A fraction is a single value, not two separate whole numbers. In $\dfrac{3}{4}$, the denominator ($4$) tells you the size of the pieces (quarters), and the numerator ($3$) tells you how many of those pieces you have. The denominator never changes just because you're adding or subtracting — it's a unit label, like "cm" or "kg", not a number to be operated on separately.

• Proper fraction: numerator smaller than denominator, value $<1$ (e.g. $\frac{3}{4}$).
• Improper fraction: numerator $\ge$ denominator, value $\ge 1$ (e.g. $\frac{11}{4}$).
• Mixed number: a whole number plus a proper fraction (e.g. $2\frac{3}{4}$).

To convert mixed → improper: multiply the whole number by the denominator, add the numerator, keep the same denominator. $2\frac{3}{4} = \dfrac{(2\times4)+3}{4} = \dfrac{11}{4}$.

To convert improper → mixed: divide numerator by denominator; the quotient is the whole number, the remainder sits over the original denominator. $\dfrac{11}{4} = 11 \div 4 = 2$ remainder $3$, so $2\frac{3}{4}$.

Equivalent fractions represent the same value — multiply (or divide) numerator and denominator by the same non-zero number, since that's equivalent to multiplying by $1$ (e.g. $\frac{3}{3}=1$). $\dfrac{4}{8}=\dfrac{1}{2}$.

To simplify, divide numerator and denominator by their highest common factor (HCF) until no common factor remains. This isn't optional — "simplest form" is often an explicit mark on the paper.

They must share a denominator first. If denominators differ, find the LCM of the denominators and convert both fractions to it, then add or subtract just the numerators — the denominator stays fixed.

Multiply: multiply the numerators together, and the denominators together. $\dfrac{4}{5}\times\dfrac{2}{3}=\dfrac{8}{15}$.

Divide: keep the first fraction, change ÷ to ×, flip the second fraction (its reciprocal). $\dfrac{2}{7}\div\dfrac{3}{4}=\dfrac{2}{7}\times\dfrac{4}{3}=\dfrac{8}{21}$.

A quick sanity check for "invert and multiply": dividing $8$ by $2$ gives the same answer as multiplying $8$ by $\frac{1}{2}$ — dividing by a number is always the same as multiplying by its reciprocal.

1. Find the LCM of all the denominators (the lowest common denominator).
2. Convert every fraction to that denominator.
3. Order by comparing the numerators only.

To order $\frac{2}{5}, \frac{1}{3}, \frac{7}{15}$: LCD is $15$, giving $\frac{6}{15}, \frac{5}{15}, \frac{7}{15}$ — so the order is $\frac{1}{3} < \frac{2}{5} < \frac{7}{15}$.

Calculate $5\frac{2}{3} - 2\frac{3}{4}$, giving your answer as a mixed number in its simplest form.

1. Convert to improper fractions: $5\frac{2}{3}=\frac{17}{3}$, $2\frac{3}{4}=\frac{11}{4}$.
2. Common denominator of $3$ and $4$ is $12$: $\frac{17}{3}=\frac{68}{12}$, $\frac{11}{4}=\frac{33}{12}$.
3. Subtract: $\frac{68}{12}-\frac{33}{12}=\frac{35}{12}$.
4. Convert back to a mixed number: $35 \div 12 = 2$ remainder $11$, so $2\frac{11}{12}$.