SolveMathAI

Fraction calculator

Two fractions in, a simplified answer out, with the common denominator shown.

Enter to solve · Shift+Enter for a new line

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  • Every step shown
  • Arithmetic to calculus

Multiplying fractions is the easy one: multiply across. Adding is the one that needs care, because the pieces have to be the same size first. The calculator above does both instantly and shows the common denominator it used; the solver below handles mixed numbers, three or more fractions, and fractions inside word problems.

How to do it by hand

Four operations, two different habits: multiply and divide straight across, add and subtract only after the denominators match.

  1. Multiply

    Multiply the tops, multiply the bottoms, then simplify. Cancel common factors before multiplying and the numbers stay small.

  2. Divide

    Flip the second fraction and multiply. Dividing by 2/5 is multiplying by 5/2.

  3. Add or subtract

    Find the least common denominator, rewrite both fractions with it, then add or subtract the tops only. The bottom never changes.

  4. Simplify

    Divide top and bottom by their greatest common divisor. 84/126 both divide by 42, giving 2/3.

Worked examples

Each one is solved the way the solver solves it: the rule first, then the line.

Add 2/3 + 1/4 and simplify

23+14\frac{2}{3} + \frac{1}{4}
  1. Least common denominator
    lcm⁡(3,4)=12\operatorname{lcm}(3,4) = 12
  2. Rewrite both fractions
    23=812,14=312\frac{2}{3} = \frac{8}{12}, \qquad \frac{1}{4} = \frac{3}{12}
  3. Add the numerators
    812+312=1112\frac{8}{12} + \frac{3}{12} = \frac{11}{12}
  4. Already in lowest terms
    gcd⁡(11,12)=1\gcd(11,12) = 1
Answer
1112\frac{11}{12}

Divide 3/4 by 2/5 and simplify

34÷25\frac{3}{4} \div \frac{2}{5}
  1. Multiply by the reciprocal
    34÷25=34×52\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2}
  2. Multiply across
    3×54×2=158\frac{3 \times 5}{4 \times 2} = \frac{15}{8}
  3. As a mixed number
    158=178\frac{15}{8} = 1\tfrac{7}{8}
Answer
158=178\frac{15}{8} = 1\tfrac{7}{8}

Simplify the fraction 84/126

84126\frac{84}{126}
  1. Greatest common divisor
    84=22⋅3⋅7,126=2⋅32⋅7  ⇒  gcd⁡=4284 = 2^2 \cdot 3 \cdot 7, \quad 126 = 2 \cdot 3^2 \cdot 7 \;\Rightarrow\; \gcd = 42
  2. Divide top and bottom
    84÷42126÷42=23\frac{84 \div 42}{126 \div 42} = \frac{2}{3}
Answer
23\frac{2}{3}

Multiply the mixed numbers 2 1/2 and 1 3/5

212×1352\tfrac{1}{2} \times 1\tfrac{3}{5}
  1. Improper fractions first
    212=52,135=852\tfrac{1}{2} = \frac{5}{2}, \qquad 1\tfrac{3}{5} = \frac{8}{5}
  2. Cancel, then multiply
    52×85=82=4\frac{5}{2} \times \frac{8}{5} = \frac{8}{2} = 4
Answer
44

Never multiply mixed numbers as they stand: 2 × 1 plus ½ × ⅗ is not the answer.

Solved problems

Asked here and worked through in full. Open one for every step.

  1. Convert 0.45 to a fraction in simplest form.
    Answer 920\frac{9}{20}
  2. Convert 0.45 to a fraction in simplest form. Which set of lengths forms a right triangle? A. 5, 6, 7 B. 6, 8, 10 C. 3, 4, 8 D. 4, 5, 6
    Answer Fraction: 920; Right triangle: B (6, 8, 10)\text{Fraction: } \frac{9}{20}; \text{ Right triangle: B (6, 8, 10)}

Where marks get lost

Adding the denominators

Two thirds plus one quarter is not three sevenths. Only the numerators are added, and only once the denominators match.

Forgetting to flip when dividing

Dividing by a fraction multiplies by its reciprocal. Dividing by 1/2 makes a number bigger, not smaller.

Leaving the answer unsimplified

Most marking schemes want lowest terms. Divide top and bottom by their greatest common divisor at the end.

Questions

How do I divide fractions?

Keep the first fraction, flip the second, and multiply. 3/4 ÷ 2/5 becomes 3/4 × 5/2 = 15/8. Dividing by a fraction is the same as multiplying by its reciprocal.

How do I add fractions?

Give both the same denominator, add the numerators, and leave the denominator alone. 1/4 + 2/4 = 3/4. Only the top changes.

How do I add fractions with different denominators?

Rewrite both over a common denominator first. For 1/3 + 1/4 use 12: 4/12 + 3/12 = 7/12. Multiplying the two denominators always gives a denominator that works.

How do I simplify a fraction?

Divide the top and the bottom by the largest number that goes into both. 18/24 divides by 6 to give 3/4. When the only common factor left is 1, it is in lowest terms.

How do I divide a fraction by a whole number?

Write the whole number as a fraction over 1 and then flip it. 3/5 ÷ 4 is 3/5 × 1/4 = 3/20. Dividing by a whole number always makes a fraction smaller.

What is the least common denominator?

The smallest number both denominators divide into. For 3 and 4 it is 12. Multiplying the two denominators always works too, but it leaves bigger numbers to simplify afterwards.

How do I turn a mixed number into a fraction?

Multiply the whole number by the denominator and add the numerator. 2 1/2 becomes (2 × 2 + 1)/2 = 5/2. Convert before multiplying or dividing, never after.

How do I compare fractions?

Put them over the same denominator and compare the numerators, or cross-multiply: 3/4 against 5/7 gives 21 against 20, so 3/4 is larger. Comparing the decimals works too.

Can it handle negative fractions?

Yes. Put the sign on the numerator and carry it through as normal; the calculator accepts a minus on either fraction.