Multiply numerators and denominators
For a/b×c/d, multiply ac over bd, with b and d nonzero. Unlike addition, multiplication does not require a common denominator. Simplify before or after multiplication.
Cross-cancel only factors, not terms joined by addition. Preserve negative signs and reduce the final fraction to lowest terms.
Divide by the reciprocal
For a/b÷c/d, multiply a/b by d/c. The divisor c/d must be nonzero, so c cannot be zero. The reciprocal changes the divisor only.
Keep-change-flip is a memory aid, but the meaning is multiplication by the reciprocal. This supports verification by multiplying quotient×divisor to recover the dividend.
Three complete examples
Input: 3/4×10/21. Formula: ac/bd. Calculation: cancel 3 with 21 to 1 and 7; cancel 10 with 4 to 5 and 2; 5/(2×7)=5/14. Result: 5/14. Interpretation: cancellation reduced intermediate values.
Input: 5/6÷15/8. Formula: 5/6×8/15. Calculation: cancel 5 with 15 and 8 with 6; 1×4/(3×3)=4/9. Result: 4/9. Interpretation: multiplying 4/9×15/8 returns 5/6.
Input: −2 1/3×3/7. Formula: convert mixed number. Calculation: −7/3×3/7=−1. Result: −1. Interpretation: one negative factor gives a negative product.
Handle mixed numbers and signs
Convert mixed numbers to improper fractions before operating. A negative mixed number such as −2 1/3 means −7/3, not −5/3.
An even number of negative factors gives a positive product; an odd number gives a negative. Denominators are conventionally written positive.
Interpret fraction operations
Multiplication can represent a part of a part: 2/3 of 3/4 is 1/2. Division asks how many groups of the divisor fit in the dividend.
Units matter. Dividing 3/4 liter into 1/8-liter servings gives 6 servings; the numeric quotient needs the contextual unit.
Check reasonableness
Multiplying two positive proper fractions produces a value smaller than either. Dividing by a positive proper fraction increases a positive value. These checks reveal inverted reciprocals.
Use exact fraction arithmetic for verification rather than rounded decimals. Confirm denominators are nonzero and the result is reduced.
Cross-cancellation is division by a common factor in a product. For 14/15×25/21, cancel 14 with 21 by 7 and 25 with 15 by 5, leaving 2/3×5/3=10/9. Every cancellation must divide a numerator and denominator by the same nonzero factor.
Zero multiplied by a valid fraction is zero. Zero divided by a nonzero fraction is zero. A nonzero fraction divided by zero is undefined, and 0÷0 is also undefined. Check the divisor before applying the reciprocal shortcut.
For area scaling, fractional length factors multiply in each dimension. A rectangle scaled to 1/2 width and 2/3 height has area factor 1/3. This illustrates why multiplying fractions can make a quantity smaller without representing subtraction.
For repeated division, parentheses and left-to-right order matter: 1/2÷1/3÷1/4 equals (1/2÷1/3)÷1/4=6, while 1/2÷(1/3÷1/4)=3/8. Write the grouping explicitly.
A decimal check should be secondary. For 5/14, approximately 0.3571 confirms magnitude but cannot prove the exact fraction. Exact multiplication by the divisor is the stronger verification for a division result.
Use the Fraction Calculator
Enter two valid fractions and select multiplication or division. The calculator displays an exact simplified result and decimal reference.
It cannot decide the correct operation in a word problem. Translate the situation, track units, and reject division by zero before calculating.
Multiplying whole numbers and fractions uses the whole number over one. For 6×5/8, write 6/1×5/8, cancel 6 with 8 by two, and obtain 15/4=3 3/4. Converting to a mixed number is a display choice after the exact improper fraction is known; do not mix formats midway and lose the operation.
To divide a mixed quantity, convert every mixed number first. For 2 1/4÷3/8, use 9/4×8/3. Cancel 8 with 4 to get 2, and 9 with 3 to get 3, producing 6. A decimal approximation is unnecessary. Substitution check: six groups of 3/8 total 18/8=9/4.
Fraction bars act as grouping symbols. The expression 1+1/2×3/4 follows multiplication before addition, whereas (1+1/2)×3/4 first makes 3/2 and then yields 9/8. In typed work, add parentheses whenever a whole sum or difference is a numerator, denominator, dividend, or divisor. Layout ambiguity is not an arithmetic rule.
Units can cancel just like numeric factors when the setup is valid. Dividing 3/4 mile by 1/8 mile per lap gives (3/4 mile)÷(1/8 mile/lap)=6 laps. Writing units through the reciprocal shows why the result is a count. If the units do not reduce to the expected type, the ratio may be reversed.
Error diagnosis can work backward. If a quotient is unexpectedly tiny when dividing a positive quantity by a proper fraction, check whether the divisor was multiplied without taking its reciprocal. If numerator and denominator share a factor in the final answer, simplification is unfinished. If a denominator is zero, stop: no amount of cancellation makes the expression defined.
Worked audit: calculate 1 3/5÷(−2/3). Convert the mixed number to 8/5, multiply by the reciprocal −3/2, cancel 8 with 2 to obtain −4, then multiply −4×3/5=−12/5=−2 2/5. Verification: −12/5×−2/3=24/15=8/5, recovering the dividend. The negative quotient is reasonable because signs differ, and its magnitude exceeds 8/5 because division by 2/3 increases a positive magnitude. Before finalizing any result, confirm all mixed numbers were converted, the divisor is nonzero, only the divisor was reciprocated, cancellation used factors across multiplication, signs were preserved, and the answer is reduced. Attach units and interpret whether the quotient represents group count, rate, or measure. Keep the exact fraction through verification, then provide a decimal only as a secondary approximation. These steps expose the most common shortcut errors without requiring oversized intermediate products.