Fractions
Fractions
Least Common Multiple (LCM) – The smallest multiple that two or more numbers have in common.
Number Multiples
4 4, 8, 12, 16, 20, 24, 28, 32, 36
6 6, 12, 18, 24, 30, 36, 42
8 6, 16, 24, 32, 40, 48
LCM = 24
Greatest Common Factor (GCF) – The largest factor that two or more numbers have in common.
Number Factors
4. 4, 2, 1 [pic]
6. 6, 3, 2, 1
8 8, 4, 2, 1
GCF = 2
Simplifying Fractions – To simplify a fraction divide the numerator and the denominator by their GCF.
[pic]÷ [pic] = [pic]
Or, you can keep dividing the numerator and denominator by a common factor until there are no more factors in common.
[pic] ÷ [pic] = [pic] [pic] ÷ [pic] = [pic]
Change Improper Fraction to Mixed Number – Divide the numerator by the denominator. Place the remainder over the denominator. Simplify the fraction.
[pic]
[pic]
[pic] = 3[pic] = 3[pic]
Change Mixed Number to Improper Fraction – Multiply the denominator by the whole number. Add the product to the numerator. Place the sum over the denominator.
4[pic]
5 x 4 = 20 + 3 = 23
4[pic] = [pic]
Adding or Subtracting Fractions – Fractions must have a common denominator.
[pic] + [pic] =
1. Find the LCM of 6 and 4.
LCM = 12
2. Convert each fraction to an equivalent fraction with denominator of 12.
[pic] x [pic] = [pic] [pic] x [pic] = [pic]
3. Rewrite the equation using the new fractions.
[pic] + [pic] =
4. Add or subtract the numerators only. Denominators do not change.
[pic] + [pic] = [pic] = [pic]
5. Simplify answer if possible.
Multiplying Fractions –
Multiply the numerators.
Multiply the denominators.
[pic] x [pic] = [pic] = [pic]
Simplify the resulting fraction.
[pic] [pic] [pic] = [pic]
Dividing Fractions –
Keep the first fraction as it is.
Change the operation to multiplication.
Invert (flip) the second fraction.
[pic] ÷ [pic]
[pic] x [pic]
Multiply the fractions.
[pic] x [pic] = [pic] = [pic]
Simplify your answer.
[pic][pic][pic] = [pic]
Operations with Mixed Numbers –
Change each fraction to a mixed number.
Simplify each fraction.
Perform the operation.
Simplify your answer.
Examples:
Addition and Subtraction
2[pic] + 1[pic] =
[pic] + [pic] = [pic] + [pic] = [pic]
Simplify [pic] = [pic] = 3[pic]
Multiplication and Division
3[pic] x 2[pic] =
[pic] x [pic] = [pic]
Simplify [pic] = 8[pic]
Comparing Fractions – In order to compare fractions, the fractions must have a common denominator.
Compare [pic] and [pic]
LCM = 15
[pic] = [pic] and [pic] = [pic]
[pic] > [pic]
Prime Factorization – Factoring a number to the point where all of the factors are prime numbers.[pic]
120
5 24
3 8 120 = 2³ x 3 x 5
2. 4
2 2
Scientific Notation
Scientific Notation uses powers of ten to express very large and very small numbers.
Scientific notation using positive exponents.
Notice the pattern in the powers of 10
10 = 10 x 1 = [pic]
100 = 10 x 10 = [pic]
1,000 = 10 x 10 x 10 = [pic]
10,000 = 10 x 10 x 10 x 10 = [pic]
100,000 = 10 x 10 x 10 x 10 x 10 = [pic]
1,000,000 = 10 x 10 x 10 x 10 x 10 x 10 = [pic]
Example 1:
Express 23500 in scientific notation.
23500 = 2350 x [pic]
Move the decimal one place to the left.
= 235.0 x [pic]
Move the decimal one place to the left.
= 23.5 x [pic]
Move the decimal one place to the left.
= 2.35 x [pic]
Now your number is in scientific notation.
Example 2:
Express 3.65 x [pic] in standard notation.
3.65 x [pic] = 3.65 x 10 x 10 x 10 x 10 = 36,500
OR
Simply move the decimal point 4 spaces to the right, filling in with zeros
as place holders.
3.65 = 36,500
Scientific notation using negative exponents.
Scientific notation with negative exponents of ten are used to write very small decimal numbers.
Notice the patterns in the table below.
Number Decimal Form Exponential Form
100 100 [pic]
10 10 [pic]
1 1 [pic]
[pic] 0.1 [pic]
[pic] 0.01 [pic]
[pic] 0.001 [pic]
[pic] 0.0001 [pic]
Use this information to write in scientific notation.
Example 1:
Express 0.0004 in scientific notation.
0.0004 = [pic] = 4 x [pic] = 4 x [pic]
Example 2:
Express 5.23 x [pic] in standard notation.
5.23 x [pic] = 5.23 x [pic] = 5.23 x 0.001 = 0.00523
OR
Simply move the decimal point 3 spaces to the left, filling in zeros as
place holders between the decimal point and the first non-zero digit.
5.23 = 0.00523
Fractions to Decimals to Percents –
Fraction to Decimal – Divide the numerator by the denominator.
Ex: ¼ = 1 ÷ 4 = 0.25
Decimal to Percent – Move the decimal two spaces to the right
or
Multiply the decimal by 100.
Ex: 0.24 = 24% or 0.24 x 100 = 24%
Percent to Fraction – Drop the percent sign, put the percent
over 100 and reduce.
or
Change the percent to a decimal and the
decimal to a fraction.
Ex: 35% = [pic] = [pic] or 35% = .35 = [pic] = [pic]
Decimal to Fraction –
Terminating Decimal –
Read the decimal using place values and write as a
fraction then reduce
or
Count the number of decimal places in the number.
Place the number (without the decimal point) over
10 to the power that equals the number of decimal
places.
Ex: .456 = 45 thousandths = [pic] = [pic]
or
.456 has 3 decimal places = [pic] = [pic] = [pic]
Repeating Decimal –
Count the number of decimal places in the number.
Place the number (without the decimal point) over
the same number of 9’s that are in the repeat.
Ex: [pic] has 2 decimals in the repeat so [pic] = [pic]
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