Equivalent Fractions and Mixed Numbers

Updated 4 October 2026

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?

Two fractions are equivalent when they name the same amount, and multiplying or dividing the top and bottom by the same number turns one into the other. Equivalent Fractions shows it with a bar: cut each fifth of $\frac{3}{5}$3/5 into 3 equal pieces, and the bar now shows 9 shaded pieces out of 15, so $\frac{3}{5}=\frac{9}{15}$3/5 = 9/15. The shaded amount has not moved; there are three times as many pieces, each a third of the size. That is why the rule works: multiplying top and bottom by 3 is multiplying by $\frac{3}{3}$3/3, which equals 1. Dividing works the same way in reverse, so $\frac{9}{15}$9/15 goes back to $\frac{3}{5}$3/5 when both parts are divided by 3. It breaks if you do something different to each part, or add: $\frac{3}{5}$3/5 with 1 added to each part becomes $\frac{4}{6}$4/6, which is more. To check, cross-multiply: 3 × 15 = 45 and 5 × 9 = 45, so the two fractions are equal.

$\frac{2}{3}=\frac{8}{12}\left(\text{both}\times 4\right)$2/3 = 8/12 (both × 4). $\frac{1}{2}=\frac{2}{4}=\frac{3}{6}=\frac{50}{100}$1/2 = 2/4 = 3/6 = 50/100.

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. $\frac{18}{24}÷6\to \frac{3}{4}$18/24 ÷ 6 → 3/4. 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?

To compare two fractions, rewrite them with the same denominator and compare the numerators: the larger numerator names the larger fraction. For $\frac{4}{5}$4/5 and $\frac{5}{7}$5/7, use 35ths, since 5 × 7 = 35: $\frac{4}{5}=\frac{28}{35}$4/5 = 28/35 and $\frac{5}{7}=\frac{25}{35}$5/7 = 25/35, so $\frac{4}{5}$4/5 is larger. Same-size pieces can simply be counted. When the numerators already match, the rule turns around: the smaller denominator names the larger fraction, because fewer cuts leave bigger pieces, so $\frac{3}{4}$3/4 is more than $\frac{3}{10}$3/10. A pair on opposite sides of a half needs no rewriting. Double each numerator and compare it with its denominator: for $\frac{3}{7}$3/7 and $\frac{5}{9}$5/9, 3 × 2 = 6 is less than 7 and 5 × 2 = 10 is more than 9, so $\frac{3}{7}$3/7 is under a half, $\frac{5}{9}$5/9 is over, and $\frac{5}{9}$5/9 is larger. To check the first pair, cross-multiply: 4 × 7 = 28 and 5 × 5 = 25, and the bigger product sits with $\frac{4}{5}$4/5.

Same bottom. Same Bottom Number: $\frac{5}{8}>\frac{3}{8}$5/8 > 3/8, because five eighths is more eighths than three.

Same top. Same Top Number: $\frac{3}{5}>\frac{3}{8}$3/5 > 3/8. Three pieces either way, but fifths are the bigger piece.

Neither. More Than a Half is the fastest first check. $\frac{7}{16}$7/16 — double 7 is 14, under 16, so it is less than a half.

Ordering Fractions puts three or more in order, which is where a common denominator stops being optional: $\frac{2}{3}$2/3, $\frac{3}{5}$3/5, $\frac{7}{10}\to$7/10 → fifteenths and tenths do not agree, so use thirtieths $\to \frac{20}{30}$→ 20/30, $\frac{18}{30}$18/30, $\frac{21}{30}\to \frac{3}{5}<\frac{2}{3}<\frac{7}{10}$21/30 → 3/5 < 2/3 < 7/10.

Adding and subtracting

Matching bottoms add by counting. Adding Fractions with the Same Denominator: $\frac{2}{7}+\frac{3}{7}=\frac{5}{7}$2/7 + 3/7 = 5/7, for the same reason two meters plus three meters is five meters. The unit is not being added; it is what makes the counts addable.

The full explanation, with its figures and questions: How do you add and subtract fractions?.

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 $\frac{7}{4}$7/4 — seven quarters exist, they are just more than a whole.

Mixed Numbers is the other way of writing the same amount: $\frac{7}{4}=1\frac{3}{4}$7/4 = 1 3/4, 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. $\frac{17}{5}\to 17÷5=3$17/5 → 17 ÷ 5 = 3 remainder $2\to 3\frac{2}{5}$2 → 3 2/5.
• Mixed to improper: whole × bottom, plus the top. $3\frac{2}{5}\to 3\times 5+2=17\to \frac{17}{5}$3 2/5 → 3 × 5 + 2 = 17 → 17/5.

Adding and subtracting mixed numbers

Handle the wholes and the parts separately, then tidy up. $2\frac{1}{2}+1\frac{3}{4}\to 2+1=3$2 1/2 + 1 3/4 → 2 + 1 = 3, and $\frac{1}{2}+\frac{3}{4}=\frac{5}{4}=1\frac{1}{4}$1/2 + 3/4 = 5/4 = 1 1/4, so $4\frac{1}{4}$4 1/4.

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\frac{1}{5}-1\frac{3}{5}\to$3 1/5 − 1 3/5 → borrow one whole as five fifths $\to 2\frac{6}{5}-1\frac{3}{5}=1\frac{3}{5}$→ 2 6/5 − 1 3/5 = 1 3/5. Converting both numbers to improper fractions first — $\frac{16}{5}-\frac{8}{5}=\frac{8}{5}$16/5 − 8/5 = 8/5 — avoids the borrow entirely and is the safer route when the numbers are awkward.

How do you find a fraction of an amount?

To find a fraction of an amount, divide the amount by the bottom number, then multiply by the top number. A Fraction of an Amount works it through: $\frac{3}{4}$3/4 of $60\to 60÷4=15\to 15\times 3=45$60 → 60 ÷ 4 = 15 → 15 × 3 = 45. Dividing by 4 finds one quarter of 60, which is 15, and three quarters is three of those, 45. The bottom number says how many equal parts to make, and the top says how many to take. When the top is 1, dividing is the whole job: $\frac{1}{5}$1/5 of 35 is 35 ÷ 5 = 7. When the amount does not divide exactly, multiply first instead: $\frac{2}{3}$2/3 of 10 is 2 × 10 = 20, then 20 ÷ 3, which is $6\frac{2}{3}$6 2/3. Either order gives the same answer. To check, add the parts back: the quarter that is left is 15, and 45 + 15 = 60.

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.

How do you multiply and divide fractions?

To multiply fractions, multiply the numerators together and the denominators together, with no common denominator needed: $\frac{5}{6}\times \frac{3}{10}=\frac{15}{60}=\frac{1}{4}$5/6 × 3/10 = 15/60 = 1/4. Multiplying by a fraction takes that part of the amount, so five sixths of three tenths is smaller than either fraction. To divide by a fraction, flip that fraction upside down and multiply: $\frac{2}{3}÷\frac{1}{9}=\frac{2}{3}\times \frac{9}{1}=\frac{18}{3}=6$2/3 ÷ 1/9 = 2/3 × 9/1 = 18/3 = 6. The flip works because dividing asks how many ninths fit into two thirds, and each third holds 3 ninths, so two thirds hold 6. Dividing by a fraction smaller than 1 always gives a bigger answer, because a small piece fits many times. A mixed number must become an improper fraction first: $1\frac{1}{3}÷\frac{1}{3}$1 1/3 ÷ 1/3 becomes $\frac{4}{3}\times \frac{3}{1}=\frac{12}{3}=4$4/3 × 3/1 = 12/3 = 4. To check a division, multiply the answer by the fraction you divided by: $6\times \frac{1}{9}=\frac{6}{9}=\frac{2}{3}$6 × 1/9 = 6/9 = 2/3.

Multiplying Fractions: straight across. $\frac{2}{3}\times \frac{4}{5}=\frac{8}{15}$2/3 × 4/5 = 8/15.

Dividing by a Fraction reads best as a question about fitting. $\frac{1}{2}÷\frac{1}{6}$1/2 ÷ 1/6 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: $\frac{1}{2}\times \frac{6}{1}=3$1/2 × 6/1 = 3.

Dividing a Fraction by a Fraction handles the general case, where the fit is not a whole number. $\frac{3}{4}÷\frac{2}{5}=\frac{3}{4}\times \frac{5}{2}=\frac{15}{8}=1\frac{7}{8}$3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8. Two fifths fits into three quarters not quite twice, and the fraction is the part-fit.

The four mistakes worth naming

• Adding the bottoms. $\frac{1}{2}+\frac{1}{2}$1/2 + 1/2 is 1, not $\frac{2}{4}$2/4. The denominator names the unit; it does not accumulate.
• Multiplying a mixed number in place. $2\frac{1}{2}\times 3$2 1/2 × 3 is not $6\frac{1}{2}$6 1/2. Convert to $\frac{5}{2}$5/2 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.