps as follows:
(1) Reduce all numbers to values between 1 and 10 multiplied by 10 to the proper power.
(2) Perform the indicated operations.
(3) Change the result to a number between 1 to 10 multiplied by 10 multiplied by 10 to the proper power.
COMPUTING AREA Mensuration formulas deal with the dimensions, areas, and volumes of geometric figures. There are five geometric figures with which you should be familiar, and there is a separate formula for finding the area of each. The area of a plane figure is equal to the number of square units it contains. Areas are measured in different units as compared to measuring length. An area that is square and 1 inch on each side is called a square inch. All area units are square units, such as square inch, square foot, square yard, square rod, and square mile. Other area units are the square centimeter, the square meter, et cetera, found in the metric system of measurement.
TABLES OF AREAS 144 square inches (sq. in.) = 1 square foot (sq. ft.).
9 square feet = 1 square yard (sq. yd.).
30% square yards = 1 square rod (sq. rd.).
160 square rods = 1 acre (A).
640 acres = 1 square mile (sq. mile).
The technique for determining the area of any geometric shape is based upon the use of formulas.
To solve a problem by formula, it is necessary to-
(1) select the formula that covers the problem situation,
(2) insert the known values in the selected formula, and 15
AREA = LW S S AREA = S' S FIGURE 1-5. A rectangle.
(3) then make the necessary mathematical manipulations to find the unknown quantity.
The Rectangle A rectangle is a four-sided plane figure whose opposite sides are equal and all of whose angles are right angles (90°). The rectangle is a very familiar area in mechanics. It is the cross-sectional area oi many beams, rods, fittings, etc. (See figure 1-5.)
The area of a rectangle is the product of the measures of the length and width when they are expressed in the same units of linear measure.
The area may be expressed by the formula:
A = LW where: A = area.
L = length of rectangle.
W = width of rectangle.
EXAMPLE A certain aircraft panel is in the form of a rectangle having a length of 24 inches and a width of 12 inches. What is the area of the panel expressed in square inches?
FIRST: Determine the known values and substitute them in the formula.
A = LW A = 24 X 12 NEXT: Perform the indicated multiplication;
the answer will be the total area in square inches.
A = 24 X 12 = 288 sq. in.
FIGURE 1-6. A square.
The Square A square is a plane figure having four equal sides and four right angles (figure 1-6).
To determine the area of a square, find the product of the length of any two sides. Since a square is a figure whose sides are equal, the formula can be expressed as the square of the sides or:
A = S where A is the area and S is the length of a side.
• EXAMPLE What is the area of a square plate whose side measures 25 inches?
FIRST: Determine the known value and substitute it in the formula A = S A = 25% NEXT: Perform the indicated multiplication;
the answer will be the total area in square inches.
A = 25 X 25 = 625 sq. in.
Triangles A triangle is a three-sided polygon. There are three basic types of triangle: scalene, equilateral or equiangular, and isosceles. A scalene triangle is one in which all sides and angles are unequal, whereas the equilateral triangle, being just the opposite, has equal sides and equal angles. A triangle that has two equal sides and angles is known as an isosceles triangle.
Triangles may be further classified as to right, obtuse, or acute. These terms are descriptive of the included angles of the triangle. A right triangle is one that has one angle measuring 90°. In an obtuse triangle, one angle is greater than 90°, 16
SCALENE EQUILATERAL OR ISOSCELES Unequal Sides and EQUIANGULAR Angles Equal Sides and Angles Two Equal Sides and Angles
RIGHT RIGHT OBTUSE Right 4 - one L = 90° Obtuse A - one L > 90° Acute 4 - all 4 < 90°
ACUTE ACUTE FIGURE 1-7. Types of triangles.
while in an acute triangle all the angles are less than 90°. The various types of triangles are shown in figure 1-7.
The altitude of a triangle is the perpendicular line drawn from the vertex to the base. In some triangles, as in figure 1-8, it may be necessary to extend the base so that the altitude will meet it.
The base of a triangle is the side upon which 17
Circumference 2'6"
ALTITUDE Diameter d
- 3'2" BASE - FIGURE 1-8. Triangle.
the triangle is supposed to stand. Any side may be taken as the base.
The area of any triangle may be calculated by using the formula:
A = ab where A is equal to Area; § is a given constant;
a is the altitude of the triangle; and b is the base.
EXAMPLE Find the area of the triangle shown in figure 1-8.
FIRST: Substitute the known values in the area formula.
1 = jab = 1 = = × 26" × 32" NEXT: Solve the formula for the unknown value.
1140 1== × 30 × 38 = 1 = 570 sq. in..
Circumference and Area of a Circle To find the circumference (distance around) or the area of a circle it is necessary to use a number called pi (r). This number represents the ratio of the circumference to the diameter of any circle.
Pi cannot be found exactly because it is a neverending decimal, but expressed to four decimal places it is 3.1416, which is accurate enough for most computations. (See figure 1-9.)
Circumference The circumference of a circle may be found by using the formula:
FIGURE 1-9. A circle.
EXAMPLE The diameter of a certain piston is 5 inches.
What is the circumference of the piston?
FIRST: Substitute the known values in the formula, C = rd.
C = 3.1416 × 5 NEXT: Solve the formula for the unknown value.
C = 15.7080 inches.
Area The area of a circle, as in a rectangie or triangle, must be expressed in square units. The distance that is one-half the diameter of a circle is known as the radius. The area of any circle is found by squaring the radius and multiplying by ». The formula is expressed thus:
A = Tr?
where A is the area of a circle; 7 is the given constant; and r is the radius of the circle.
EXAMPLE The bore (inside diameter) of a certain aircraft engine cylinder is 5 inches. Find the cross sectional area of this bore.
FIRST: Substitute the known values in the formula, A = mr, 1 = 3.1416 X 2.52 NEXT: Solve the formula for the unknown value.
where C is the circumference; « is the given con- 4 = 3.1416 × 6.25 stant, 3.1416; and d is the diameter of the circle. 4 = 19.635 sq. in.
10' bz FIGURE 1-10. A trapezoid.
14" The Trapezoid FIGURE 1-11. Computing the area of a trapezoid.
A trapezoid (figure 1-10) is a quadrilateral having one pair of parallel sides. The area of a trapezoid is determined by using the formula:
1 = = (02 + 62) 12 where A is the area; ¿ is the given constant;
b and b are the lengths of the two parallel sides;
and h is the height.
EXAMPLE What is the area of a trapezoid whose bases are 14 inches and 10 inches, and whose altitude is 6 inches? (See fig. 1-11.)
FIRST: Substitute the known values in the formula.
#= (01+142) 2 1 = =(10 + 11)6.
NEXT: Solve the formula for the unknown value.
A = 2(21)% edge to trailing edge. If the wing is a tapered wing, the average width or chord, known as the mean chord, must be known in finding the area.
The formula for calculating wing area is:
A = SC where A is the area expressed in square feet, S is the wing span, and C is the average chord.
The process used in calculating wing area will depend upon the shape of the wing. In some instances it will be necessary to use the formula for finding wing area in conjunction with one of the formulas for the area of a quadrilateral or a
EXAMPLES
1. Find the area of the wing illustrated in To determine the area, it is necessary to decide what formula to use. It can be seen that the wing tips would form a 7-foot-diameter circle; the remainder of the wing planform is then in the shape of a rectangle. By combining the formulas for wing area and area of a circle, the area of a wing having circular tips can be calculated.
FIRST: Substitute the known value in the formula.
1 1 = 2X111 1 = 72 sq. in.
Wing Area To deseribe the planform of a wing (figure 1-12), several terms are required. Lo calculate wing area, it will be necessary to consider the meaning of the terms-span and chord. The wing span is the length of the wing from wing tip to wing tip.
The chord is the width of the wing from leading A = SC + TRE A = (25 - 7) (7) + (3.1416) (3.52).
The value for S is represented by the original wing span less the diameter of the circular tips.
NEXT: Solve the formula for the unknown value.
1 = (18 × 7) + (3.1416 × 12.25) 1 = 126 + 38.5 A = 161.5 sq. ft.
A = Wing Area, ft.* C = Average chord, ft.
S = Span, ft.
S - FIGURE 1-12. Wing planform.
2. Find the area of a tapered wing (figure 1-14) COMPUTING THE VOLUME OF SOLIDS whose structural span is 50 feet and whose mean Solids are objects with three dimensions-.
chord is 6'8". length, breadth, and thickness. They are of many FIRST: Substitute the known values in the shapes, the most common of which are prisms, formula. cylinders, pyramids, cones, and spheres. Occasionally, it is necessary to determine the volume A = SC of a rectangle, a cube, a cylinder, or a sphere.
A = 50' X 6'8". Since all volu les are not measured in the same NEXT: Solve the formula for the unknown value.
A = 50' × 6.67' A = 333.5 sq. ft.
3. Find the area of a trapezoidal wing (shown in figure 1-15) whose leading edge span measures 30 feet, whose trailing edge span measures 34 feet, and whose chord is 5 feet.
units, it is necessary to know all the common units of volume and how they are related to each other. For example, the mechanic may know the volume of a tank in cubic feet or cubic inches, but when the tank is full of gasoline, he will be interested in how many gallons it contains. The following table shows the relationship between some of the common units of volume.
UNITS OF SPACE MEASURE FIRST: Substitute the known values in the 1,728 cu. in. = 1 cu. ft.
formula. 27 cu. ft. = 1 cu. yd.
A = - (61 + b2) h 231 cu. in. = 1 gal.
7.5 gals. = 1 cu. ft.
1 = = (30 + 34)5. 2 pts. = 1 qt.
4 qts. = 1 gal.
NEXT: Solve the formula for the unknown value. Volume of a Rectangular Solid 1 A rectangular solid is a solid bounded by rec- 4 = 2(04)S tangles. In other words, it is a square-cornered 4 = 2(320) 4 = 160 sq. ft. 25' 6'8" 25' FIGURE 1-13. Wing with circular tips. FIGURE 1-14. Tapered wing with sweepback.
30 feet 5 feet 34 feet FIGURE 1-15. Trapezoid wing.
volume such as a box (figure 1-16). If the solid has equal dimensions, it is called a cube.
The formula for determining the volume of a rectangular solid may be expressed thus:
V = Iwh where: V = Volume.
1 = length.
w - width.
h = height.
EXAMPLE A rectangular-shaped baggage compartment measures 5 feet 6 inches in length, 3 feet 4 inches in width, and 2 feet 3 inches in height. How many cubic feet of baggage will it hold?
FIRST: Substitute the known values into the formula.
V = Loh V = 5'6" X 3'4" X 2'3".
NEXT: Solve the formula for the unknown value.
- S FIGURE 1-17. A cube.
If the rectangular solid is in the shape of a cube (figure 1-17), the formula can be expressed as the cube of the sides:
where Vis the volume and S is the side measurement of the cube.
Area and Volume of a Cylinder A solid having the shape of a can, length of pipe, or other such object is called a cylinder.
The ends of a cylinder are identical circles as shown in figure 1-18.
Surface Area The surface area of a cylinder is found by multiplying the circumference of the base by the altitude. The formula may be expressed as:
1 = Tah where A is the area, » is the given constant, d is the diameter, and h is the height of the cylinder.
V = A 105 = 41.25 cu. ft.
FIGURE 1-16. A rectangular solid. FIGURE 1-18. A cylinder.