Whole Numbers, Fractions, and Decimals

3-1 Mathematics in Aviation Maintenance Chapter 3

Introduction

Mathematics is woven into many areas of everyday life. Performing mathematical calculations with success requires an understanding of the correct methods, procedures, practice, and review of these principles. Mathematics may be thought of as a set of tools. The aviation mechanic needs these tools to successfully complete the maintenance, repair, installation, or certification of aircraft equipment. Many examples of using mathematical principles by the aviation mechanic are available. Tolerances in turbine engine components are critical, making it necessary to measure within a ten-thousandth of an inch. Because of these close tolerances, it is important that the aviation mechanic can make accurate measurements and mathematical calculations.

An aviation mechanic working on aircraft fuel systems also uses mathematical principles to calculate volumes and capacities of fuel tanks. The use of fractions and surface area calculations are required to perform sheet metal repair on aircraft structures.

Whole Numbers

Whole numbers are the numbers 0, 1, 2, 3, 4, 5, and so on. Whole numbers can be thought of as counting numbers. Addition of Whole Numbers Addition is the process where the value of one number is added to the value of another. The result is called the sum. When working with whole numbers, it is important to understand the principle of the place value. The place value in a whole number is the value of the position of each individual digit within the entire number. For example, in the number 512, the 5 is in the hundreds column, the 1 is in the tens column, and the 2 is in the ones column. Examples of place values of three whole numbers are shown in Figure 3-1.

When adding several whole numbers, such as 4,314, 122, 93,132, and 10, align them into columns according to place value and then add. 4,314 122 93,132 + 10 97,578 Therefore, 97,578 is the sum of the four whole numbers. Subtraction of Whole Numbers Subtraction is the process where the value of one number is taken from the value of another. The result is called the difference. When subtracting two whole numbers, such as 3,461 from 97,564, align them into columns according to place value and then subtract. 97,564 – 3,461 94,103 The difference of the two whole numbers is 94,103. Multiplication of Whole Numbers Multiplication is the process of repeated addition. For example, 4 × 3 is the same as 4 + 4 + 4. The result is called the product.

Example: How many hydraulic system filters do you have if there are 35 cartons in the supply room and each carton contains 18 filters? 18 × 35 90 54 630 Therefore, there are 630 filters in the supply room. Division of Whole Numbers Division is the process of finding how many times one number (called the divisor) is contained in another number (called the dividend). The result is the quotient, and any amount left over is called the remainder. quotient divisor dividend Example: 218 landing gear bolts need to be divided between 7 aircraft. How many bolts will each aircraft receive? 3-2 Place Value Ten Thousands Thousands Hundreds Tens Ones 3 5 2 6 9 1 2 7 4 9 35 shown as 269 shown as 12,749 shown as 31 7 218 − 21 8 − 7 1 In this case, there are 31 bolts for each of the seven aircraft with one extra remaining.

Fractions

A fraction is a number written in the form N⁄D where N is called the numerator and D is called the denominator. The fraction bar between the numerator and denominator shows that division is taking place. 17 , 2 , 5Some examples of fractions are: 18 3 8 The denominator of a fraction cannot be a zero. For example, the fraction 2⁄0 is not allowed, because dividing by zero is undefined. An improper fraction is a fraction in which the numerator is equal to or larger than the denominator. For example, 4⁄4 or 15⁄8 are examples of improper fractions. Finding the Least Common Denominator To add or subtract fractions, they must have a common denominator. In math, the least common denominator (LCD) is generally used. One way to find the LCD is to list the multiples of each denominator and then choose the smallest number that they all have in common (can be divided by).

Example: Add 1⁄5 + 1⁄10 by finding the LCD. Multiples of 5 are: 5, 10, 15, 20, 25, and so on. Multiples of 10 are: 10, 20, 30, 40, and so on. Notice that 10, 20, and 30 are in both lists, but 10 is the smallest or LCD. The advantage of finding the LCD is that the final answer should be in the simplest form. A common denominator can also be found for any group of fractions by multiplying all the denominators together. This number is not always the LCD, but it can still be used to add or subtract fractions. Example: Add 2⁄3 + 3⁄5 + 4⁄7 by finding a common denominator. A common denominator can be found by multiplying the denominators 3 × 5 × 7 to get 105.

( 2 + 3 + 4 )=( 70 + 63 + 60 )= 193 = 1 88 3 5 7 105 105 105 105 105 Addition of Fractions In order to add fractions, the denominators must be the same number. This is referred to as having “common denominators.” Example: Add 1⁄7 to 3⁄7 1 + 3 = 1 + 3 = 4 7 7 7 7 If the fractions do not have the same denominator, then one or all the denominators must be changed so that every fraction has a common denominator. Example: Find the total thickness of a panel made from 3⁄32-inch thick aluminum, that has a 1⁄64-inch thick paint coating. To add these fractions, determine a common denominator. The LCD for this example is 1, so only the first fraction must be changed since the denominator of the second fraction is already in 64ths.

( 3 + 1 )=( 3 × 2 + 1 )=( 6 + 1 )=( 6 + 1 )= 7 32 64 32 × 2 64 64 64 64 64 Therefore, 7⁄64 is the total thickness. Subtraction of Fractions To subtract fractions, they must have a common denominator. Example: Subtract 2⁄17 from 10⁄17 10 – 2 = 10 – 2 = 8 17 17 17 17 If the fractions do not have the same denominator, then one or all the denominators must be changed so that every fraction has a common denominator. 3-3 Example: The tolerance for rigging the aileron droop of an airplane is 7⁄8 inch ± 1⁄5 inch. What is the minimum droop to which the aileron can be rigged? To subtract these fractions, first change both to common denominators. The common denominator in this example is 40. Change both fractions to 1⁄40, as shown, then subtract.

( 7 – 1 )=( 7 × 5 – 1 × 8 )=( 35 – 8 )=( 35 – 8 ) = 27 8 5 8 × 5 5 × 8 40 40 40 40 Therefore, 27⁄40 is the minimum droop. Multiplication of Fractions Multiplication of fractions does not require a common denominator. To multiply fractions, first multiply the numerators. Then, multiply the denominators. Example: 3 × 7 × 1 = 3 × 7 × 1 = 21 5 8 2 5 × 8 × 2 80 The use of cancellation when multiplying fractions is a helpful technique. Cancellation divides out or cancels all common factors that exist between the numerators and denominators. When all common factors are cancelled before the multiplication, the final product is in the simplest form.

Example: 2 1 ( 14 × 3 ) = ( 14 × 3 ) = ( 2 × 1 ) = 2 15 7 15 7 5 × 1 5 5 1 Division of Fractions Division of fractions does not require a common denominator. To divide fractions, first change the division symbol to multiplication. Next, invert the second fraction. Then, multiply the fractions. Example: Divide 7⁄8 by 4⁄3 ( 7 ÷ 4 ) = ( 7 × 3 ) = ( 7 × 3 ) = 21 8 3 8 4 8 × 4 32 Example: In Figure 3-2, the center of the hole is in the center of the plate. Find the distance that the center of the hole is from the edges of the plate. To find the answer, the length and width of the plate should each be divided in half. First, change the mixed numbers to improper fractions: 5 7⁄16 inches = 87⁄16 inches 3 5⁄8 inches = 29⁄8 inches Then, divide each improper fraction by 2 to find the center of the plate.

87 ÷ 2 = 87 × 1 = 87 inches 16 1 16 2 32 29 ÷ 2 = 29 × 1 = 29 inches 8 1 8 2 16 Finally, convert each improper fraction to a mixed number: 87 = 87 ÷ 32 = 2 23 inches 32 32 29 = 29 ÷ 16 = 1 13 inches 16 16 Therefore, the distance to the center of the hole from each of the plate edges is 2 23⁄32 inches and 1 13⁄16 inches. Reducing Fractions A fraction needs to be reduced when it is not in the simplest form or “lowest terms.” Lowest term means that the numerator and denominator do not have any factors in common. That is, they cannot be divided by the same number (or factor). To reduce a fraction, determine what the common factor(s) are and divide these out of the numerator and denominator.

For example, when both the numerator and denominator are even numbers, they can both be divided by 2. Example: The total travel of a jackscrew is 13⁄16 inch. If the travel in one direction from the neutral position is 7⁄16 inch, what is the travel in the opposite direction? 13 – 7 = 13 – 7 = 6 16 16 16 16 The fraction 6⁄16 is not in lowest terms because the numerator (6) and the denominator (16) have a common factor of 2. To reduce 6⁄16, divide the numerator and the denominator by 2. The final reduced fraction is 3⁄8 as shown below. 6 = 6 ÷ 2 = 3 16 16 ÷ 2 8 Therefore, the travel in the opposite direction is 3⁄8 inch.

Mixed Numbers A mixed number is a combination of a whole number and a fraction. 3-4 3 5 8 5 7 16 Addition of Mixed Numbers To add mixed numbers, add the whole numbers together. Then add the fractions together by finding a common denominator. The final step is to add the sum of the whole numbers to the sum of the fractions for the final answer. Example: The cargo area behind the rear seat of a small airplane can handle solids that are 4 3⁄4 feet long. If the rear seats are removed, then 2 1⁄3 feet is added to the cargo area. What is the total length of the cargo area when the rear seats are removed?

4 3 + 2 1 = (4 + 2) +( 3 + 1 ) = 6 +( 9 + 4 ) = 6 13 = 4 3 4 3 12 12 12 7 1 feet of cargo room 12 Subtraction of Mixed Numbers To subtract mixed numbers, find a common denominator for the fractions. Subtract the fractions from each other. It may be necessary to borrow from the larger whole number when subtracting the fractions. Subtract the whole numbers from each other. The final step is to combine the final whole number with the final fraction. Example: What is the length of the grip of the bolt shown in Figure 3-3? The overall length of the bolt is 3 1⁄2 inches, the shank length is 3 1⁄8 inches, and the threaded portion is 15⁄16 inches long. To find the grip, subtract the length of the threaded portion from the length of the shank.

31⁄8 inches – 15⁄16 inches = grip length To subtract, start with the fractions. Borrowing is necessary because 5⁄16 is larger than 1⁄8 (or 2⁄16). From the whole number 3, borrow 1, which is actually 16⁄16. After borrowing, the first mixed number is now 2 18⁄16. This is because, 3 1⁄8 = 3 2⁄16 = 2 + 1 + 2⁄16 = 2 + 16⁄16 + 2⁄16 = 2 18⁄16. 3 1 – 1 5 = 3 2 – 1 5 = 2 18 – 1 5 = 113 8 16 16 16 16 16 16 Therefore, the grip length of the bolt is 1 13⁄16 inches. (Note: The value for the overall length of the bolt was given in the example, but it was not needed to solve the problem. This type of information is sometimes referred to as a “distracter,” because it distracts from the information needed to solve the problem.)

The Decimal Number System

Origin and Definition The number system that we use every day is called the decimal system. The prefix in the word decimal, dec, is a Latin root for the word “ten.” The decimal system probably originated from the fact that we have ten fingers (or digits). The decimal system has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. The decimal system is a base 10 system and has been in use for over 5,000 years. A decimal is a number with a decimal point. For example, 0.515, 0.10, and 462.625 are all decimal numbers. Like whole numbers, decimal numbers also have place value. The place values are based on powers of 10, as shown in Figure 3-4.

Addition of Decimal Numbers To add decimal numbers, they must first be arranged so that the decimal points are aligned vertically and according to place value. That is, adding tenths with tenths, ones with ones, hundreds with hundreds, and so forth. Example: Find the total resistance for the circuit diagram shown in Figure 3-5. The total resistance of a series circuit is equal to the sum of the individual resistances. To find the total resistance, RT, the individual resistances are added together. RT = 2.34 + 37.5 + 0.09 Arrange the resistance values in a vertical column so that the decimal points are aligned and then add.

2.34 37.5 + 0.09 39.93 Therefore, the total resistance, RT = 39.93 ohms. Subtraction of Decimal Numbers To subtract decimal numbers, they must first be arranged so that the decimal points are aligned vertically and according to place value. That is, subtracting tenths from tenths, ones 3-5 Grip Shank Overall length 37.5 Ohms 2.34 Ohms .09 Ohms M Place Value Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones Tenths Hundredths Thousandths Ten Thousandths 1 6 2 3 0 5 1 0 0 5 3 1 3 2 4 1,623,051 0.0531 32.4 from ones, hundreds from hundreds, and so forth. Example: A series circuit containing two resistors has a total resistance (RT) of 37.272 ohms. One of the resistors (R 1) has a value of 14.88 ohms. What is the value of the other resistor (R2)?

R2 = RT – R1 = 37.272 – 14.88 Arrange the decimal numbers in a vertical column so that the decimal points are aligned and then subtract. 37.272 –14.88 22.392 Therefore, the second resistor, R2 = 22.392 ohms. Multiplication of Decimal Numbers To multiply decimal numbers, vertical alignment of the decimal point is not required. Instead, align the numbers to the right in the same way that whole numbers are multiplied (with no regard to the decimal points or place values) and then multiply. The last step is to place the decimal point in the correct place in the answer. To do this, count the number of decimal places in each of the numbers, add the total, and then assign that number of decimal places to the result.

Example: To multiply 0.2 × 6.03, arrange the numbers vertically and align them to the right. Multiply the numbers, ignoring the decimal points for now. 6.03 × 0.2 1206 (ignore the decimal points, for now) After multiplying the numbers, count the total number of decimal places in both numbers. For this example, 6.03 has 2 decimal places and 0.2 has 1 decimal place. Together there are a total of 3 decimal places. The decimal point for the answer is placed 3 decimal places from the right. Therefore, the answer is 1.206. 6.03 2 decimal places × 0.2 1 decimal place 1.206 3 decimal places Example: Using the formula watts = amperes × voltage, what is the wattage of an electric drill that uses 9.45 amperes from a 120-volt source? Align the numbers to the right and multiply.

After multiplying the numbers, count the total number of decimal places in both numbers. For this example, 9.45 has 2 decimal places and 120 has no decimal place. Together there are 2 decimal places. The decimal point for the answer is placed 2 decimal places from the right. Therefore, the answer is 1,134.00 watts, or simplified to 1,134 watts. 9.45 2 decimal places × 120 no decimal place 000 1890 + 945 1,134.00 2 decimal places 3-6 Division of Decimal Numbers Division of decimal numbers is performed the same way as whole numbers, unless the divisor is a decimal. quotient divisor dividend When the divisor is a decimal, it must be changed to a whole number before dividing. To do this, move the decimal in the divisor to the right until there are no decimal places. At the same time, move the decimal point in the dividend to the right the same number of places. Then divide. The decimal in the quotient is placed directly above the decimal in the dividend.

Example: Divide 0.144 by 0.12 1.2 0.12 0.144 = 12. 14.4 – 12 24 – 24 0 Move the decimal in the divisor (0.12) two places to the right. The result is 12.0. Next, move the decimal in the dividend (0.144) two places to the right. The result is 14.4. Now divide. The result is 1.2. Example: The wing area of an airplane is 262.6 square feet and its span is 40.4 feet. Find the mean chord of its wing using the formula: area ÷ span = mean chord. 6.5 40.4 262.6 = 404. 2626.0 – 2424 2020 – 2020 0 Move the decimal in the divisor (40.4) one place to the right. Next, move the decimal in the dividend (262.6) one place to the right. Then divide. The mean chord length is 6.5 feet.

Rounding Off Decimal Numbers Occasionally, it is necessary to round off a decimal number to some value that is practical to use. For example, a measurement is calculated to be 29.4948 inches. To use this measurement, we can use the process of “rounding off.” A decimal is “rounded off” by keeping the digits for a certain number of places and discarding the rest. The degree of accuracy desired determines the number of digits to be retained. When the digit immediately to the right of the last retained digit is 5 or greater, round up by 1. When the digit immediately to the right of the last retained digit is less than 5, leave the last retained digit unchanged.

Example: An actuator shaft is 2.1938 inches in diameter. Round to the nearest tenth. The digit in the tenths column is a 1. The digit to the right of the 1 is a 9. Since 9 is greater than or equal to 5, “round up” the 1 to a 2. Therefore, 2.1938 rounded to the nearest tenth is 2.2. Example: The outside diameter of a bearing is 3.1648 centimeters. Round to the nearest hundredth. The digit in the hundredths column is a 6. The digit to the right of the 6 is a 4. Since 4 is less than 5, do not round up the 6. Therefore, 3.1648 to the nearest hundredth is 3.16. Example: The length of a bushing is 3.7487 feet. Round to the nearest thousandth.

The digit in the thousandths column is an 8. The digit to the right of the 8 is a 7. Since 7 is greater than or equal to 5, “round up” the 8 to a 9. Therefore, 3.7487 to the nearest thousandth is 3.749. Converting Decimal Numbers to Fractions To change a decimal number to a fraction, “read” the decimal out loud, and then write it into a fraction just as it is read as shown below. Example: One oversized rivet has a diameter of 0.52 inches. Convert 0.52 to a fraction. The decimal 0.52 is read as “fifty-two hundredths.” 0.52 = 52 “fifty-two” 100 “hundredths” In the above fraction of 52⁄100, we can divide 4 into each number resulting in a fraction of 13⁄25.

A dimension often appears in a maintenance manual or on a blueprint as a decimal instead of a fraction. To use the dimension, it may need to be converted to a fraction. An aviation mechanic frequently uses a steel rule that is calibrated in units of 1⁄64 of an inch. To change a decimal to the nearest equivalent common fraction, multiply the decimal by 64. The product of the decimal and 64 is the numerator of the fraction and 64 is the denominator. Reduce the fraction, if needed. 3-7 Example: The width of a hex head bolt is 0.3123 inches. Convert the decimal 0.3123 to a common fraction to decide which socket would be the best fit for the bolt head. First, multiply the 0.3123 decimal by 64: 0.3123 × 64 = 19.9872 Next, round the product to the nearest whole number: 19.98722 ≈ 20.

Use this whole number (20) as the numerator and 64 as the denominator: 20⁄64. Now, reduce 20⁄64 to 5⁄16 as 4 is common to both the numerator and denominator. Therefore, the correct socket would be the 5⁄16 inch socket (20⁄64 reduced). Example: When accurate holes of uniform diameter are required for aircraft structures, they are first drilled approximately 1⁄64 inch undersized and then reamed to the final desired diameter. What size drill bit should be selected for the undersized hole if the final hole is reamed to a diameter of 0.763 inches? First, multiply the 0.763 decimal by 64. 0.763 × 64 = 48.832 Next, round the product to the nearest whole number: 48.832 ≈ 49.

Use this number (49) as the numerator and 64 as the denominator: 49⁄64 is the closest fraction to the final reaming diameter of 0.763 inches. To determine the drill size for the initial undersized hole, subtract 1⁄64 inch from the finished hole size. 49 – 1 = 48 = 3 64 64 64 4 Therefore, a 3⁄4-inch drill bit should be used for the initial undersized holes. Converting Fractions to Decimals To convert any fraction to a decimal, simply divide the top number (numerator) by the bottom number (denominator). Every fraction has an approximate decimal equivalent. Example: 0.51 2 = 1 ÷ 2 = 2 1.0 Therefore, 1 2 = 0.5 – 1.0 0 0.375 3 8 = 3 ÷ 8 = 8 3.000 Therefore, 3 8 = 0.375 – 24 60 – 56 40 – 40 0 Calculator tip: numerator (top number) ÷ denominator (bottom number) = the decimal equivalent of the fraction.

Some fractions when converted to decimals produce a repeating decimal. Example: 0.33 1 3 = 1 ÷ 3 = 3 1.00 = 0.3 or 0.33 – 9 This decimal can be represented with a bar, or can be rounded. (A bar indicates that the number(s) beneath it are repeated to infinity.) 10 – 9 1 Other examples of repeating decimals: 0.212121… = 0.21 0.6666… = 0.7 or 0.67 0.254254… = 0.254 Decimal Equivalent Chart chart. Measurements starting at 1⁄64 inch and up to 3 inches have been converted to decimal numbers and to millimeters.

Ratio

A ratio is the comparison of two numbers or quantities. A ratio may be expressed in three ways: as a fraction, with a colon, or with the word “to.” For example, a gear ratio of 5:7 can be expressed as any of the following: 5⁄7 or 5:7 or 5 to 7 Aviation Applications Ratios have widespread application in the field of aviation. Example: Compression ratio on a reciprocating engine is the ratio of the volume of a cylinder with the piston at the bottom of its stroke to the volume of the cylinder with the piston at the top of its stroke. For example, a typical compression ratio might be 10:1 (or 10 to 1).

Aspect ratio is the ratio of the length (or span) of an airfoil to