Fractions for Firefighter Exam Study Guide (page 4)
Problems involving fractions may be straightforward calculation questions, or they may be word problems. Typically, they ask you to add, subtract, multiply, divide, or compare fractions.
Working with Fractions
A fraction is a part of something.
Example:Let's say that a pizza was cut into 8 equal slices and you ate 3 of them. The fraction tells you what part of the pizza you ate. The pizza below shows this: 3 of the 8 pieces (the ones you ate) are shaded.
Three Kinds of Fractions
|Proper fraction:||The top number is less than the bottom number:|
|The value of a proper fraction is less than 1.|
|Improper fraction:||The top number is greater than or equal to the bottom number:|
|The value of an improper fraction is 1 or more.|
|Mixed number:||A fraction is written to the right of a whole number:|
|The value of a mixed number is more than 1: It is the sum of the whole number plus the fraction.|
Changing Improper Fractions into Mixed or Whole Numbers
It's easier to add and subtract fractions that are mixed numbers rather than improper fractions. To change an improper fraction, say into a mixed number, follow these steps:
- Divide the bottom number (2) into the top number (13) to get the whole number portion (6) of the mixed number:
- Write the remainder of the division (1) over the old bottom number (2):
- Check: Change the mixed number back into an improper fraction (see the following steps).
Changing Mixed Numbers into Improper Fractions
It's easier to multiply and divide fractions when you are working with improper fractions rather than mixed numbers. To change a mixed number, say , into an improper fraction, follow these steps:
|1. Multiply the whole number (2) by the bottom number (4).||2 × 4 = 8|
|2. Add the result (8) to the top number (3).||8 + 3 = 11|
|3. Put the total (11) over the bottom number (4).|
|4. Check: Reverse the process by changing the improper fraction into a mixed number. If you get back the number you started with, your answer is right.|
Reducing a fraction means writing it in lowest terms, that is, with smaller numbers. For instance, 50¢ is of a dollar, or of a dollar. In fact, if you have a 50¢ piece in your pocket, you say that you have a half dollar. Reducing a fraction does not change its value.
Follow these steps to reduce a fraction:
- Find a whole number that divides evenly into both numbers that make up the fraction.
- Divide that number into the top of the fraction, and replace the top of the fraction with the quotient (the answer you got when you divided).
- Do the same thing to the bottom number.
- Repeat the first 3 steps until you can't find a number that divides evenly into both numbers of the fraction.
For example, let's reduce We could do it in 2 steps: then Or we could do it in a single step:
Shortcut:When the top and bottom numbers both end in zeros, cross out the same number of zeros in both numbers to begin the reducing process. For example, reduces to when you cross out 2 zeros in both numbers. Also, remember that any number that ends with an even number (0, 2, 4, 6, or 8) can be divided by 2. Any fraction in which both the top and bottom numbers end with even numbers can be reduced
Whenever you do arithmetic with fractions, reduce your answer. On a multiple-choice test, don't panic if your answer isn't listed. Try to reduce it and then compare it to the choices. Reduce the following fractions to lowest terms. The answers appear at the end of this section on page 162.
Raising Fractions to Higher Terms
Before you can add and subtract fractions, you have to know how to raise a fraction to higher terms. This is actually the opposite of reducing a fraction.
Follow these steps to raise to 24ths:
|1. Divide the old bottom number (3) into the new one (24):|
|2. Multiply the answer (8) by the old top number (2):||2 × 8 = 16|
|3. Put the answer (16) over the new bottom number (24):|
|4. Check: Reduce the new fraction to see if you get back the original one:|
Raise these fractions to higher terms:
If the fractions have the same bottom numbers, just add the top numbers together and write the total over the bottom number.
Examples: Reduce the sum: Change the sum to a mixed number: then reduce:
There are a few extra steps to add mixed numbers with the same bottom numbers, say
- 1. Add the fractions:
- 2. Change the improper fraction into a mixed number:
- 3. Add the whole numbers: 2 +1= 3
- 4. Add the results of steps 2 and 3:
Finding the Least Common Denominator
If the fractions you want to add don't have the same bottom number, you will have to raise some or all of the fractions to higher terms so that they all have the same bottom number, called the common denominator. All of the original bottom numbers divide evenly into the common denominator. If it is the smallest number that they all divide evenly into, it is called the least common denominator (LCD).
Here are a few tips for finding the LCD, the smallest number that all the bottom numbers evenly divide into:
- See if all the bottom numbers divide evenly into the biggest bottom number.
- Check out the multiplication table of the largest bottom number until you find a number that all the other bottom numbers evenly divide into.
- When all else fails, multiply all the bottom numbers together.
|1. Find the LCD. Multiply the bottom numbers:||3 × 5 = 15|
|2. Raise each fraction to 15ths:|
|3. Add as usual:|
Try these addition problems:
If the fractions have the same bottom numbers, just subtract the top numbers and write the difference over the bottom number.
If the fractions you want to subtract don't have the same bottom number, you will have to raise some or all of the fractions to higher terms so that they all have the same bottom number, or LCD. If you forgot how to find the LCD, just read the section on adding fractions with different bottom numbers.
|1. Raise each fraction to 12ths because 12 is the LCD, the smallest number|
|that 6 and 4 both divide into evenly:|
|2. Subtract as usual:|
Subtracting mixed numbers with the same bottom number is similar to adding mixed numbers.
- Subtract the fractions:
- Subtract the whole numbers: 4 – 1 = 3
- Add the results of steps 1 and 2:
Sometimes there is an extra borrowing step when you subtract mixed numbers with the same bottom numbers, say
- You can't subtract the fractions the way they are because is bigger than . So you borrow 1 from the 7, making it 6, and change that 1 to because 5 is the bottom number:
- Add the numbers from step 1:
- Now you have a different version of the original problem:
- Subtract the fractional parts of the two mixed numbers:
- Subtract the whole number parts of the two mixed numbers: 6 – 2 = 4
- Add the results of the last two steps together:
Try these subtraction problems:
- Officer Peterson drove miles to the police station. Then he drove miles to his first assignment. When he left there, he drove 2 miles to his next assignment. Then he drove miles back to the police station for a meeting. Finally, he drove miles home. How many miles did he travel in total?
- Before leaving the fire station, Firefighter Sorensen noted that the mileage gauge on Engine 2 registered miles. When she arrived at the scene of the fire, the mileage gauge registered miles. How many miles did she drive from the station to the fire scene?
Now let's put what you've learned about adding and subtracting fractions to work in some real-life problems.
Multiplying fractions is actually easier than adding them. All you do is multiply the top numbers and then multiply the bottom numbers.
Sometimes you can cancel before multiplying. Cancelling is a shortcut that makes the multiplication go faster because you are multiplying with smaller numbers. It's very similar to reducing: If there is a number that divides evenly into a top number and bottom number, do that division before multiplying. If you forget to cancel, you will still get the right answer, but you will have to reduce it.
|1. Cancel the 6 and the 9 by dividing 3 into both of them: 6 ÷ 3 = 2 and 9 ÷ 3 = 3. Cross out the 6 and the 9.|
|2. Cancel the 5 and the 20 by dividing 5 into both of them: 5 ÷ 5 = 1 and 20 ÷ 5 = 4. Cross out the 5 and the 20.|
|3. Multiply across the new top numbers and the new bottom numbers:|
Try these multiplication problems:
To multiply a fraction by a whole number, first rewrite the whole number as a fraction with a bottom number of 1:
Example: (Optional: Convert to a mixed number: )
To multiply with mixed numbers, it's easier to change them to improper fractions before multiplying.
- Convert to an improper fraction:
- Convert to an improper fraction:
- Cancel and multiply the fractions:
- Optional: Convert the improper fraction to a mixed number:
Now try these multiplication problems with mixed numbers and whole numbers:
Here are a few more real-life problems to test your skills:
- After driving of the 15 miles to work, Mr. Stone stopped to make a phone call. How many miles had he driven when he made his call?
- If Henry worked of a 40-hour week, how many hours did he work?
- Technician Chin makes $14.00 an hour. When she works more than 8 hours in a day, she gets overtime pay of times her regular hourly wage for the extra hours. How much did she earn for working 11 hours in one day?
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