Add, Simplify & Convert
Last reviewed: May 2026
Perform all four operations on fractions: add, subtract, multiply, divide. Also simplify fractions to lowest terms, convert between fractions, decimals, and percentages, and handle mixed numbers. Fractions are the foundation of rational number arithmetic and appear everywhere from cooking recipes to construction measurements to financial calculations.1
| Operation | Method | Example |
|---|---|---|
| Addition | Find LCD, add numerators | 1/3 + 1/4 = 4/12 + 3/12 = 7/12 |
| Subtraction | Find LCD, subtract numerators | 3/4 − 1/3 = 9/12 − 4/12 = 5/12 |
| Multiplication | Multiply across | 2/3 × 3/4 = 6/12 = 1/2 |
| Division | Flip and multiply | 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6 |
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/8 | 0.125 | 12.5% |
| 1/4 | 0.25 | 25% |
| 1/3 | 0.333 | 33.3% |
| 3/8 | 0.375 | 37.5% |
| 1/2 | 0.5 | 50% |
| 5/8 | 0.625 | 62.5% |
| 3/4 | 0.75 | 75% |
A fraction represents a part of a whole — the numerator (top) tells how many parts you have, and the denominator (bottom) tells how many equal parts make up the whole. Fractions, decimals, and percentages are three ways to express the same concept: ¾ = 0.75 = 75%. Understanding fractions is essential for cooking, construction, finance, statistics, and everyday math — yet fractions are consistently rated as one of the most challenging math topics by students and adults alike.
Addition/Subtraction: Requires a common denominator. ²⁄₃ + ¹⁄₄ → find LCM(3,4) = 12 → ⁸⁄₁₂ + ³⁄₁₂ = ¹¹⁄₁₂. Multiplication: Multiply straight across — numerator × numerator, denominator × denominator. ²⁄₃ × ⁴⁄₅ = ⁸⁄₁₅. Division: Flip the second fraction and multiply. ²⁄₃ ÷ ⁴⁄₅ = ²⁄₃ × ⁵⁄₄ = ¹⁰⁄₁₂ = ⁵⁄₆. Simplifying: Divide numerator and denominator by their GCF. ¹²⁄₁₈ → GCF(12,18) = 6 → ²⁄₃.
U.S. construction relies heavily on fractional measurements. Tape measures are divided into 1/16" increments. Common construction fractions: ½", ¼", ⅛", 1/16", ¾", ⅜", 5/16", 7/32". Adding measurements requires fraction arithmetic: a board at 5-³⁄₄" plus another at 3-⁷⁄₈" = 9-⅝" (requires converting ³⁄₄ and ⁷⁄₈ to eighths: ⁶⁄₈ + ⁷⁄₈ = ¹³⁄₈ = 1-⁵⁄₈, plus 5+3 = 8, total = 9-⁵⁄₈). These calculations happen constantly on job sites, which is why experienced carpenters can add and subtract fractions mentally at speed.
Fraction to decimal: Divide numerator by denominator. ³⁄₈ = 3 ÷ 8 = 0.375. Decimal to fraction: Use place value. 0.375 = 375/1000. Simplify by GCF(375,1000) = 125 → ³⁄₈. Fraction to percentage: Convert to decimal, multiply by 100. ³⁄₈ = 0.375 × 100 = 37.5%. Common equivalents worth memorizing: ¹⁄₃ ≈ 33.3%, ²⁄₃ ≈ 66.7%, ¹⁄₈ = 12.5%, ³⁄₈ = 37.5%, ⁵⁄₈ = 62.5%, ⁷⁄₈ = 87.5%.
A mixed number combines a whole number and fraction: 3-¼ means "3 and one quarter." An improper fraction has a numerator larger than its denominator: ¹³⁄₄. They represent the same value: 3-¼ = ¹³⁄₄. Converting mixed to improper: multiply whole × denominator + numerator, keep denominator. 3-¼ = (3×4+1)/4 = ¹³⁄₄. Converting improper to mixed: divide numerator by denominator. ¹³⁄₄ = 13÷4 = 3 remainder 1 = 3-¼. Improper fractions are easier to calculate with; mixed numbers are easier to visualize and communicate.
A fraction represents a division operation: 3/4 means 3 divided by 4, or 0.75. The numerator (top number) counts how many parts you have; the denominator (bottom number) tells you how many equal parts make the whole. Proper fractions (numerator < denominator, like 3/8) represent values less than 1. Improper fractions (numerator ≥ denominator, like 7/4) represent values of 1 or greater and can be converted to mixed numbers: 7/4 = 1 3/4. Neither form is more "correct" — improper fractions are easier to compute with, while mixed numbers are more intuitive for everyday measurement (1 3/4 cups of flour is clearer than 7/4 cups). Negative fractions place the minus sign in front of the fraction, in the numerator, or in the denominator — all three positions are mathematically equivalent: -3/4 = (-3)/4 = 3/(-4).
Adding fractions with the same denominator is straightforward: add the numerators and keep the denominator. 2/7 + 3/7 = 5/7. Different denominators require finding a common denominator — ideally the LCD (least common denominator). For 1/3 + 1/4: LCD(3,4) = 12. Convert: 4/12 + 3/12 = 7/12. A universal (but messier) approach: cross-multiply. For a/b + c/d: the result is (ad + bc)/(bd). So 1/3 + 1/4 = (1×4 + 1×3)/(3×4) = 7/12. This always works but may produce fractions that need simplifying — 2/4 + 1/6 = (12 + 4)/24 = 16/24, which simplifies to 2/3. Subtraction follows the same rules with minus instead of plus: 5/6 - 1/4 = (20 - 6)/24 = 14/24 = 7/12. The most common error is adding denominators: 1/3 + 1/4 ≠ 2/7. This mistake persists because it seems intuitive but violates what fractions represent — you can't combine thirds and fourths without first converting to the same size pieces.
Multiplication is the simplest fraction operation: multiply numerators together and denominators together. 2/3 × 4/5 = 8/15. No common denominator needed. Cross-canceling before multiplying reduces work: in 4/9 × 3/8, the 4 and 8 share a factor of 4 (giving 1 and 2), and the 3 and 9 share a factor of 3 (giving 1 and 3): 1/3 × 1/2 = 1/6. Division flips the second fraction and multiplies: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. The logic: dividing by 4/5 asks "how many groups of 4/5 fit in 2/3?" — which is the same as multiplying 2/3 by the reciprocal of 4/5. Dividing by a fraction always gives a larger result than the original (when dividing by a proper fraction), which confuses students who associate division with "making smaller." Thinking of it as "how many half-cups fit in 3 cups?" (3 ÷ 1/2 = 6) makes the concept concrete.
US construction trades measure in fractions of an inch down to 1/16" or 1/32". Reading a tape measure requires recognizing the fraction hierarchy: the longest mark between inches is 1/2", the next longest marks are 1/4" and 3/4", then 1/8" marks, and the smallest marks are 1/16". Adding measurements requires fraction arithmetic: a board at 23 5/8" plus a 1/4" kerf (saw blade width) plus another board at 15 3/16" totals 23 5/8 + 1/4 + 15 3/16 = 23 10/16 + 4/16 + 15 3/16 = 38 17/16 = 39 1/16". Machinists work in thousandths of an inch (0.001") but convert to fractions for standard tooling: a 3/8" drill bit is 0.375", a 7/16" wrench fits a 0.4375" bolt head. Pipe sizing adds confusion — a "1/2 inch pipe" has an internal diameter of approximately 0.622" and an external diameter of 0.840", because pipe sizes refer to nominal bore, not actual dimensions. These measurement conventions, while initially confusing, become second nature with practice in the respective trades.
A fraction represents a division operation: 3/4 means 3 divided by 4, or 0.75. The numerator (top number) counts how many parts you have; the denominator (bottom number) tells you how many equal parts make the whole. Proper fractions (numerator < denominator, like 3/8) represent values less than 1. Improper fractions (numerator ≥ denominator, like 7/4) represent values of 1 or greater and can be converted to mixed numbers: 7/4 = 1 3/4. Neither form is more "correct" — improper fractions are easier to compute with, while mixed numbers are more intuitive for everyday measurement (1 3/4 cups of flour is clearer than 7/4 cups). Negative fractions place the minus sign in front of the fraction, in the numerator, or in the denominator — all three positions are mathematically equivalent: -3/4 = (-3)/4 = 3/(-4).
Adding fractions with the same denominator is straightforward: add the numerators and keep the denominator. 2/7 + 3/7 = 5/7. Different denominators require finding a common denominator — ideally the LCD (least common denominator). For 1/3 + 1/4: LCD(3,4) = 12. Convert: 4/12 + 3/12 = 7/12. A universal (but messier) approach: cross-multiply. For a/b + c/d: the result is (ad + bc)/(bd). So 1/3 + 1/4 = (1×4 + 1×3)/(3×4) = 7/12. This always works but may produce fractions that need simplifying — 2/4 + 1/6 = (12 + 4)/24 = 16/24, which simplifies to 2/3. Subtraction follows the same rules with minus instead of plus: 5/6 - 1/4 = (20 - 6)/24 = 14/24 = 7/12. The most common error is adding denominators: 1/3 + 1/4 ≠ 2/7. This mistake persists because it seems intuitive but violates what fractions represent — you can't combine thirds and fourths without first converting to the same size pieces.
Multiplication is the simplest fraction operation: multiply numerators together and denominators together. 2/3 × 4/5 = 8/15. No common denominator needed. Cross-canceling before multiplying reduces work: in 4/9 × 3/8, the 4 and 8 share a factor of 4 (giving 1 and 2), and the 3 and 9 share a factor of 3 (giving 1 and 3): 1/3 × 1/2 = 1/6. Division flips the second fraction and multiplies: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. The logic: dividing by 4/5 asks "how many groups of 4/5 fit in 2/3?" — which is the same as multiplying 2/3 by the reciprocal of 4/5. Dividing by a fraction always gives a larger result than the original (when dividing by a proper fraction), which confuses students who associate division with "making smaller." Thinking of it as "how many half-cups fit in 3 cups?" (3 ÷ 1/2 = 6) makes the concept concrete.
US construction trades measure in fractions of an inch down to 1/16" or 1/32". Reading a tape measure requires recognizing the fraction hierarchy: the longest mark between inches is 1/2", the next longest marks are 1/4" and 3/4", then 1/8" marks, and the smallest marks are 1/16". Adding measurements requires fraction arithmetic: a board at 23 5/8" plus a 1/4" kerf (saw blade width) plus another board at 15 3/16" totals 23 5/8 + 1/4 + 15 3/16 = 23 10/16 + 4/16 + 15 3/16 = 38 17/16 = 39 1/16". Machinists work in thousandths of an inch (0.001") but convert to fractions for standard tooling: a 3/8" drill bit is 0.375", a 7/16" wrench fits a 0.4375" bolt head. Pipe sizing adds confusion — a "1/2 inch pipe" has an internal diameter of approximately 0.622" and an external diameter of 0.840", because pipe sizes refer to nominal bore, not actual dimensions. These measurement conventions, while initially confusing, become second nature with practice in the respective trades.
A fraction represents a division operation: 3/4 means 3 divided by 4, or 0.75. The numerator (top number) counts how many parts you have; the denominator (bottom number) tells you how many equal parts make the whole. Proper fractions (numerator < denominator, like 3/8) represent values less than 1. Improper fractions (numerator ≥ denominator, like 7/4) represent values of 1 or greater and can be converted to mixed numbers: 7/4 = 1 3/4. Neither form is more "correct" — improper fractions are easier to compute with, while mixed numbers are more intuitive for everyday measurement (1 3/4 cups of flour is clearer than 7/4 cups). Negative fractions place the minus sign in front of the fraction, in the numerator, or in the denominator — all three positions are mathematically equivalent: -3/4 = (-3)/4 = 3/(-4).
Adding fractions with the same denominator is straightforward: add the numerators and keep the denominator. 2/7 + 3/7 = 5/7. Different denominators require finding a common denominator — ideally the LCD (least common denominator). For 1/3 + 1/4: LCD(3,4) = 12. Convert: 4/12 + 3/12 = 7/12. A universal (but messier) approach: cross-multiply. For a/b + c/d: the result is (ad + bc)/(bd). So 1/3 + 1/4 = (1×4 + 1×3)/(3×4) = 7/12. This always works but may produce fractions that need simplifying — 2/4 + 1/6 = (12 + 4)/24 = 16/24, which simplifies to 2/3. Subtraction follows the same rules with minus instead of plus: 5/6 - 1/4 = (20 - 6)/24 = 14/24 = 7/12. The most common error is adding denominators: 1/3 + 1/4 ≠ 2/7. This mistake persists because it seems intuitive but violates what fractions represent — you can't combine thirds and fourths without first converting to the same size pieces.
Multiplication is the simplest fraction operation: multiply numerators together and denominators together. 2/3 × 4/5 = 8/15. No common denominator needed. Cross-canceling before multiplying reduces work: in 4/9 × 3/8, the 4 and 8 share a factor of 4 (giving 1 and 2), and the 3 and 9 share a factor of 3 (giving 1 and 3): 1/3 × 1/2 = 1/6. Division flips the second fraction and multiplies: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. The logic: dividing by 4/5 asks "how many groups of 4/5 fit in 2/3?" — which is the same as multiplying 2/3 by the reciprocal of 4/5. Dividing by a fraction always gives a larger result than the original (when dividing by a proper fraction), which confuses students who associate division with "making smaller." Thinking of it as "how many half-cups fit in 3 cups?" (3 ÷ 1/2 = 6) makes the concept concrete.
US construction trades measure in fractions of an inch down to 1/16" or 1/32". Reading a tape measure requires recognizing the fraction hierarchy: the longest mark between inches is 1/2", the next longest marks are 1/4" and 3/4", then 1/8" marks, and the smallest marks are 1/16". Adding measurements requires fraction arithmetic: a board at 23 5/8" plus a 1/4" kerf (saw blade width) plus another board at 15 3/16" totals 23 5/8 + 1/4 + 15 3/16 = 23 10/16 + 4/16 + 15 3/16 = 38 17/16 = 39 1/16". Machinists work in thousandths of an inch (0.001") but convert to fractions for standard tooling: a 3/8" drill bit is 0.375", a 7/16" wrench fits a 0.4375" bolt head. Pipe sizing adds confusion — a "1/2 inch pipe" has an internal diameter of approximately 0.622" and an external diameter of 0.840", because pipe sizes refer to nominal bore, not actual dimensions. These measurement conventions, while initially confusing, become second nature with practice in the respective trades.
A fraction represents a division operation: 3/4 means 3 divided by 4, or 0.75. The numerator (top number) counts how many parts you have; the denominator (bottom number) tells you how many equal parts make the whole. Proper fractions (numerator < denominator, like 3/8) represent values less than 1. Improper fractions (numerator ≥ denominator, like 7/4) represent values of 1 or greater and can be converted to mixed numbers: 7/4 = 1 3/4. Neither form is more "correct" — improper fractions are easier to compute with, while mixed numbers are more intuitive for everyday measurement (1 3/4 cups of flour is clearer than 7/4 cups). Negative fractions place the minus sign in front of the fraction, in the numerator, or in the denominator — all three positions are mathematically equivalent: -3/4 = (-3)/4 = 3/(-4).
Adding fractions with the same denominator is straightforward: add the numerators and keep the denominator. 2/7 + 3/7 = 5/7. Different denominators require finding a common denominator — ideally the LCD (least common denominator). For 1/3 + 1/4: LCD(3,4) = 12. Convert: 4/12 + 3/12 = 7/12. A universal (but messier) approach: cross-multiply. For a/b + c/d: the result is (ad + bc)/(bd). So 1/3 + 1/4 = (1×4 + 1×3)/(3×4) = 7/12. This always works but may produce fractions that need simplifying — 2/4 + 1/6 = (12 + 4)/24 = 16/24, which simplifies to 2/3. Subtraction follows the same rules with minus instead of plus: 5/6 - 1/4 = (20 - 6)/24 = 14/24 = 7/12. The most common error is adding denominators: 1/3 + 1/4 ≠ 2/7. This mistake persists because it seems intuitive but violates what fractions represent — you can't combine thirds and fourths without first converting to the same size pieces.
Multiplication is the simplest fraction operation: multiply numerators together and denominators together. 2/3 × 4/5 = 8/15. No common denominator needed. Cross-canceling before multiplying reduces work: in 4/9 × 3/8, the 4 and 8 share a factor of 4 (giving 1 and 2), and the 3 and 9 share a factor of 3 (giving 1 and 3): 1/3 × 1/2 = 1/6. Division flips the second fraction and multiplies: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. The logic: dividing by 4/5 asks "how many groups of 4/5 fit in 2/3?" — which is the same as multiplying 2/3 by the reciprocal of 4/5. Dividing by a fraction always gives a larger result than the original (when dividing by a proper fraction), which confuses students who associate division with "making smaller." Thinking of it as "how many half-cups fit in 3 cups?" (3 ÷ 1/2 = 6) makes the concept concrete.
US construction trades measure in fractions of an inch down to 1/16" or 1/32". Reading a tape measure requires recognizing the fraction hierarchy: the longest mark between inches is 1/2", the next longest marks are 1/4" and 3/4", then 1/8" marks, and the smallest marks are 1/16". Adding measurements requires fraction arithmetic: a board at 23 5/8" plus a 1/4" kerf (saw blade width) plus another board at 15 3/16" totals 23 5/8 + 1/4 + 15 3/16 = 23 10/16 + 4/16 + 15 3/16 = 38 17/16 = 39 1/16". Machinists work in thousandths of an inch (0.001") but convert to fractions for standard tooling: a 3/8" drill bit is 0.375", a 7/16" wrench fits a 0.4375" bolt head. Pipe sizing adds confusion — a "1/2 inch pipe" has an internal diameter of approximately 0.622" and an external diameter of 0.840", because pipe sizes refer to nominal bore, not actual dimensions. These measurement conventions, while initially confusing, become second nature with practice in the respective trades.
A fraction represents a division operation: 3/4 means 3 divided by 4, or 0.75. The numerator (top number) counts how many parts you have; the denominator (bottom number) tells you how many equal parts make the whole. Proper fractions (numerator < denominator, like 3/8) represent values less than 1. Improper fractions (numerator ≥ denominator, like 7/4) represent values of 1 or greater and can be converted to mixed numbers: 7/4 = 1 3/4. Neither form is more "correct" — improper fractions are easier to compute with, while mixed numbers are more intuitive for everyday measurement (1 3/4 cups of flour is clearer than 7/4 cups). Negative fractions place the minus sign in front of the fraction, in the numerator, or in the denominator — all three positions are mathematically equivalent: -3/4 = (-3)/4 = 3/(-4).
Adding fractions with the same denominator is straightforward: add the numerators and keep the denominator. 2/7 + 3/7 = 5/7. Different denominators require finding a common denominator — ideally the LCD (least common denominator). For 1/3 + 1/4: LCD(3,4) = 12. Convert: 4/12 + 3/12 = 7/12. A universal (but messier) approach: cross-multiply. For a/b + c/d: the result is (ad + bc)/(bd). So 1/3 + 1/4 = (1×4 + 1×3)/(3×4) = 7/12. This always works but may produce fractions that need simplifying — 2/4 + 1/6 = (12 + 4)/24 = 16/24, which simplifies to 2/3. Subtraction follows the same rules with minus instead of plus: 5/6 - 1/4 = (20 - 6)/24 = 14/24 = 7/12. The most common error is adding denominators: 1/3 + 1/4 ≠ 2/7. This mistake persists because it seems intuitive but violates what fractions represent — you can't combine thirds and fourths without first converting to the same size pieces.
Multiplication is the simplest fraction operation: multiply numerators together and denominators together. 2/3 × 4/5 = 8/15. No common denominator needed. Cross-canceling before multiplying reduces work: in 4/9 × 3/8, the 4 and 8 share a factor of 4 (giving 1 and 2), and the 3 and 9 share a factor of 3 (giving 1 and 3): 1/3 × 1/2 = 1/6. Division flips the second fraction and multiplies: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. The logic: dividing by 4/5 asks "how many groups of 4/5 fit in 2/3?" — which is the same as multiplying 2/3 by the reciprocal of 4/5. Dividing by a fraction always gives a larger result than the original (when dividing by a proper fraction), which confuses students who associate division with "making smaller." Thinking of it as "how many half-cups fit in 3 cups?" (3 ÷ 1/2 = 6) makes the concept concrete.
US construction trades measure in fractions of an inch down to 1/16" or 1/32". Reading a tape measure requires recognizing the fraction hierarchy: the longest mark between inches is 1/2", the next longest marks are 1/4" and 3/4", then 1/8" marks, and the smallest marks are 1/16". Adding measurements requires fraction arithmetic: a board at 23 5/8" plus a 1/4" kerf (saw blade width) plus another board at 15 3/16" totals 23 5/8 + 1/4 + 15 3/16 = 23 10/16 + 4/16 + 15 3/16 = 38 17/16 = 39 1/16". Machinists work in thousandths of an inch (0.001") but convert to fractions for standard tooling: a 3/8" drill bit is 0.375", a 7/16" wrench fits a 0.4375" bolt head. Pipe sizing adds confusion — a "1/2 inch pipe" has an internal diameter of approximately 0.622" and an external diameter of 0.840", because pipe sizes refer to nominal bore, not actual dimensions. These measurement conventions, while initially confusing, become second nature with practice in the respective trades.
→ Simplify always. Divide both parts by their GCD.
→ For division: Flip and multiply — never divide directly.
→ Memorize common equivalents: 1/4=0.25, 1/3=0.333, 1/2=0.5, 3/4=0.75.
→ For construction: Tape measures use 1/16" increments. Know your fractions.
See also: Percentage · Ratio · Average · Scientific