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Fractions

Prime=20 numbers
Greatest=20 common factor
Least=20 common multiple
What=20 is a fraction?
Equivalent=20 fractions
Comparing=20 fractions
Converting=20 and reducing fractions
Lowest=20 terms
Improper=20 fractions
Mixed=20 numbers
Converting=20 mixed numbers to improper fractions
Converting=20 improper fractions to mixed numbers
Writing=20 a fraction as a decimal
Rounding=20 a fraction to the nearest hundredth
Adding=20 and subtracting fractions
Adding=20 and subtracting mixed numbers
Multiplying=20 fractions and whole numbers
Multiplying=20 fractions and fractions
Multiplying=20 mixed numbers
Reciprocal
Dividing=20 fractions
Dividing=20 mixed numbers
Simplifying=20 complex fractions
Repeating=20 decimals=20

=20

Math=20 Contests 
School League Competitions =
Contest Problem=20 Books 
Challenging, fun math practice =
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Comprehensive Learning Tools=20

3DMath=20 Visit the Math League

Prime Numbers

A whole number greater than one that is divisible by only 1 and = itself. The=20 numbers 2, 3, 5, 37, and 101 are some examples of prime numbers.=20

Greatest Common Factor

The greatest common factor of two or more whole numbers is the = largest whole=20 number that divides each of the numbers.=20

There are two methods of finding the greatest common factor of two = numbers.=20

Method 1: List all the factors of each number, then list the = common=20 factors and choose the largest one.=20

Example:=20

36: 1, 2, 3, 4, 6, 9, 12, = 18, 36=20

54: 1, 2, 3, 6, 9, 18, 27, = 54=20

The common factors are: 1, 2, 3, 6, 9, and 18.=20

The greatest common factor is: 18.=20

Method 2: List the prime factors, then multiply the common = prime=20 factors.=20

Example:=20

36 =3D 2 =D7 2 =D7 3 = =D7 3=20

54 =3D 2 =D7 3 =D7 3 = =D7 3=20

The common prime factors are 2, 3, and 3.=20

The greatest common factor is=20 2 =D7 3 =D7 3 =3D 18..= =20

Least Common Multiple

The least common multiple of two or more nonzero whole numbers is the = smallest whole number that is divisible by each of the numbers. There = are two=20 common methods for finding the least common multiple of 2 numbers.=20

Method 1:=20

List the multiples of each number, and look for the smallest number = that=20 appears in each list.=20

Example:=20

Find the least common multiple of 12 and 42. We list the multiples of = each=20 number:=20

12: 12, 24, 36, 48, 60, 72, 84, ...=20

42: 42, 84, 126, 168, 190, ...=20

We see that the number 84 is the smallest number that appears in each = list.

Method 2:=20

Factor each of the numbers into primes. For each different prime = number in=20 either of the factorizations, follow these steps:=20

1. Count the number of times it appears in each of the = factorizations.=20

2. Take the largest of these two counts.=20

3. Write down that prime number as many times as the count in step 2. =

To find the least common multiple take the product of all of the = prime=20 numbers written down in steps 1, 2, and 3.

Example:=20

Find the least common multiple of 24 and 90. First, we find the prime = factorization of each number.=20

24 =3D 2 =D7 2 =D7 2 =D7 3=20

90 =3D 2 =D7 3 =D7 3 =D7 5=20

The prime numbers 2, 3, and 5 appear in the factorizations. We follow = steps 1=20 through 3 for each of these primes.=20

The number 2 occurs 3 times in the first factorization and 1 time in = the=20 second, so we will use three 2's.=20

The number 3 occurs 1 time in the first factorization and 2 times in = the=20 second, so we will use two 3's.=20

The number 5 occurs 0 times in the first factorization and 1 time in = the=20 second factorization, so we will use one 5.=20

The least common multiple is the product of three 2's, two 3's, and = one 5.=20

2 =D7 2 =D7 2 =D7 3 =D7 3 = ;=D7 5 =3D 360=20

Example:=20

Find the least common multiple of 14 and 49. First, we find the prime = factorization of each number.=20

14 =3D 2 =D7 7=20

49 =3D 7 =D7 7=20

The prime numbers 2 and 7 appear in the factorizations. We follow = steps 1=20 through 3 for each of these primes.=20

The number 2 occurs 1 times in the first factorization and 0 times in = the=20 second, so we will use one 2.=20

The number 7 occurs 1 time in the first factorization and 2 times in = the=20 second, so we will use two 7's.=20

The least common multiple is the product of one 2 and two 7's.=20

2 =D7 7 =D7 7 =3D 98=20

Examples:=20

Some other least common multiples are listed below.=20

The least common multiple of 12 and 9 is 36.=20

The least common multiple of 6 and 18 is 18.=20

The least common multiple of 2, 3, 4, and 5 is 60.=20

What is a Fraction?

A fraction is a number that expresses part of a group.=20

Fractions are written in the form or=20 a/b, where a and b are whole numbers, and the number b is = not 0.=20 For the purposes of these web pages, we will denote fractions using the = notation=20 a/b, though the preferred notation is generally .=20

The number a is called the numerator, and the number b = is=20 called the denominator.
=20

Examples:=20

The following numbers are all fractions
1/2, 3/7, 6/10, 4/99=20

Example:=20

The fraction 4/6 represents the shaded portion of the circle below. = There are=20 6 pieces in the group, and 4 of them are shaded.=20

=20

Example:=20

The fraction 3/8 represents the shaded portion of the circle below. = There are=20 8 pieces in the group, and 3 of them are shaded.=20

=20

Example:=20

The fraction 2/3 represents the shaded portion of the circle below. = There are=20 3 pieces in the group, and 2 of them are shaded.=20

=20

Equivalent Fractions

Equivalent fractions are different fractions which name the same = amount.=20

Examples:=20

The fractions 1/2, 2/4, 3/6, 100/200, and 521/1042 are all equivalent = fractions.
The fractions 3/7, 6/14, and 24/56 are all equivalent = fractions.=20
We can test if two fractions are equivalent by cross-multiplying = their=20 numerators and denominators. This is also called taking the = cross-product.=20

Example:=20

Test if 3/7 and 18/42 are equivalent fractions.
The first = cross-product is=20 the product of the first numerator and the second denominator:=20 3 =D7 42 =3D 126.
The second cross-product is = the product of=20 the second numerator and the first denominator:=20 18 =D7 7 =3D 126.
Since the cross-products are = the same, the=20 fractions are equivalent.=20

Example:=20

Test if 2/4 and 13/20 are equivalent fractions.
The first = cross-product is=20 the product of the first numerator and the second denominator:=20 2 =D7 20 =3D 40.
The second cross-product is the = product of=20 the second numerator and the first denominator: = 4 =D7 13 =3D 52.=20
Since the cross-products are different, the fractions are not = equivalent.=20 Since the second cross-product is larger than the first, the second = fraction is=20 larger than the first.=20

Comparing Fractions

1. To compare fractions with the same denominator, look at their = numerators.=20 The larger fraction is the one with the larger numerator.
2. To = compare=20 fractions with different denominators, take the cross product. The first = cross-product is the product of the first numerator and the second = denominator.=20 The second cross-product is the product of the second numerator and the = first=20 denominator. Compare the cross products using the following rules:
a. = If the=20 cross-products are equal, the fractions are equivalent.
b. If the = first=20 cross product is larger, the first fraction is larger.
c. If the = second cross=20 product is larger, the second fraction is larger.=20

Example:=20

Compare the fractions 3/7 and 1/2.
The first cross-product is the = product=20 of the first numerator and the second denominator:=20 3 =D7 2 =3D 6.
The second cross-product is the = product of=20 the second numerator and the first denominator: = 7 =D7 1 =3D 7.=20
Since the second cross-product is larger, the second fraction is = larger.=20

Example:=20

Compare the fractions 13/20 and 3/5.
The first cross-product is = the=20 product of the first numerator and the second denominator:=20 5 =D7 13 =3D 65.
The second cross-product is the = product of=20 the second numerator and the first denominator: = 20 =D7 3 =3D 60.=20
Since the first cross-product is larger, the first fraction is = larger.=20

Converting and Reducing=20 Fractions

For any fraction, multiplying the numerator and denominator by the = same=20 nonzero number gives an equivalent fraction. We can convert one fraction = to an=20 equivalent fraction by using this method.=20

Examples:=20

1/2 =3D (1 =D7 3)/(2 =D7 3) =3D&nbs= p;3/6=20

2/3 =3D (2 =D7 2)/(3 =D7 2) =3D&nbs= p;4/6=20

3/5 =3D (3 =D7 4)/(5 =D7 4) =3D&nbs= p;12/20=20

Another method of converting one fraction to an equivalent fraction = is by=20 dividing the numerator and denominator by a common factor of the = numerator and=20 denominator.=20

Examples:=20

20/42 =3D (20 =F7 2)/(42 =F7 2) =3D=  10/21=20

36/72 =3D (36 =F7 3)/(72 =F7 3) =3D=  12/24=20

9/27 =3D (9 =F7 3)/(27 =F7 3) =3D&n= bsp;3/9=20

When we divide the numerator and denominator of a fraction by their = greatest=20 common factor, the resulting fraction is an equivalent fraction in = lowest terms.=20

Lowest Terms

A fraction is in lowest terms when the greatest common factor of its=20 numerator and denominator is 1. There are two methods of reducing a = fraction to=20 lowest terms.=20

Method 1:=20

Divide the numerator and denominator by their greatest common factor. =

12/30 =3D (12 =F7 6)/(30 =F7 6) =3D=  2/5=20

Method 2:=20

Divide the numerator and denominator by any common factor. Keep = dividing=20 until there are no more common factors.=20

12/30 =3D (12 =F7 2)/(30 =F7 2) =3D=  6/15 =3D (6 =F7 3)/(15 =F7 3) =3D=  2/5=20

Improper Fractions

Improper fractions have numerators that are larger than or equal to = their=20 denominators.=20

Examples:=20

11/4, 5/5, and 13/2 are improper fractions.=20

Mixed Numbers

Mixed numbers have a whole number part and a fraction part.=20

Examples:=20

are mixed numbers also written as 2 3/4 and = 6 1/2. In=20 these web pages, we denote mixed numbers in the form=20 a b/c.=20

Converting Mixed Numbers = to=20 Improper Fractions

To change a mixed number into an improper fraction, multiply the = whole number=20 by the denominator and add it to the numerator of the fractional part.=20

Examples:=20

2 3/4 =3D ((2 =D7 4) + 3)/4 =3D= 11/4=20

6 1/2 =3D ((6 =D7 2) + 1)/2 =3D=  13/2=20

Converting Improper = Fractions to=20 Mixed Numbers

To change an improper fraction into a mixed number, divide the = numerator by=20 the denominator. The remainder is the numerator of the fractional part.=20

Examples:=20

11/4 =3D 11 =F7 4 =3D 2 r3&n= bsp;=3D 2 3/4=20

13/2 =3D 13 =F7 2 =3D 6 r1&n= bsp;=3D 6 1/2=20

Writing a Fraction as a = Decimal

Method 1 - Convert to an equivalent fraction whose denominator is a = power of=20 10, such as 10, 100, 1000, 10000, and so on, then write in decimal form. =

Examples:=20

1/4 =3D (1 =D7 25)/(4 =D7 25) =3D&n= bsp;25/100 =3D 0.25=20

3/20 =3D (3 =D7 5)/(20 =D7 5) =3D&n= bsp;15/100 =3D 0.15=20

9/8 =3D (9 =D7 125)/(8 =D7 125) =3D=  1125/1000 =3D 1.125=20

Method 2 - Divide the numerator by the denominator. Round to the = decimal=20 place asked for, if necessary.=20

Example:=20

13/4 =3D 13 =F7 4 =3D 3.25=20

Example:=20

Convert 3/7 to a decimal.=20

Round to the nearest thousandth.=20

We divide one decimal place past the place we need to round to, then = round=20 the result.=20

3/7 =3D 3 =F7 7 =3D 0.4285=85=20

which equals 0.429 when rounded to the nearest thousandth.=20

Example:=20

Convert 4/9 to a decimal.=20

Round to the nearest hundredth.=20

We divide one decimal place past the place we need to round to, then = round=20 the result.=20

4/9 =3D 4 =F7 9 =3D 0.4444=85=20

which equals 0.44 when rounded to the nearest hundredth.=20

Rounding a Fraction = to the=20 Nearest Hundredth

Divide to the thousandths place. If the last digit is less than 5, = drop it.=20 This is particularly useful for converting a fraction to a percent, if = we want=20 to convert to the nearest percent.=20

1/3 =3D 1 =F7 3 =3D 0.333=85 which = rounds to 0.33=20

If the last digit is 5 or greater, drop it and round up.=20

2/7 =3D 2 =F7 7 =3D 0.285 which rounds = to 0.29=20

Adding and Subtracting=20 Fractions

If the fractions have the same denominator, their sum is the sum of = the=20 numerators over the denominator. If the fractions have the same = denominator,=20 their difference is the difference of the numerators over the = denominator. We do=20 not add or subtract the denominators! Reduce if necessary.=20

Examples:=20

3/8 + 2/8 =3D 5/8=20

9/2 - 5/2 =3D 4/2 =3D 2=20

If the fractions have different denominators:
1) First, find the = least=20 common denominator.
2) Then write equivalent fractions using this=20 denominator.
3) Add or subtract the fractions. Reduce if necessary.=20

Example:=20

3/4 + 1/6 =3D ?=20

The least common denominator is 12.=20

3/4 + 1/6 =3D 9/12 + 2/12 =3D = 11/12.=20

Example:=20

9/10 - 1/2 =3D ?=20

The least common denominator is 10.=20

9/10 - 1/2 =3D 9/10 - 5/10 =3D = ;4/10 =3D 2/5.=20

Example:=20

2/3 + 2/7 =3D ?=20

The least common denominator is 21=20

2/3 + 2/7 =3D 14/21 + 6/21 =3D = ;20/21.=20

Adding and Subtracting = Mixed=20 Numbers

To add or subtract mixed numbers, simply convert the mixed numbers = into=20 improper fractions, then add or subtract them as fractions.=20

Example:=20

9 1/2 + 5 3/4 =3D ?=20

Converting each number to an improper fraction, we have=20 9 1/2 =3D 19/2 and 5 3/4 =3D 23/4.=20

We want to calculate 19/2 + 23/4. The LCM of 2 and 4 is 4, = so=20

19/2 + 23/4 =3D 38/4 + 23/4 =3D&nbs= p;(38 + 23)/4 =3D 61/4.=20

Converting back to a mixed number, we have = 61/4 =3D 15 1/4.=20

The strategy of converting numbers into fractions when adding or = subtracting=20 is often useful, even in situations where one of the numbers is whole or = a=20 fraction.=20

Example:=20

13 - 1 1/3 =3D ?=20

In this situation, we may regard 13 as a mixed number without a = fractional=20 part. To convert it into a fraction, we look at the denominator of the = fraction=20 4/3, which is 1 1/3 expressed as an improper fraction. The = denominator is=20 3, and 13 =3D 39/3. So=20 13 - 1 1/3 =3D 39/3 - 4/3 =3D&nbs= p;(39-4)/3 =3D 35/3,=20 and 35/3 =3D 11 2/3.=20

Example:=20

5 1/8 - 2/3 =3D ?=20

This time, we may regard 2/3 as a mixed number with 0 as its whole = part.=20 Converting the first mixed number to an improper fraction, we have=20 5 1/8 =3D 41/8. The problem becomes=20

5 1/8 - 2/3 =3D 41/8 - 2/3 =3D=  123/24 - 16/24 =3D (123 - 16)/24 = ;=3D 107/24.=20

Converting back to a mixed number, we have = 107/24 =3D 4 11/24.=20

Example:=20

92 + 4/5 =3D ?=20

This is easy. To express this as a mixed number, just put the whole = number=20 and the fraction side by side. The answer is 92 4/5.=20

Multiplying Fractions = and Whole=20 Numbers

To multiply a fraction by a whole number, write the whole number as = an=20 improper fraction with a denominator of 1, then multiply as fractions.=20

Example:=20

8 =D7 5/21 =3D ?=20

We can write the number 8 as 8/1. Now we multiply the fractions.=20

8 =D7 5/21 =3D 8/1 =D7 5/21 =3D&nbs= p;(8 =D7 5)/(1 =D7 21) =3D 40/21=20

Example:=20

2/15 =D7 10 =3D ?=20

We can write the number 10 as 10/1. Now we multiply the fractions.=20

2/15 =D7 10 =3D 2/15 =D7 10/1 =3D&n= bsp;(2 =D7 10)/(15 =D7 1) =3D 20/15 =3D=  4/3=20

Multiplying Fractions and = Fractions

When two fractions are multiplied, the result is a fraction with a = numerator=20 that is the product of the fractions' numerators and a denominator that = is the=20 product of the fractions' denominators.=20

Example:=20

4/7 =D7 5/11 =3D ?=20

The numerator will be the product of the numerators: = 4 =D7 5, and the=20 denominator will be the product of the denominators: 7 =D7 11. =

The answer is = (4 =D7 5)/(7 =D7 11) =3D 20/77.=20

Remember that like numbers in the numerator and denominator cancel = out.=20

Example:=20

14/15 =D7 15/17 =3D ?=20

Since the 15's in the numerator and denominator cancel, the answer is =

14/15 =D7 15/17 =3D 14/1 =D7 1/17 =3D=  (14 =D7 1)/(1 =D7 17) =3D 14/17=20

Example:=20

4/11 =D7 22/36 =3D ?=20

In the solution below, first we cancel the common factor of 11 in the = top and=20 bottom of the product, then we cancel the common factor of 4 in the top = and=20 bottom of the product.=20

4/11 =D7 22/36 =3D 4/1 =D7 2/36 =3D=  1/1 =D7 2/9 =3D 2/9=20

Multiplying Mixed Numbers =

To multiply mixed numbers, convert them to improper fractions and = multiply.=20

Example:=20

4 1/5 =D7 2 2/3 =3D ?.=20

Converting to improper fractions, we get = 4 1/5 =3D 21/5 and=20 2 2/3 =3D 8/3. So the answer is=20

4 1/5 =D7 2 2/3 =3D 21/5 =D7 8= /3 =3D (21 =D7 8)/(5 =D7 3) =3D 1= 68/15 =3D 11 3/15.=20

Examples:=20

3/4 =D7 1 1/8 =3D 3/4 =D7 9/8 = =3D 27/32.=20

3 =D7 7 3/4 =3D 3 =D7 31/4 =3D=  (3 =D7 31)/4 =3D 93/4 =3D 23=20 1/4.=20

Reciprocal

The reciprocal of a fraction is obtained by switching its numerator = and=20 denominator. To find the reciprocal of a mixed number, first convert the = mixed=20 number to an improper fraction, then switch the numerator and = denominator of the=20 improper fraction. Notice that when you multiply a fraction and its = reciprocal,=20 the product is always 1.=20

Example:=20

Find the reciprocal of 31/75. We switch the numerator and denominator = to find=20 the reciprocal: 75/31.=20

Example:=20

Find the reciprocal of 12 1/2. First, convert the mixed number = to an=20 improper fraction: 12 1/2 =3D 25/2. Next, we switch the = numerator=20 and denominator to find the reciprocal: 2/25.=20

Dividing Fractions

To divide a number by a fraction, multiply the number by the = reciprocal of=20 the fraction.=20

Examples:=20

7 =F7 1/5 =3D 7 =D7 5/1 =3D 7&= nbsp;=D7 5 =3D 35=20

1/5 =F7 16 =3D 1/5 =F7 16/1 =3D&nbs= p;1/5 =D7 1/16 =3D (1 =D7 1)/(5 =D7&nb= sp;16) =3D 1/80=20

3/5 =F7 7/12 =3D 3/5 =D7 12/7 =3D&n= bsp;(3 =D7 12)/(5 =D7 7) =3D 36/35=20 or 1 1/35=20

Dividing Mixed Numbers

To divide mixed numbers, you should always convert to improper = fractions,=20 then multiply the first number by the reciprocal of the second.=20

Examples:=20

1 1/2 =F7 3 1/8 =3D 3/2 =F7 25= /8 =3D 3/2 =D7 8/25 =3D (3 =D7 8)= /(2 =D7 25) =3D 24/50=20

1 =F7 3 3/5 =3D 1/1 =F7 18/5 =3D=  1/1 =D7 5/18 =3D (1 =D7 5)/(1 =D7=  18) =3D 5/18=20

3 1/8 =F7 2 =3D 25/8 =F7 2/1 =3D=  25/8 =D7 1/2 =3D (25 =D7 1)/(8 =D7=  2) =3D 25/16=20 or 1 9/16.=20

Simplifying Complex = Fractions

A complex fraction is a fraction whose numerator or denominator is = also a=20 fraction or mixed number.=20

Example of complex fractions:=20

=20

otherwise written as (1/4)/(2/3), (3/7)/100, 11/(2/3), and=20 (23 1/5)/(2/3).=20

To simplify complex fractions, change the complex fraction into a = division=20 problem: divide the numerator by the denominator.=20

The first of these examples becomes=20

(1/4)/(2/3) =3D 1/4 =F7 2/3 =3D 1/4 = ;=D7 3/2 =3D 3/8.=20

The second of these becomes=20

(3/7)/100 =3D 3/7 =F7 100 =3D 3/7 =D7=  1/100 =3D 3/700.=20

The third of these becomes=20

11/(2/3) =3D 11 =F7 2/3 =3D 11 =D7&= nbsp;3/2 =3D 33/2 =3D 16=20 1/2.=20

The fourth of these becomes=20

(23 1/5)/(2/3) =3D 23 1/5 =F7 2/3 =3D=  116/5 =F7 2/3 =3D 116/5 =D7 3/2 = =3D 174/5 =3D 34 4/5.=20

Repeating Decimals

Every fraction can be written as a decimal.=20

For example, 1/3 is 1 divided by 3.=20

If you use a calculator to find 1 =F7 3, the calculator = returns=20 0.333333... This is called a repeating decimal. To represent the = idea=20 that the 3's repeat forever, one uses a horizontal bar (overstrike) as = shown=20 below:=20

=20

Example:

What is the repeating decimal for 1/7 ? Dividing 7 into 1, we get=20 0.142857142..., and we see the pattern begin to repeat with the second = 1, so=20 .=20



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=A9 1997-2006 by Math League Press
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Last updated August 2006 by = Steve=20 Conrad and Dan Flegler.=20
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