Whole numbers are the numbers 0, 1, 2, 3, 4, 5, and so on.
Addition of Whole Numbers
Addition is the process in which 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 the digit within the 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. The place values of three whole numbers are shown in Figure 1-1.
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.
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 are in the supply room if there are 35 cartons and each carton contains 18 filters?
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.
Example: 218 landing gear bolts need to be divided between 7 aircraft. How many bolts will each aircraft receive?
The solution is 31 bolts per aircraft with a remainder of 1 bolt left over.
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.
Some examples of fractions are: 17/18, 2/3, 5/8
The denominator of a fraction cannot be a zero. For example, the fraction 2⁄0 is not allowed. 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 commonly used. One way to find the LCD is to list the multiples of each denominator and then choose the smallest one that they have in common.
Example: Add 1/5 + 1/10 by finding the least common denominator.
Multiples of 5 are: 5, 10, 15, 20, 25, and on. Multiples of 10 are: 10, 20, 30, 40, and on. Notice that 10, 20, and 30 are in both lists, but 10 is the smallest or least common denominator (LCD). The advantage of finding the LCD is that the final answer is more likely to be in lowest terms.
A common denominator can also be found for any group of fractions by multiplying all of the denominators together. This number will not always be 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.
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
If the fractions do not have the same denominator, then one or all of 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, which has a paint coating that is 1⁄64-inch thick. To add these fractions, determine a common denominator. The least common denominator for this example is 1, so only the first fraction must be changed since the denominator of the second fraction is already 64.
Subtraction of Fractions
In order to subtract fractions, they must have a common denominator.
Example: Subtract 2/17 from 10/17
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.
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.
The use of cancellation when multiplying fractions is a helpful technique which 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 will be in lowest terms.
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
Example: In Figure 1-2, 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:
57/16 inches = 87/16 inches
35/8 inches = 29/8 inches
Then, divide each improper fraction by 2 to find the center of the plate.
Finally, convert each improper fraction to a mixed number:
Therefore, the distance to the center of the hole from each of the plate edges is 2 23/32 inches and 113/16 inches.
A fraction needs to be reduced when it is not in “lowest terms.” Lowest terms 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?
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.
Therefore, the travel in the opposite direction is 3/8 inch.