Positive Rational Number Operations
Fractions, decimals and percents are three costumes for the same number. Choose whichever makes the arithmetic easiest.
Half a pizza is left on one tray, and a third of a pizza on another. How much pizza is that?
Two slices — but that is not an amount. A half and a third are different sizes.
"Two out of five" is wrong too: the five pieces are not equal, so counting them measures nothing.
Cut both pizzas into sixths and the problem dissolves. The half is 3 sixths, the third is 2 sixths.
Now the pieces are the same size, so counting works: 5 sixths.
That is the whole reason for a common denominator. It is not a rule about fractions — it is the act of cutting both amounts into pieces of the same size so they can be counted together.
Adding and subtracting needs a common denominator
You can only add pieces that are the same size. Thirds and fourths are different sizes, so first rename both as twelfths.
23 + 14 = 812 + 312 = 1112.
To find the LCD, list multiples of the larger denominator until one is divisible by the other: 4, 8, 12 — and 12 works for 3.
A picture that helps
Imagine slicing a pizza into thirds and another into fourths. You cannot say how much pizza you have in one clean fraction until both pizzas are cut the same way — into twelfths.
Multiplying is easier than adding
Multiplying fractions needs no common denominator. Just multiply across the top and across the bottom.
23 × 14 = 212 = 16.
Better: cancel before you multiply. 23 × 14 — the 2 and the 4 share a factor of 2, so it becomes 13 × 12 = 16. Smaller numbers, less to simplify later.
Watch out
The rule "common denominator" belongs to + and - only. Students often drag it into multiplication, where it just makes extra work.
Dividing: multiply by the reciprocal
ab ÷ cd = ab × dc.
Why? Because "how many 14s fit inside 12?" is asking how many small pieces make a big one — the answer 2 is bigger than either number.
Keep it, Change it, Flip it: keep the first fraction, change ÷ to ×, flip the second.
Fractions to decimals and back
To make a fraction a decimal, divide the top by the bottom: 58 = 5 ÷ 8 = 0.625.
A fraction terminates exactly when its denominator (in lowest terms) has only 2s and 5s as prime factors. 58 terminates because 8 = 23. 13 repeats because 3 is neither.
To make a terminating decimal a fraction, read it aloud: 0.625 is "six hundred twenty-five thousandths" = 6251000 = 58.
Order of operations still rules
Parentheses, then exponents, then multiply/divide left to right, then add/subtract left to right.
Inside a complex fraction, the bar acts like a giant pair of parentheses: finish the whole top and the whole bottom before dividing.
Worked examples
Mixed numbers
Compute 312 - 134.
- Turn both into improper fractions: 72 and 74 312 = 3 × 2 + 12 = 72.
- Common denominator 4: 144 - 74 Rename halves as quarters.
- 14 - 74 = 74 Subtract the numerators; the denominator stays.
- 74 = 134 7 ÷ 4 = 1 remainder 3.
134
Division that surprises people
How many 38-cup scoops are in 214 cups?
- This is 214 ÷ 38 "How many fit inside" is division.
- = 94 ÷ 38 = 94 × 83 Keep, change, flip.
- Cancel: 94 × 83 = 31 × 21 = 6 9 and 3 share 3; 8 and 4 share 4.
6 scoops.
Mixing forms
Compute 34 + 0.35 + 20% as a decimal.
- Put everything in one costume — decimals. Mixing forms invites mistakes.
- 34 = 0.75, 20% = 0.20 Divide 3 by 4; move the percent point two left.
- 0.75 + 0.35 + 0.20 = 1.30 Line up the decimal points and add.
1.30
Words to know
- Rational number
- Any number you can write as a fraction ab with b ≠ 0.
- Reciprocal
- The flip of a fraction. The reciprocal of 35 is 53.
- LCD
- Least common denominator — the smallest number both denominators divide into.
- Terminating / repeating
- Every fraction becomes a decimal that either stops or repeats forever.
Mistakes to watch for
Adding the denominators. 12 + 13 ≠ 25 — and a sense check proves it, since 25 is smaller than the half you started with. You add numerators only, after the denominators match.
Finding a common denominator before multiplying. It is legal but pointless: 23 × 45 is just 815. Multiplication needs no matching, and the extra step usually causes a slip.
Flipping the wrong fraction when dividing. Only the second one flips, because dividing by 12 asks "how many halves fit in?" — which doubles the answer, not halves it.
Leaving an improper fraction where the question wanted a mixed number, or the reverse. 114 and 234 are the same amount; read what form the question asked for.
Ancient Egyptians wrote almost every fraction as a sum of unit fractions — ones with a 1 on top.
So 34 was written 12 + 14, and 27 as 14 + 128.
The Rhind Papyrus, from about 1650 BC, contains a table converting dozens of fractions into this form — presumably because sharing loaves in equal pieces was the practical problem.