Repeating Decimal: A decimal where one or more digits repeat infinitely

Notation: \(0.\overline{3} = 0.333...\) or \(0.\overline{27} = 0.272727...\)

Bar notation: The bar shows which digits repeat

Step 1: Let \(x\) = the repeating decimal

Step 2: Multiply both sides by \(10^n\) where \(n\) = number of repeating digits

Step 3: Subtract original equation from new equation

Step 4: Solve for \(x\) and simplify

One Digit Repeating: \(0.\overline{d} = \frac{d}{9}\)

Example: \(0.\overline{3} = \frac{3}{9} = \frac{1}{3}\)

Two Digits Repeating: \(0.\overline{ab} = \frac{ab}{99}\)

Example: \(0.\overline{27} = \frac{27}{99} = \frac{3}{11}\)

Three Digits Repeating: \(0.\overline{abc} = \frac{abc}{999}\)

Example: \(0.\overline{123} = \frac{123}{999} = \frac{41}{333}\)

Convert \(0.\overline{36}\) to a fraction:
Let \(x = 0.363636...\)
Multiply by 100: \(100x = 36.363636...\)
Subtract: \(100x - x = 36\)
Simplify: \(99x = 36\)
Solution: \(x = \frac{36}{99} = \frac{4}{11}\)

Step 1: Count decimal places \((n)\)

Step 2: Write as \(\frac{\text{number without decimal}}{10^n}\)

Step 3: Simplify to lowest terms

1 decimal place: \(0.a = \frac{a}{10}\)

Example: \(0.7 = \frac{7}{10}\)

2 decimal places: \(0.ab = \frac{ab}{100}\)

Example: \(0.25 = \frac{25}{100} = \frac{1}{4}\)

3 decimal places: \(0.abc = \frac{abc}{1000}\)

Example: \(0.125 = \frac{125}{1000} = \frac{1}{8}\)

Formula: \(\frac{a}{b} = a \div b\)

Example: \(\frac{3}{4} = 3 \div 4 = 0.75\)

Decimal to Mixed Number:
\(3.25 = 3 + 0.25 = 3 + \frac{25}{100} = 3\frac{1}{4}\)

Mixed Number to Decimal:
\(2\frac{3}{5} = 2 + \frac{3}{5} = 2 + 0.6 = 2.6\)

Rational Number: Any number that can be expressed as \(\frac{p}{q}\) where \(p, q\) are integers and \(q \neq 0\)

Examples: \(\frac{3}{4}, -\frac{2}{5}, 0.5, -3, 2\frac{1}{2}, 0.\overline{3}\)

Method 1: Common Denominator

Convert fractions to have the same denominator, then compare numerators

Example: Compare \(\frac{2}{3}\) and \(\frac{3}{4}\)
LCD = 12: \(\frac{8}{12}\) and \(\frac{9}{12}\)
Since \(8 < 9\), therefore \(\frac{2}{3} < \frac{3}{4}\)

Method 2: Convert to Decimals

Convert all numbers to decimal form and compare

Example: \(\frac{2}{3} = 0.\overline{6}\) and \(\frac{3}{4} = 0.75\)
Since \(0.666... < 0.75\), therefore \(\frac{2}{3} < \frac{3}{4}\)

Method 3: Cross Multiplication

For \(\frac{a}{b}\) and \(\frac{c}{d}\): Compare \(a \times d\) with \(b \times c\)

✓ All positive numbers > 0 > all negative numbers

✓ For negative fractions: smaller absolute value is greater

✓ Use number line: numbers to the right are greater

Step 1: Convert all numbers to the same form (all decimals OR all fractions with common denominator)

Step 2: Compare the values

Step 3: Arrange in required order (ascending or descending)

Order from least to greatest: \(\frac{3}{4}, 0.6, -\frac{1}{2}, 0.8, -0.3\)

Step 1 - Convert to decimals:
\(\frac{3}{4} = 0.75\), \(0.6 = 0.6\), \(-\frac{1}{2} = -0.5\), \(0.8 = 0.8\), \(-0.3 = -0.3\)

Step 2 - Order: \(-0.5, -0.3, 0.6, 0.75, 0.8\)

Step 3 - Original form: \(-\frac{1}{2}, -0.3, 0.6, \frac{3}{4}, 0.8\)

• Ascending Order: Smallest to largest (least to greatest)

• Descending Order: Largest to smallest (greatest to least)

• Always place negative numbers before positive numbers

Multiplicative Inverse (Reciprocal): Two numbers whose product is 1

Formula: If \(a \times b = 1\), then \(b\) is the reciprocal of \(a\)

For Fractions: Flip the numerator and denominator
Formula: Reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\)
Example: Reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\)

For Whole Numbers: Write as \(\frac{1}{n}\)
Formula: Reciprocal of \(n\) is \(\frac{1}{n}\)
Example: Reciprocal of \(5\) is \(\frac{1}{5}\)

For Negative Numbers: Reciprocal is also negative
Example: Reciprocal of \(-\frac{2}{3}\) is \(-\frac{3}{2}\)

For Mixed Numbers: Convert to improper fraction first
Example: Reciprocal of \(2\frac{1}{3} = \frac{7}{3}\) is \(\frac{3}{7}\)

✓ Reciprocal of \(1\) is \(1\)

✓ Reciprocal of \(-1\) is \(-1\)

✓ Zero has NO reciprocal \((\frac{1}{0}\) is undefined)

Same Denominator:

Formula: \(\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}\)

Example: \(\frac{3}{7} + \frac{2}{7} = \frac{5}{7}\)

Different Denominators:

Formula: \(\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\) or find LCD

Example: \(\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}\)

Step 1: Find the Least Common Denominator (LCD)

Step 2: Convert each fraction to equivalent fraction with LCD

Step 3: Add or subtract the numerators

Step 4: Keep the common denominator

Step 5: Simplify the result

Step 1: Align decimal points vertically

Step 2: Add zeros as placeholders if needed

Step 3: Add or subtract as with whole numbers

Example: \(3.45 + 2.8 = 3.45 + 2.80 = 6.25\)

Step 1: Read carefully and identify what is being asked

Step 2: Identify the given information

Step 3: Determine the operation (addition or subtraction)

Step 4: Set up the equation

Step 5: Solve and check reasonableness

Addition: total, sum, combined, altogether, increased, more than

Subtraction: difference, less than, decreased, left, remaining, how much more

Problem: Sarah ran \(2\frac{3}{4}\) miles on Monday and \(3\frac{1}{2}\) miles on Tuesday. How many total miles did she run?
Solution: \(2\frac{3}{4} + 3\frac{1}{2} = \frac{11}{4} + \frac{7}{2} = \frac{11}{4} + \frac{14}{4} = \frac{25}{4} = 6\frac{1}{4}\) miles

Same Signs: Add absolute values, keep the common sign
Example: \(-\frac{2}{3} + (-\frac{1}{3}) = -\frac{3}{3} = -1\)

Different Signs: Subtract smaller from larger, use sign of larger absolute value
Example: \(\frac{5}{6} + (-\frac{1}{3}) = \frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}\)

Key Rule: Subtracting is the same as adding the opposite

Formula: \(a - b = a + (-b)\)

Example: \(\frac{3}{4} - \frac{1}{2} = \frac{3}{4} + (-\frac{1}{2}) = \frac{3}{4} - \frac{2}{4} = \frac{1}{4}\)

Example: \(\frac{2}{5} - (-\frac{1}{5}) = \frac{2}{5} + \frac{1}{5} = \frac{3}{5}\)

Formula: \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)

Multiply numerators together and denominators together

Example: \(\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\)

Formula: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)

"Keep-Change-Flip": Keep first fraction, change to multiplication, flip second fraction

Example: \(\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}\)

Same Signs → Positive Result

\((+) \times (+) = (+)\) and \((-) \times (-) = (+)\)

\((+) \div (+) = (+)\) and \((-) \div (-) = (+)\)

Different Signs → Negative Result

\((+) \times (-) = (-)\) and \((-) \times (+) = (-)\)

\((+) \div (-) = (-)\) and \((-) \div (+) = (-)\)

Multiplication: of, times, product, each, per, twice

Division: per, each, split, shared, quotient, ratio

Multiplication Problem:
A recipe needs \(\frac{2}{3}\) cup of sugar. If you make \(\frac{3}{4}\) of the recipe, how much sugar do you need?
Solution: \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\) cup

Division Problem:
You have \(\frac{3}{4}\) pound of cheese to divide among 6 people. How much does each person get?
Solution: \(\frac{3}{4} \div 6 = \frac{3}{4} \times \frac{1}{6} = \frac{3}{24} = \frac{1}{8}\) pound

Multiplication Rules:

✓ Multiply numerators, multiply denominators

✓ Same signs = positive, different signs = negative

✓ Simplify before or after multiplying

Division Rules:

✓ Multiply by the reciprocal

✓ Same signs = positive, different signs = negative

✓ Cannot divide by zero

\(-\frac{2}{5} \times \frac{3}{7} = -\frac{6}{35}\) (different signs = negative)

\(-\frac{4}{9} \div (-\frac{2}{3}) = -\frac{4}{9} \times (-\frac{3}{2}) = \frac{12}{18} = \frac{2}{3}\) (same signs = positive)

Addition: Find common denominator, add numerators

Subtraction: Find common denominator, subtract numerators OR add the opposite

Multiplication: Multiply numerators, multiply denominators

Division: Multiply by the reciprocal of the divisor

Addition/Subtraction:

• Same signs: Add, keep sign

• Different signs: Subtract, use sign of larger absolute value

Multiplication/Division:

• Same signs: Result is positive

• Different signs: Result is negative

P - Parentheses / B - Brackets

E - Exponents / O - Orders

MD - Multiplication and Division (left to right)

AS - Addition and Subtraction (left to right)

Step 1: Solve inside parentheses first

Step 2: Calculate exponents

Step 3: Multiply and divide from left to right

Step 4: Add and subtract from left to right

Evaluate: \(\frac{1}{2} + \frac{2}{3} \times \frac{3}{4}\)
Step 1: Multiply first → \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\)
Step 2: Add → \(\frac{1}{2} + \frac{1}{2} = \frac{2}{2} = 1\)
Answer: \(1\)

Step 1: Read the problem carefully multiple times

Step 2: Identify what you need to find

Step 3: List all given information

Step 4: Break into smaller steps

Step 5: Solve each step using appropriate operations

Step 6: Check if answer makes sense

Problem: A recipe calls for \(2\frac{1}{4}\) cups of flour. You want to make \(1\frac{1}{2}\) times the recipe. You already have \(\frac{3}{4}\) cup. How much more flour do you need?

Step 1: Find total flour needed
\(2\frac{1}{4} \times 1\frac{1}{2} = \frac{9}{4} \times \frac{3}{2} = \frac{27}{8} = 3\frac{3}{8}\) cups

Step 2: Subtract what you have
\(3\frac{3}{8} - \frac{3}{4} = \frac{27}{8} - \frac{6}{8} = \frac{21}{8} = 2\frac{5}{8}\) cups

Answer: You need \(2\frac{5}{8}\) more cups of flour

⚡ Remember PEMDAS! Always follow the order of operations! ⚡