Proper Fraction

A fraction where the numerator is less than the denominator.

Examples: \(\frac{3}{4}\), \(\frac{2}{5}\), \(\frac{5}{8}\)

Improper Fraction

A fraction where the numerator is greater than or equal to the denominator.

Examples: \(\frac{7}{4}\), \(\frac{9}{5}\), \(\frac{11}{3}\)

Mixed Number (Mixed Fraction)

A whole number combined with a proper fraction.

Examples: \(2\frac{1}{4}\), \(3\frac{2}{5}\), \(1\frac{7}{8}\)

Improper Fraction → Mixed Number

Steps:

1. Divide the numerator by the denominator
2. The quotient becomes the whole number
3. The remainder becomes the new numerator
4. The divisor remains as the denominator

Mixed Number → Improper Fraction

Steps:

1. Multiply the whole number by the denominator
2. Add the numerator to the product
3. Write the sum over the original denominator

Adding Fractions (Same Denominator)

Add the numerators, keep the denominator the same!

Adding Fractions (Different Denominators)

📝 Steps for Adding Mixed Numbers

1. Convert mixed numbers to improper fractions
2. Find the Least Common Multiple (LCM) if denominators are different
3. Create equivalent fractions with the same denominator
4. Add the numerators
5. Simplify and convert back to mixed number if needed

💡 Examples

Subtracting Fractions (Same Denominator)

Subtract the numerators, keep the denominator the same!

Subtracting Fractions (Different Denominators)

📝 Steps for Subtracting Mixed Numbers

1. Convert mixed numbers to improper fractions
2. Find the LCM if denominators are different
3. Create equivalent fractions with the same denominator
4. Subtract the numerators
5. Simplify and convert back to mixed number if needed

💡 Examples

Multiplying Fractions Formula

Multiply numerators together, multiply denominators together!

📝 Steps for Multiplying Mixed Numbers

1. Convert mixed numbers to improper fractions
2. Multiply the numerators together
3. Multiply the denominators together
4. Simplify and convert to mixed number if needed

💡 Examples

Dividing Fractions Formula

Keep → Change → Flip (Multiply by the reciprocal)

📝 Steps for Dividing Mixed Numbers

1. Convert mixed numbers to improper fractions
2. KEEP the first fraction as it is
3. CHANGE division (÷) to multiplication (×)
4. FLIP the second fraction (find its reciprocal)
5. Multiply and simplify

💡 Examples

🎯 Steps to Solve Word Problems

1. READ the problem carefully and identify key information
2. DETERMINE which operation to use (add, subtract, multiply, or divide)
3. WRITE an equation to represent the problem
4. SOLVE using the correct steps for that operation
5. CHECK if the answer makes sense

Problem 1: Addition Word Problem

Step 1: Write equation → \(2\frac{1}{3} + 4\frac{1}{6}\)
Step 2: Convert to improper fractions → \(\frac{7}{3} + \frac{25}{6}\)
Step 3: Find LCM (6) and create equivalent fractions → \(\frac{14}{6} + \frac{25}{6}\)
Step 4: Add → \(\frac{14 + 25}{6} = \frac{39}{6}\)
Step 5: Convert to mixed number and simplify → \(6\frac{3}{6} = 6\frac{1}{2}\)

✓ Answer: \(6\frac{1}{2}\) pounds of apples

Problem 2: Subtraction Word Problem

Step 1: Write equation → \(5\frac{5}{8} - 3\frac{1}{4}\)
Step 2: Convert to improper fractions → \(\frac{45}{8} - \frac{13}{4}\)
Step 3: Find LCM (8) and create equivalent fractions → \(\frac{45}{8} - \frac{26}{8}\)
Step 4: Subtract → \(\frac{45 - 26}{8} = \frac{19}{8}\)
Step 5: Convert to mixed number → \(2\frac{3}{8}\)

✓ Answer: \(2\frac{3}{8}\) cups left

Problem 3: Multiplication Word Problem

Step 1: Write equation → \(\frac{2}{3} \times 2\frac{1}{2}\)
Step 2: Convert to improper fractions → \(\frac{2}{3} \times \frac{5}{2}\)
Step 3: Multiply numerators → \(2 \times 5 = 10\)
Step 4: Multiply denominators → \(3 \times 2 = 6\)
Step 5: Simplify → \(\frac{10}{6} = \frac{5}{3} = 1\frac{2}{3}\)

✓ Answer: \(1\frac{2}{3}\) cups of flour

Problem 4: Division Word Problem

Step 1: Write equation → \(11 \div \frac{1}{5}\)
Step 2: Convert 11 to fraction → \(\frac{11}{1} \div \frac{1}{5}\)
Step 3: Keep-Change-Flip → \(\frac{11}{1} \times \frac{5}{1}\)
Step 4: Multiply → \(\frac{11 \times 5}{1 \times 1} = \frac{55}{1} = 55\)

✓ Answer: 55 seeds can be planted

🎯 Strategy for Multi-Step Problems

1. IDENTIFY all the steps needed to solve the problem
2. BREAK DOWN the problem into smaller parts
3. SOLVE each part in the correct order
4. USE the result from one step in the next step
5. CHECK your final answer makes sense

Multi-Step Problem 1

Step 1: Find total fruit → \(2\frac{1}{2} + 1\frac{3}{4}\)
    Convert: \(\frac{5}{2} + \frac{7}{4} = \frac{10}{4} + \frac{7}{4} = \frac{17}{4} = 4\frac{1}{4}\) pounds

Step 2: Find \(\frac{2}{3}\) of total → \(4\frac{1}{4} \times \frac{2}{3}\)
    Convert: \(\frac{17}{4} \times \frac{2}{3} = \frac{34}{12} = \frac{17}{6} = 2\frac{5}{6}\) pounds

✓ Answer: Sarah used \(2\frac{5}{6}\) pounds of fruit

Multi-Step Problem 2

Step 1: Find remaining ribbon → \(12\frac{1}{2} - 3\frac{1}{4}\)
    Convert: \(\frac{25}{2} - \frac{13}{4} = \frac{50}{4} - \frac{13}{4} = \frac{37}{4} = 9\frac{1}{4}\) feet

Step 2: Divide into 4 pieces → \(9\frac{1}{4} \div 4\)
    Convert: \(\frac{37}{4} \div \frac{4}{1} = \frac{37}{4} \times \frac{1}{4} = \frac{37}{16} = 2\frac{5}{16}\) feet

✓ Answer: Each piece is \(2\frac{5}{16}\) feet long

Multi-Step Problem 3

Step 1: Find fabric used → \(2\frac{1}{3} + 1\frac{1}{2}\)
    Convert: \(\frac{7}{3} + \frac{3}{2} = \frac{14}{6} + \frac{9}{6} = \frac{23}{6}\) yards

Step 2: Find remaining fabric → \(8\frac{3}{4} - \frac{23}{6}\)
    Convert: \(\frac{35}{4} - \frac{23}{6} = \frac{105}{12} - \frac{46}{12} = \frac{59}{12}\) yards

Step 3: Divide into \(\frac{1}{4}\) yard strips → \(\frac{59}{12} \div \frac{1}{4}\)
    Keep-Change-Flip: \(\frac{59}{12} \times \frac{4}{1} = \frac{236}{12} = \frac{59}{3} = 19\frac{2}{3}\)
    Since we can only make whole strips → 19 strips

✓ Answer: Maria can make 19 complete strips

🎯 Remember the Order of Operations

Parentheses → Exponents → Multiplication/Division → Addition/Subtraction