Formula:

$$a\frac{b}{c} = \frac{(a \times c) + b}{c}$$

Steps:

1. Multiply the whole number by the denominator
2. Add the numerator to the result
3. Place the result over the original denominator

Example:

$2\frac{3}{5} = \frac{(2 \times 5) + 3}{5} = \frac{10 + 3}{5} = \frac{13}{5}$

Formula:

$$\frac{a}{b} = q\frac{r}{b}$$

(where $q$ is the quotient and $r$ is the remainder when dividing $a$ by $b$)

Steps:

1. Divide the numerator by the denominator
2. The quotient becomes the whole number
3. The remainder becomes the new numerator
4. Keep the same denominator

Example:

$\frac{17}{5} = 3\frac{2}{5}$ (because $17 \div 5 = 3$ remainder $2$)

Formula:

$$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$$

Steps:

1. Add the numerators
2. Keep the same denominator
3. Simplify if possible

Example:

$\frac{2}{7} + \frac{3}{7} = \frac{2 + 3}{7} = \frac{5}{7}$

Formula:

$$\frac{a}{b} + \frac{c}{d} = \frac{(a \times d) + (c \times b)}{b \times d}$$

Steps:

1. Find the least common denominator (LCD)
2. Convert fractions to equivalent fractions with LCD
3. Add the numerators
4. Keep the common denominator
5. Simplify if possible

Example:

$\frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12}$

Steps:

1. Method 1: Convert to improper fractions, add, then convert back
2. Method 2: Add whole numbers and fractions separately

Example (Method 2):

$2\frac{1}{4} + 1\frac{2}{4} = (2 + 1) + (\frac{1}{4} + \frac{2}{4}) = 3 + \frac{3}{4} = 3\frac{3}{4}$

Formula:

$$\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$$

Steps:

1. Subtract the numerators
2. Keep the same denominator
3. Simplify if possible

Example:

$\frac{5}{8} - \frac{2}{8} = \frac{5 - 2}{8} = \frac{3}{8}$

Formula:

$$\frac{a}{b} - \frac{c}{d} = \frac{(a \times d) - (c \times b)}{b \times d}$$

Steps:

1. Find the least common denominator (LCD)
2. Convert fractions to equivalent fractions with LCD
3. Subtract the numerators
4. Keep the common denominator
5. Simplify if possible

Example:

$\frac{3}{4} - \frac{1}{3} = \frac{9}{12} - \frac{4}{12} = \frac{5}{12}$

Steps:

1. If the fraction part of the first number is smaller, borrow 1 from the whole number
2. Convert the borrowed 1 to a fraction with the same denominator
3. Add it to the fraction part
4. Subtract whole numbers and fractions separately

Example:

$5\frac{1}{4} - 2\frac{3}{4} = 4\frac{5}{4} - 2\frac{3}{4} = (4-2) + (\frac{5}{4} - \frac{3}{4}) = 2\frac{2}{4} = 2\frac{1}{2}$

Formula:

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$

Steps:

1. Multiply the numerators together
2. Multiply the denominators together
3. Simplify the result if possible
4. Tip: You can simplify before multiplying by canceling common factors

Example:

$\frac{2}{3} \times \frac{3}{5} = \frac{2 \times 3}{3 \times 5} = \frac{6}{15} = \frac{2}{5}$

Steps:

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

Example:

$2\frac{1}{2} \times 1\frac{1}{3} = \frac{5}{2} \times \frac{4}{3} = \frac{20}{6} = \frac{10}{3} = 3\frac{1}{3}$

Formula:

$$n \times \frac{a}{b} = \frac{n \times a}{b}$$

Steps:

1. Multiply the whole number by the numerator
2. Keep the same denominator
3. Simplify if possible

Example:

$4 \times \frac{2}{3} = \frac{4 \times 2}{3} = \frac{8}{3} = 2\frac{2}{3}$

Formula:

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$$

Steps (Keep, Change, Flip):

1. Keep the first fraction the same
2. Change the division sign to multiplication
3. Flip the second fraction (find its reciprocal)
4. Multiply the fractions
5. Simplify if possible

Example:

$\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$

Steps:

1. Convert all mixed numbers to improper fractions
2. Use "Keep, Change, Flip" method
3. Multiply the fractions
4. Simplify and convert back to a mixed number if needed

Example:

$3\frac{1}{2} \div 1\frac{3}{4} = \frac{7}{2} \div \frac{7}{4} = \frac{7}{2} \times \frac{4}{7} = \frac{28}{14} = 2$

Formula:

$$n \div \frac{a}{b} = n \times \frac{b}{a} = \frac{n \times b}{a}$$

Steps:

1. Write the whole number as a fraction ($\frac{n}{1}$)
2. Use "Keep, Change, Flip" method
3. Multiply and simplify

Example:

$6 \div \frac{2}{3} = \frac{6}{1} \times \frac{3}{2} = \frac{18}{2} = 9$

1. Read carefully: Understand what the problem is asking
2. Identify key information: Circle or underline numbers and important words
3. Determine the operation: Look for keywords
   • Addition: total, altogether, combined, sum, in all
   • Subtraction: difference, how much more, left, remaining
   • Multiplication: of, times, product, each
   • Division: per, each, split, shared equally, how many groups
4. Write the equation: Set up the problem with the correct operation
5. Solve: Follow the rules for that operation
6. Check: Does your answer make sense?
7. Answer in context: Include units and write a complete sentence

Problem:

Sarah walked $\frac{3}{4}$ mile in the morning and $1\frac{1}{2}$ miles in the evening. How far did she walk in total?

Solution:

$\frac{3}{4} + 1\frac{1}{2} = \frac{3}{4} + 1\frac{2}{4} = \frac{3}{4} + \frac{6}{4} = \frac{9}{4} = 2\frac{1}{4}$ miles

Answer: Sarah walked $2\frac{1}{4}$ miles in total.

Problem:

A recipe calls for $\frac{2}{3}$ cup of sugar. If you want to make $2\frac{1}{2}$ times the recipe, how much sugar do you need?

Solution:

$\frac{2}{3} \times 2\frac{1}{2} = \frac{2}{3} \times \frac{5}{2} = \frac{10}{6} = \frac{5}{3} = 1\frac{2}{3}$ cups

Answer: You need $1\frac{2}{3}$ cups of sugar.

Problem:

James had $4\frac{1}{2}$ pizzas. He ate $\frac{3}{4}$ of a pizza for lunch and gave $1\frac{1}{4}$ pizzas to his friend. How much pizza does he have left?

Solution:

Step 1: Find total pizza given away/eaten: $\frac{3}{4} + 1\frac{1}{4} = \frac{3}{4} + 1\frac{1}{4} = 2$ pizzas

Step 2: Subtract from original amount: $4\frac{1}{2} - 2 = 2\frac{1}{2}$ pizzas

Answer: James has $2\frac{1}{2}$ pizzas left.