The four operations
Multiplication is the easy one and it confuses people precisely because it is easier than addition — you multiply the tops and the bottoms and you are done, no common denominator required. Division is multiplication in disguise: dividing by 2/3 is the same as multiplying by 3/2, because dividing by a fraction asks "how many of these fit inside", and smaller pieces fit more times.
Simplifying and mixed numbers
To simplify, divide numerator and denominator by their greatest common divisor. Finding it is quick with the Euclidean algorithm: repeatedly replace the larger number with the remainder of dividing it by the smaller, until one becomes zero.
Worked example
Simplify 84/126. GCD: 126 mod 84 = 42, then 84 mod 42 = 0, so the GCD is 42. Dividing both parts gives 2/3.
An improper fraction such as 19/6 converts to a mixed number by dividing: 19 ÷ 6 = 3 remainder 1, so it is 3 1/6. Improper fractions are usually easier to compute with; mixed numbers are usually easier to picture. Convert at the end, not the beginning.
Where fractions beat decimals
Decimals feel more modern, but fractions remain the better tool in several everyday situations — and not only in maths lessons.
- They are exact. One third is precisely 1/3; as a decimal it is 0.333… forever, and any rounding introduces error that compounds through a calculation.
- Cooking scales cleanly. Halving 3/4 cup is 3/8. Halving 0.1875 litres is arithmetic nobody wants at a hob.
- Imperial measurement is built on them. Spanners, drill bits, timber and pipe fittings are sized in sixteenths and thirty-seconds of an inch.
- Music is fractional. Half notes, quarter notes and time signatures are ratios, and reading them as decimals helps nobody.
- Odds and probability are usually clearer as fractions: 1/6 for a die is more intuitive than 0.1667.
Scaling a recipe
A recipe serving 4 needs 3/4 cup of rice. For 6 people, multiply by 6/4 = 3/2: 3/4 × 3/2 = 9/8 = 1 1/8 cups. Every step stays exact, and the answer is something you can actually measure.
Fractions, decimals and percentages
All three describe the same thing. A fraction becomes a decimal by dividing top by bottom, and a decimal becomes a percentage by multiplying by 100.
- A fraction terminates as a decimal only if its simplified denominator has no prime factors other than 2 and 5. That is why 1/8 = 0.125 exactly but 1/3 = 0.333… forever.
- To convert a decimal back, put the digits over the matching power of ten and simplify: 0.35 = 35/100 = 7/20.
- Repeating decimals are handled by algebra: if x = 0.777…, then 10x = 7.777…, so 9x = 7 and x = 7/9.