Fractions Study Guide

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Fractions

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Table of contents

What is a fraction? .................................................................................................3 Mixed Fractions ....................................................................................................3 Cancelling down....................................................................................................4 Adding Fractions ...................................................................................................5 LCD ? Lowest Common Denominator ........................................................................6 Subtracting Fractions .............................................................................................7 Multiplying Fractions .............................................................................................7 Dividing Fractions .................................................................................................8

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What is a fraction?

Fractions are another way to display division. If the number above the fraction line (the numerator) is smaller than the number below the line (the denominator), it means that the total value of the fraction is between 0 and 1. These are called proper fractions.

Examples:

1 5 13 125 2 , 6 , 20 , 317 < 1

If the numerator and the denominator are equal, the value is exactly 1. Examples:

1 6 15 201 1 , 6 , 15 , 201 = 1

If the numerator is greater than the denominator, the total value is greater than 1. These are called improper fractions.

Examples:

4 7 11 34 3 , 5 , 8 , 12 > 1

Mixed Fractions

A Mixed fraction is a fraction which is composed of a whole number and a fraction. For

example: 7 is one and one sixth, or 1 1 and 9 is two and a quarter, or 2 1.

6

6

4

4

Improper fractions can always be expressed as mixed fractions, and vice versa.

Converting improper fractions to mixed fractions:

To perform this conversion, one must divide the numerator by the denominator. The number of times that the denominator is completely contained within the numerator would become the whole number. The remainder would be expressed as a proper fraction.

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Examples:

8 62 2 3 = 3+3 = 23 16 15 1 1 5 = 5 +5 = 35

Converting mixed fractions to improper fractions:

To perform this conversion, one must multiply the denominator by the whole number, and add this result to the numerator. This sum would be the new numerator, while the denominator remains the same.

Examples:

7 (1 ? 8) + 7 8 + 7 15

18 =

8

= 8 =8

2 (4 ? 5) + 2 20 + 2 22

45 =

5

= 5 =5

Cancelling down

Some fractions may look different but actually have the same value. For example: 1 = 2 = 4

4 8 16

Multiplying or dividing both numerator and denominator of a fraction by the same number maintains the value of the fraction. Sometimes we perform this operation to help our calculations. This operation is called "cancelling down". When we use a fraction, we usually give it in its simplest form. To do this we look at the numerator and the denominator and see if there is a number by which both can be divided with no remainder.

Two examples:

2 1?2 1

9 3?3 3

8 = 2 ? 4 = 4 ; 12 = 3 ? 4 = 4

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Adding Fractions

When the denominators (the bottom lines) are all the same, you simply add the top line (numerators). For example:

1 5 1+5 6 8+8= 8 =8 When the denominators are different, we need to change the fractions so that the denominators are the same. Only then we could add the top line as above. At times, all we need to do is to cancel down the fraction with the greater denominator, for example: 1 6 1 2?3 3 3+9=3+3?3=3=1 At most times this action is insufficient, and we actually need to find a new denominator. That action is called "finding a common denominator". The simplest way to find a common denominator is to multiply both current denominators, and the product would be the new denominator. The numerator from the first fraction is multiplied by the denominator of the second fraction and is added to a multiplication of the numerator of the second fraction by the denominator of the first, to form the new numerator. For example: 3 1 3 ? 4 + 1 ? 5 12 + 5 17 5 + 4 = 5 ? 4 = 20 = 20 If possible, cancel down after you finish adding.

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LCD ? Lowest Common Denominator

Multiplying both denominators may be the simplest way to get a common denominator. However, it might not be the most convenient way. In order to work with the smallest numbers, which are the easiest to perform calculations with, we look for the lowest common denominator, or LCD.

The LCD is the smallest number, which can be divided by both denominators.

For example:

24 5 + 15 = ?

Multiplying the denominators would give 75. This is not the most convenient number to work with. The number 15 can be divided by both 15 (15 : 1 = 15) and 5 (15 : 5 = 3). Therefore, we shall use 15 as the common denominator. In such cases, the coefficients for multiplying each of the numerators will be the quotients of the common denominator and the previous denominator.

Continuing with the example:

2 4 2 ? 3 + 4 ? 1 6 + 4 10 5 ? 2 2

5 + 15 =

15

= 15 = 15 = 5 ? 3 = 3

Another example:

15 4+6=?

The LCD is 12. Now we can calculate the coefficients. 12 : 4 = 3; 12 : 6 = 2;

1 5 1 ? 3 + 5 ? 2 3 + 10 13 1

4+6=

12

= 12 = 12 = 1 12

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Subtracting Fractions

Subtracting is done the same way as adding fractions, except for the fact that there will be a minus sign in the numerator, and the order of the numbers in the numerator would be important.

For example:

7 3 7 ? 4 - 3 ? 9 28 - 27 1 9 - 4 = 9 ? 4 = 36 = 36

If possible, cancel down after you finish subtracting.

Another example:

16 6 25 - 10 = ?

The LCD is 50. Now we can calculate the coefficients. 50 : 25 = 2; 50 : 10 = 5;

16 6 16 ? 2 - 6 ? 5 32 - 30 2 1

25 - 10 =

50

= 50 = 50 = 25

Multiplying Fractions

Multiplying fractions is a rather simple operation: All you have to do is to multiply the numerators to form a new numerator, and to multiply the denominators to form a new denominator. For example:

3 2 3?2 6 7 ? 5 = 7 ? 5 = 35

While multiplying fractions, it is possible and advised to cancel down the multiplication before calculating the product. It will make the action simpler. Cancelling down can be done even by using the numerator of the first fraction and the denominator of other, and vice versa. For example:

5 3 5?3 1?3 1?1 1 9 ? 20 = 9 ? 20 = 9 ? 4 = 3 ? 4 = 12

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Dividing Fractions

Dividing two fractions is quite similar to multiplying fractions with a slight change.

13 24=?

This division action has the same result as multiplying the first fraction by the inverse of the second fraction:

1 3 1 4 1?4 2 24=2?3=2?3=3

Notice that what we have done is multiplying the numerator of the first fraction by the denominator of the second fraction to form the new numerator and multiplying the denominator of the first fraction by the numerator of the second, to form the new denominator. Remember that the order of the numbers is important.

Let's look at another example: 1 7 = 1 ? 4 = 1?1 = 1

8 4 8 7 2?7 14

Dividing fractions by a whole number ? if either the dividend or the divisor is a whole number rather than a fraction, one can always perform a quick conversion of the whole number to a fraction, and then continue as explained above. For example:

3

16

3 = 1 ; 16 = 1

In conclusion, it is advised to be familiar with commonly used fractions and the ratios between them. Do remember that multiplying a denominator by a number, divides the value of the total fraction by the same value. For example:

11

1

2 = 4?2 = 8?4 =

11

1

3 = 6?2 = 9?3 =

11

1

5 = 10 ? 2 = 15 ? 3 =

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