The four operations
Multiplying is the easy one: multiply the numerators, multiply the denominators. ⅔ × ⅗ = 6/15, which simplifies to 2/5.
Dividing is multiplying by the reciprocal — flip the second fraction and multiply. ½ ÷ ¾ becomes ½ × 4/3 = 4/6 = 2/3.
Adding and subtracting need a common denominator first. The reliable method is to cross-multiply: a/b + c/d = (ad + cb) / bd. So ½ + ⅓ = (3 + 2)/6 = 5/6. It does not always give the lowest common denominator, but simplifying afterwards fixes that.
Simplifying
Divide the numerator and denominator by their greatest common divisor. For 45/60 the GCD is 15, giving 3/4. The calculator does this automatically, which is why an answer may look different from your intermediate working while being the same value.
Mixed numbers
An improper fraction like 41/24 is 24 ÷ 24 = 1 remainder 17, so it becomes 1 17/24. To go the other way, multiply the whole number by the denominator and add the numerator: 2 3/8 = (2 × 8 + 3)/8 = 19/8.
Decimals to fractions
Count the decimal places, put the digits over that power of ten, then simplify. 2.375 has three places, so it is 2375/1000 — which reduces to 19/8, or 2 3/8. Repeating decimals like 0.333… cannot be captured exactly this way; they need the algebraic method, which is why 1/3 shows as 0.3333 rather than converting back cleanly.
Working with parts of a whole more often than fractions? The percentage calculator covers the same ideas in the notation most people actually use day to day.