Equivalent Fractions and Mixed Numbers
This page follows fractions the way the app teaches them: as a ladder, from cutting something into equal parts up to dividing one fraction by another. If you arrived wanting the direct answer to how do I add fractions, the fractions guide answers exactly that and nothing else; this one is for the whole progression, and for the rungs that get skipped — equivalence, mixed numbers, and fractions of amounts.
One idea holds it together. The bottom number is a unit, not a quantity. Fifths and eighths are different-sized pieces, and almost every rule below is about making pieces the same size before counting them, or about re-cutting them on purpose.
Where fractions begin
They begin with the word equal. Equal Parts is about nothing else: three of four pieces is only three quarters if the four pieces are the same size, and a shape cut into four unequal bits is not showing quarters at all. That distinction is what makes fractions arithmetic rather than description.
Halves, Quarters, Thirds names the first few, and Top and Bottom is where the notation arrives: the bottom says how many pieces the whole was cut into, the top says how many you have. A Fraction of a Group then makes the leap that catches people — a fraction can describe a group of objects, not only a cut-up shape. Half of 12 counters is 6, and the "whole" is the twelve.
What makes two fractions equivalent?
Multiplying top and bottom by the same number does not change the value. Equivalent Fractions shows why with a picture: each piece is cut into the same number of smaller pieces and you take proportionally more of them, so the amount in your hand is untouched.
2/3 = 8/12 (both × 4). .
Run it backwards and you get Simplest Form — keep dividing until nothing divides both, which is the same as dividing once by the highest common factor. . Finding that factor quickly is a job for the factors and multiples toolkit.
A Common Denominator is equivalence used in the other direction: instead of making a fraction simpler, you make two fractions match, so their counts can finally be compared or added.
How do you compare two fractions?
Make the pieces the same size, then compare the counts. The three shortcuts below are all that method with a step skipped.
Same bottom. Same Bottom Number: 5/8 > 3/8, because five eighths is more eighths than three.
Same top. Same Top Number: 3/5 > 3/8. Three pieces either way, but fifths are the bigger piece. This is the comparison that feels backwards and is not — more cuts make smaller pieces.
Neither. More Than a Half is the fastest first check: double the top and see how it lands against the bottom. — double 7 is 14, under 16, so it is less than a half. If one fraction is over a half and the other under, you are finished.
Ordering Fractions puts three or more in order, which is where a common denominator stops being optional: , , 7/10 → fifteenths and tenths do not agree, so use thirtieths → 20/30, , 21/30 → 3/5 < 2/3 < 7/10.
Adding and subtracting
Matching bottoms add by counting. Adding Fractions with the Same Denominator: , for the same reason two metres plus three metres is five metres. The unit is not being added; it is what makes the counts addable.
Adding Fractions with Unlike Denominators is the same sum after one preparatory step. 1/3 + 1/4 → twelfths → 4/12 + 3/12 = 7/12. Multiplying the two bottoms always produces a workable common denominator; the lowest common multiple just leaves less to simplify afterwards.
Subtracting Fractions is the identical move: .
Improper fractions and mixed numbers
An improper fraction has a top at least as big as its bottom, which simply means it is worth one or more. Improper Fractions makes the point that nothing is wrong with — seven quarters exist, they are just more than a whole.
Mixed Numbers is the other way of writing the same amount: , one whole and three quarters.
Mixed Numbers and Improper Fractions gives both conversions, and both are just division and multiplication.
- Improper to mixed: divide top by bottom; the quotient is the whole part, the remainder stays on top. remainder .
- Mixed to improper: whole × bottom, plus the top. .
Adding and subtracting mixed numbers
Handle the wholes and the parts separately, then tidy up. , and , so .
Subtraction is where it goes wrong, because the fraction part sometimes does not go. The fix is borrowing, exactly as in column subtraction: take one whole and turn it into pieces. 3 1/5 − 1 3/5 → borrow one whole as five fifths → 2 6/5 − 1 3/5 = 1 3/5. Converting both numbers to improper fractions first — — avoids the borrow entirely and is the safer route when the numbers are awkward.
How do you find a fraction of an amount?
Divide by the bottom, then multiply by the top. A Fraction of an Amount works it through: of .
Dividing first keeps the numbers small, which matters when you are doing it in your head. The order is reversible when that helps — 3 × 60 = 180, then ÷ 4 = 45 — and the same answer arriving both ways is worth noticing, because it is the reason the fraction-of and multiplication rules are the same rule.
Multiplying and dividing fractions
These are the ones people brace for, and they are the easy ones — no common denominator is needed for either.
Multiplying Fractions: straight across. .
Dividing by a Fraction reads best as a question about fitting. asks how many sixths fit into a half, and the answer is three — visible before any flipping. The flip is a shortcut for that question: .
Dividing a Fraction by a Fraction handles the general case, where the fit is not a whole number. . Two fifths fits into three quarters not quite twice, and the fraction is the part-fit.
The four mistakes worth naming
- Adding the bottoms. is 1, not . The denominator names the unit; it does not accumulate.
- Multiplying a mixed number in place. is not . Convert to first, then multiply.
- Simplifying one row only. Whatever you do to the top you do to the bottom, or the value moves.
- Finding a common denominator before multiplying. Not wrong, just wasted work — multiplication does not need matching units.
Where this leads next
A fraction whose bottom is ten or a hundred is a decimal, and a fraction whose bottom is a hundred is a percentage — the same amount in three notations, followed up in decimals, percentages and interest. Comparing two quantities rather than a part to a whole is the subject of ratio, rates and proportion. And whether a fraction's decimal stops or repeats forever is decided by its denominator's prime factors, in number theory.
Practise it in the game
Math Challenge teaches every rung above as an illustrated lesson — the equivalence picture, the mixed-number conversion, the fitting argument for division — inside a catalog of 800+ lessons across the whole ladder. When a try-it question goes wrong, the app re-teaches the beat that question came from rather than a nearby one.
The Fractions topic drills adding and subtracting, starting with friendly denominators and widening as you get them right.
Your turn
Three to try — tap what you get.
Which equals 3/4?
7/2 as a mixed number
2 2/5 as an improper fraction