A fraction $\frac{a}{b}$ represents $a$ parts out of $b$ equal parts of a whole. The denominator ($b$) tells you how many equal parts the whole has been split into; the numerator ($a$) tells you how many of those parts you have. In $\frac{2}{3}$, the whole is cut into 3 equal pieces and you have 2 of them.

Find the highest common factor (HCF) of the numerator and denominator, then divide both by it. The fraction is fully simplified once the HCF of the new numerator and denominator is 1.

Example: Simplify $\frac{12}{18}$.

1. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. The HCF is 6.
2. Divide both parts by 6: $\frac{12 \div 6}{18 \div 6} = \frac{2}{3}$.

You can't compare fractions directly unless they share the same denominator. Find the lowest common multiple (LCM) of all the denominators, convert each fraction to an equivalent fraction with that denominator, then simply compare the numerators.

Example: Order $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}$ from smallest to largest.

1. The LCM of 2, 3, and 4 is 12.
2. Convert each: $\frac{1}{2}=\frac{6}{12}$, $\frac{2}{3}=\frac{8}{12}$, $\frac{3}{4}=\frac{9}{12}$.
3. Compare numerators: $6 < 8 < 9$, so the order is $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}$.

Convert both fractions to the same common denominator first (using the LCM, exactly as above), then add or subtract the numerators only — the denominator stays the same.

Example: $\frac{1}{4} + \frac{2}{5}$. The LCM of 4 and 5 is 20: $\frac{1}{4}=\frac{5}{20}$ and $\frac{2}{5}=\frac{8}{20}$. So $\frac{5}{20}+\frac{8}{20}=\frac{13}{20}$.

Three friends compare how much of their chocolate bars they've eaten: Amara has eaten $\frac{3}{5}$, Ben has eaten $\frac{7}{10}$, and Cora has eaten $\frac{13}{20}$. Who has eaten the most, and simplify any fraction that can be simplified.

1. First check for simplification: $\frac{3}{5}$ and $\frac{7}{10}$ are already in lowest terms. $\frac{13}{20}$ has HCF(13, 20) = 1, so it's also already simplified.
2. Find the LCM of the denominators 5, 10, 20: it's 20.
3. Convert: $\frac{3}{5}=\frac{12}{20}$, $\frac{7}{10}=\frac{14}{20}$, $\frac{13}{20}$ stays as it is.
4. Compare numerators: $14 > 13 > 12$, so Ben has eaten the most, then Cora, then Amara.