Fraction Decomposition Calculator
Break fractions down using three useful methods: mixed number conversion, Egyptian fraction decomposition, and partial fraction decomposition for simple rational expressions.
Choose a decomposition method
Use this calculator to convert an improper fraction into a mixed number, split a fraction into a sum of unit fractions using the Egyptian method, or decompose a rational expression into partial fractions when the denominator has simple linear factors.
Mixed number form: a/b = q + r/b, where q is the whole-number quotient.
A fraction decomposition calculator helps you break a number or expression into simpler parts. Depending on context, �decomposition� can mean different things:
- Mixed-number decomposition of an improper fraction (e.g., 175=325\tfrac{17}{5} = 3 \tfrac{2}{5}517?=352?).
- Egyptian fraction decomposition into a sum of unit fractions (e.g., 56=12+13\tfrac{5}{6} = \tfrac{1}{2} + \tfrac{1}{3}65?=21?+31?).
- Algebraic partial fraction decomposition of a rational function (e.g., 3x+5×2?1=4x?1?1x+1\tfrac{3x+5}{x^2-1} = \tfrac{4}{x-1} – \tfrac{1}{x+1}x2?13x+5?=x?14??x+11?).
This guide explains each method, the formulas behind it, and worked examples you can follow without a tool.
1) Mixed-Number Decomposition
Goal: Convert an improper fraction ab\tfrac{a}{b}ba? (with a?ba \ge ba?b) into qrbq \tfrac{r}{b}qbr?, where
q=?a/b?q = \lfloor a/b \rfloorq=?a/b? and r=a?qbr = a – qbr=a?qb with 0?r<b0 \le r < b0?r<b.
Steps
- Divide aaa by bbb to get integer quotient qqq and remainder rrr.
- Write ab=q+rb\tfrac{a}{b} = q + \tfrac{r}{b}ba?=q+br?.
- Reduce rb\tfrac{r}{b}br? if possible.
Example
175?17=5?3+2?325\tfrac{17}{5} \Rightarrow 17 = 5 \cdot 3 + 2 \Rightarrow 3 \tfrac{2}{5}517??17=5?3+2?352?.
Tip: Always simplify the fractional remainder by dividing numerator and denominator by gcd?(r,b)\gcd(r,b)gcd(r,b).
2) Egyptian Fraction Decomposition (Sum of Unit Fractions)
Goal: Express ab\tfrac{a}{b}ba? as a sum of distinct unit fractions ?1ni\sum \tfrac{1}{n_i}?ni?1?.
Greedy Algorithm (Fibonacci�Sylvester)
- Compute n=?ba?n = \left\lceil \tfrac{b}{a} \right\rceiln=?ab??.
- Write ab=1n+(ab?1n)\tfrac{a}{b} = \tfrac{1}{n} + \left(\tfrac{a}{b} – \tfrac{1}{n}\right)ba?=n1?+(ba??n1?).
- Reduce the remainder to a single fraction and repeat until the remainder is zero.
Example
Decompose 57\tfrac{5}{7}75?:
- ?7/5?=2?57=12+(57?12)=12+314.\lceil 7/5 \rceil = 2 \Rightarrow \tfrac{5}{7} = \tfrac{1}{2} + \left(\tfrac{5}{7} – \tfrac{1}{2}\right) = \tfrac{1}{2} + \tfrac{3}{14}.?7/5?=2?75?=21?+(75??21?)=21?+143?.
- Next, 314\tfrac{3}{14}143?: ?14/3?=5?314=15+(314?15)=15+170.\lceil 14/3 \rceil = 5 \Rightarrow \tfrac{3}{14} = \tfrac{1}{5} + \left(\tfrac{3}{14} – \tfrac{1}{5}\right) = \tfrac{1}{5} + \tfrac{1}{70}.?14/3?=5?143?=51?+(143??51?)=51?+701?.
- Final: 57=12+15+170.\tfrac{5}{7} = \tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{70}.75?=21?+51?+701?.
Notes
- The greedy method always terminates but not always with the fewest terms.
- Alternative methods (Engel expansion, optimized search) can minimize terms or denominators.
3) Partial Fraction Decomposition (Algebra)
Goal: Decompose a rational function P(x)Q(x)\tfrac{P(x)}{Q(x)}Q(x)P(x)? into a sum of simpler fractions that are easier to integrate or manipulate.
Preconditions
- P(x)P(x)P(x), Q(x)Q(x)Q(x) are polynomials with real coefficients.
- If deg?P?deg?Q\deg P \ge \deg QdegP?degQ, first perform polynomial long division: P(x)Q(x)=S(x)+R(x)Q(x),deg?R<deg?Q.\frac{P(x)}{Q(x)} = S(x) + \frac{R(x)}{Q(x)}, \quad \deg R < \deg Q.Q(x)P(x)?=S(x)+Q(x)R(x)?,degR<degQ.
- Factor Q(x)Q(x)Q(x) into linear factors (x?a)(x-a)(x?a) and irreducible quadratics (x2+bx+c,??b2?4c<0)(x^2+bx+c,\; b^2-4c<0)(x2+bx+c,b2?4c<0).
Templates
- Distinct linear factors: P(x)(x?a1)?(x?an)=?k=1nAkx?ak.\frac{P(x)}{(x-a_1)\cdots(x-a_n)} = \sum_{k=1}^n \frac{A_k}{x-a_k}.(x?a1?)?(x?an?)P(x)?=k=1?n?x?ak?Ak??.
- Repeated linear (x?a)m(x-a)^m(x?a)m: P(x)(x?a)m=?k=1mAk(x?a)k.\frac{P(x)}{(x-a)^m} = \sum_{k=1}^{m} \frac{A_k}{(x-a)^k}.(x?a)mP(x)?=k=1?m?(x?a)kAk??.
- Irreducible quadratic x2+bx+cx^2+bx+cx2+bx+c: P(x)x2+bx+c=Ax+Bx2+bx+c.\frac{P(x)}{x^2+bx+c} = \frac{Ax+B}{x^2+bx+c}.x2+bx+cP(x)?=x2+bx+cAx+B?.
- Repeated irreducible quadratic (x2+bx+c)m(x^2+bx+c)^m(x2+bx+c)m: ?k=1mAkx+Bk(x2+bx+c)k.\sum_{k=1}^{m} \frac{A_k x + B_k}{(x^2+bx+c)^k}.k=1?m?(x2+bx+c)kAk?x+Bk??.
Solving Coefficients (Two Common Ways)
- Multiply both sides by Q(x)Q(x)Q(x), expand, and equate coefficients.
- Use the cover-up method for distinct linear terms to find residues quickly; finish with coefficient matching if needed.
Worked Example (Distinct Linear) 3x+5×2?1=3x+5(x?1)(x+1)=Ax?1+Bx+1.\frac{3x+5}{x^2-1}=\frac{3x+5}{(x-1)(x+1)}=\frac{A}{x-1}+\frac{B}{x+1}.x2?13x+5?=(x?1)(x+1)3x+5?=x?1A?+x+1B?.
Multiply by (x?1)(x+1)(x-1)(x+1)(x?1)(x+1): 3x+5=A(x+1)+B(x?1)=(A+B)x+(A?B).3x+5=A(x+1)+B(x-1)=(A+B)x+(A-B).3x+5=A(x+1)+B(x?1)=(A+B)x+(A?B).
Match coefficients:
?A+B=3, A?B=5?A=4, B=?1\,A+B=3,\ A-B=5 \Rightarrow A=4,\ B=-1A+B=3, A?B=5?A=4, B=?1.
Hence: 3x+5×2?1=4x?1?1x+1.\frac{3x+5}{x^2-1}=\frac{4}{x-1}-\frac{1}{x+1}.x2?13x+5?=x?14??x+11?.
4) Prime-Factor-Based Decomposition (Optional)
Sometimes �decomposition� refers to factoring the numerator/denominator and simplifying:
- Compute prime factorizations of aaa and bbb.
- Cancel common factors to reduce ab\tfrac{a}{b}ba? to lowest terms.
- Use the simplified form for any of the methods above.
Example
84210=22?3?72?3?5?7=25.\tfrac{84}{210} = \tfrac{2^2 \cdot 3 \cdot 7}{2 \cdot 3 \cdot 5 \cdot 7} = \tfrac{2}{5}.21084?=2?3?5?722?3?7?=52?.
Comparison of Methods
| Method | Input Type | Output | Typical Use |
|---|---|---|---|
| Mixed Number | Improper fraction a/ba/ba/b | qrbq \tfrac{r}{b}qbr? | Basic arithmetic, education |
| Egyptian Fraction | Proper fraction a/ba/ba/b | ?1/ni\sum 1/n_i?1/ni? | Number theory, historical forms |
| Partial Fractions | Rational function P(x)/Q(x)P(x)/Q(x)P(x)/Q(x) | Sum of simple rational terms | Calculus, Laplace transforms |
| Prime-Factor Simplification | Any fraction | Reduced form | Preprocessing, simplification |
Common Mistakes and How to Avoid Them
- Skipping reduction first: Always simplify ab\tfrac{a}{b}ba? before decomposing.
- Mixing units or formats: Keep everything in exact fractions when possible to prevent rounding errors.
- Partial fractions without division: If deg?P?deg?Q\deg P \ge \deg QdegP?degQ, perform long division first.
- Egyptian fractions with duplicates: Terms must be distinct unit fractions.
- Sign errors: Track minus signs carefully when recombining to verify the original fraction.
Frequently Asked Questions
Is there only one Egyptian fraction for a given rational number?
No. There are many valid decompositions; the greedy algorithm yields one correct solution but not necessarily the shortest.
Do I always need to factor the denominator for partial fractions?
Yes, at least conceptually. You must know the factor structure (linear vs irreducible quadratic) to set up the right template.
What if I can�t factor Q(x)Q(x)Q(x) over the reals?
Use irreducible quadratic factors over R\mathbb{R}R. Over C\mathbb{C}C, everything factors linearly.
Can mixed-number and Egyptian methods be combined?
Yes. Convert an improper fraction to a mixed number first; then decompose the proper fractional remainder as Egyptian fractions.
Conclusion
A fraction decomposition calculator can mean different tools depending on your goal: converting to a mixed number, expressing a rational as a sum of unit fractions, or performing algebraic partial fractions. Understanding each method lets you verify a calculator�s output, choose the right approach for your task, and avoid common pitfalls.
