Adding Simple Unlike Fractions

Calculate $\frac{1}{2} + \frac{1}{3}$

Find the LCD of 2 and 3: Multiples of 2: 2, 4, 6, 8... Multiples of 3: 3, 6, 9... The smallest common multiple is 6 = LCD = 6

Convert $\frac{1}{2}$ to sixths: $\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$ = $\frac{3}{6}$

Convert $\frac{1}{3}$ to sixths: $\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$ = $\frac{2}{6}$

Add the numerators: $\frac{3}{6} + \frac{2}{6} = \frac{3 + 2}{6} = \frac{5}{6}$ = $\frac{5}{6}$

Check if simplification is needed: 5 and 6 share no common factors other than 1 = Already simplified

Answer: $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$

When One Denominator is a Multiple of the Other

Calculate $\frac{2}{3} + \frac{1}{6}$

Find the LCD of 3 and 6: 6 is already a multiple of 3, so LCD = 6 = LCD = 6

Convert $\frac{2}{3}$ to sixths: $\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$ = $\frac{4}{6}$

Keep $\frac{1}{6}$ as is: $\frac{1}{6}$ already has denominator 6 = $\frac{1}{6}$

Add the numerators: $\frac{4}{6} + \frac{1}{6} = \frac{4 + 1}{6} = \frac{5}{6}$ = $\frac{5}{6}$

Answer: $\frac{2}{3} + \frac{1}{6} = \frac{5}{6}$

Adding Fractions That Need Simplifying

Calculate $\frac{3}{4} + \frac{1}{6}$

Find the LCD of 4 and 6: Multiples of 4: 4, 8, 12, 16... Multiples of 6: 6, 12, 18... LCD = 12 = LCD = 12

Convert $\frac{3}{4}$ to twelfths: $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$ = $\frac{9}{12}$

Convert $\frac{1}{6}$ to twelfths: $\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}$ = $\frac{2}{12}$

Add the numerators: $\frac{9}{12} + \frac{2}{12} = \frac{11}{12}$ = $\frac{11}{12}$

Check simplification: 11 is prime, 12 = 2 × 2 × 3. No common factors. = Already simplified

Answer: $\frac{3}{4} + \frac{1}{6} = \frac{11}{12}$

Result Greater Than One

Calculate $\frac{3}{4} + \frac{2}{3}$

Find the LCD of 4 and 3: Multiples of 4: 4, 8, 12... Multiples of 3: 3, 6, 9, 12... LCD = 12 = LCD = 12

Convert $\frac{3}{4}$ to twelfths: $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$ = $\frac{9}{12}$

Convert $\frac{2}{3}$ to twelfths: $\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$ = $\frac{8}{12}$

Add the numerators: $\frac{9}{12} + \frac{8}{12} = \frac{17}{12}$ = $\frac{17}{12}$

Convert to mixed number: $\frac{17}{12} = 1\frac{5}{12}$ (17 ÷ 12 = 1 remainder 5) = $1\frac{5}{12}$

Answer: $\frac{3}{4} + \frac{2}{3} = \frac{17}{12} = 1\frac{5}{12}$

Mistake: Adding both numerators AND denominators: $\frac{1}{2} + \frac{1}{3} = \frac{2}{5}$

Why: This is the most common error! Denominators tell you the SIZE of each piece. You cannot add pieces of different sizes directly.

Correct: Find a common denominator first, then add only the numerators: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

Mistake: Using any common multiple instead of simplifying at the end

Why: While using 24 instead of 12 works, your answer will need simplifying: $\frac{18}{24} + \frac{16}{24} = \frac{34}{24}$

Correct: Either use the LCD from the start, or remember to simplify your final answer: $\frac{34}{24} = \frac{17}{12}$

Mistake: Forgetting to multiply both numerator and denominator

Why: If you only change the denominator, you change the value of the fraction!

Correct: Always multiply top AND bottom by the same number to create an equivalent fraction

Cooking and Recipes

Recipes often require combining different fractional amounts of ingredients.

A recipe needs $\frac{1}{3}$ cup of oil and you want to add $\frac{1}{4}$ cup more for extra moisture. Total oil = $\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$ cup.

Time Management

Adding time spent on different activities often involves unlike fractions.

You spend $\frac{1}{2}$ hour on math and $\frac{1}{3}$ hour on reading. Total: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ hour (50 minutes).

DIY and Construction

Measuring materials often requires adding fractions with different denominators.

You need a piece of wood that is $\frac{3}{8}$ inch plus $\frac{1}{4}$ inch thick. Total: $\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$ inch.

Q: Why can't I just add the denominators?

A: Denominators represent the SIZE of each piece. $\frac{1}{2}$ means 1 piece when something is cut into 2 parts. $\frac{1}{3}$ means 1 piece when cut into 3 parts. These pieces are different sizes! You need same-sized pieces (same denominator) before you can count them together.

Q: Do I always need the LEAST common denominator?

A: No, any common denominator works mathematically. But using the LCD makes your numbers smaller and calculations easier. For example, with $\frac{1}{2} + \frac{1}{3}$, you could use 6, 12, 18, or any multiple of 6. Using 6 is just simpler!

Q: What if I get an improper fraction as my answer?

A: That is perfectly fine! You can leave it as an improper fraction like $\frac{7}{4}$, or convert it to a mixed number like $1\frac{3}{4}$. Both are correct — check what your teacher prefers.