Treat each ratio as one fraction
A proportion states a/b=c/d with nonzero denominators. The entries may themselves be fractions, but parentheses must preserve which expression occupies each position.
Cross-multiplication gives ad=bc because multiplying both sides by bd clears the outer denominators. Record restrictions before canceling.
Simplify exact fractions first
Convert mixed numbers to improper fractions and reduce factors when helpful. Do not replace exact fractions with rounded decimals, which can change a clean solution.
Complex fractions can be simplified by multiplying numerator and denominator by a common denominator, or by handling the outer proportion carefully.
Three complete examples
Input: (1/2)/x=(3/4)/6. Formula: (1/2)×6=x×(3/4). Calculation: 3=3x/4; x=4. Result: x=4. Interpretation: substitution gives 1/8 on both sides.
Input: x/(2/3)=9/(4/5). Formula: x×(4/5)=9×(2/3). Calculation: 4x/5=6; x=30/4=15/2. Result: 7.5. Interpretation: exact fraction is 15/2.
Input: (3/8)/(5/6)=x/10. Formula: (3/8)×10=(5/6)x. Calculation: 15/4=5x/6; x=9/2. Result: 4.5. Interpretation: both ratios equal 9/20.
Track restrictions and zero
No denominator can equal zero. If a variable appears in a denominator, exclude values that make it zero before solving.
Cross-products can both be zero in degenerate cases, producing many solutions rather than one. A contradiction produces no solution. Do not divide by a variable expression without considering whether it can be zero.
Use units and order
Corresponding quantities must appear in the same order on both sides. Convert incompatible units or include conversion factors before solving.
A proportion is valid only when the relationship is proportional. Fixed fees, thresholds, changing rates, and nonlinear geometry may invalidate the model.
Verify by substitution
Insert the exact solution into the original proportion, check every denominator, and simplify both sides. Verifying only the cross-multiplied equation can overlook an excluded value.
Estimate magnitude as a reasonableness check. Preserve the fraction until the final display and label any decimal rounding.
Clearing nested denominators can reduce clutter. Multiply an entire equation by the least common multiple of all numeric denominators, applying it to every term. Skipping one term changes the equation. Then solve the resulting integer-coefficient relationship.
If a variable appears in several denominator expressions, record every restriction separately. A candidate created after multiplying by those expressions must be checked against all restrictions, since multiplication can hide where the original statement was undefined.
Scale-factor reasoning can solve some problems faster than cross-products. If 2/3 becomes 10/9, the factor is (10/9)÷(2/3)=5/3; apply the same factor to the corresponding term. Cross-products then verify the result.
Units can form fractional rates, such as 3/4 cup per 2/3 batch. Dividing gives 9/8 cups per batch. Keep compound units attached so inverting a rate also inverts its units; otherwise a numerically correct reciprocal may answer the opposite question.
A graph of a direct proportion passes through the origin with constant slope. If data include a nonzero intercept or changing slope, forcing a proportion produces inconsistent scale factors. Check the model before solving an unknown from one pair.
Use the Proportion Calculator
Enter three known positions and one unknown. For fractional entries, simplify them first or perform the exact algebra shown in the guide.
The calculator solves the equality supplied; it cannot decide whether a word problem truly describes proportional quantities.
Consider (x+1)/(3/4)=2/(5/6). Cross-products give (x+1)(5/6)=3/2. Multiply by 6 to obtain 5(x+1)=9, so x=4/5. Verification uses the original expression: (9/5)/(3/4)=12/5 and 2/(5/6)=12/5. Parentheses protect the entire numerator.
A variable denominator example needs an exclusion: 3/(x−1)=1/2 requires x≠1. Cross-multiplication gives 6=x−1 and x=7, which satisfies the restriction. If algebra had produced x=1, it would be rejected even if a transformed equation seemed to accept it. Original-domain checks are part of the solution.
Proportion tables help preserve correspondence. Put like quantities in the same row or column and attach units, such as cups over servings on both sides. Reversing both ratios is valid; reversing only one is not. Before cross-multiplying, read each ratio aloud to confirm identical order and meaning.
Similarity problems can involve fractional side lengths, but corresponding sides must actually belong to similar figures. If scale factor is 3/2 and an original side is 5/6 meter, the matching side is 5/6×3/2=5/4 meter. The arithmetic does not establish similarity; geometric conditions supply that premise.
If both cross-products simplify to the same expression for every permitted variable value, the equation is an identity with infinitely many solutions in its domain. If constants conflict, there is no solution. A solver designed for exactly one missing numeric position may not represent these algebraic cases, so manual classification remains necessary.
Worked audit: solve (2/5)/(x−1)=(3/4)/(5/2). First record x≠1. The right side is 3/4×2/5=3/10. Thus (2/5)/(x−1)=3/10. Cross-multiply 10×2/5=3(x−1), giving 4=3x−3 and x=7/3. Substitution gives x−1=4/3, so (2/5)÷(4/3)=3/10, matching the right side; the solution respects the restriction. Before solving, parenthesize compound positions, convert mixed numbers, keep fractions exact, align corresponding units, record every forbidden denominator value, and justify proportionality. After algebra, substitute into the original statement rather than only the cleared equation. Classify identities and contradictions instead of forcing a single answer. If a decimal is requested, derive it from the verified exact fraction and label rounding. This workflow separates domain, modeling, algebra, and presentation errors that cross-multiplication alone cannot detect.