How to Compare, Add and Subtract Fractions Like a Pro

To generate equivalent fractions, multiply or divide both the numerator and denominator of a fraction by the same non-zero number; to simplify, divide both by their highest common factor (HCF).

Equivalent fractions are different fractions that represent the same value. For example, $\frac{1}{2}$, $\frac{2}{4}$, and $\frac{3}{6}$ are all equivalent because $1\times 2=2$ and $2\times 2=4$, so $\frac{1}{2}=\frac{2}{4}$. To reduce a fraction, find the HCF of the numerator and denominator and divide both by it. For example, $\frac{8}{12}$: HCF of 8 and 12 is 4, so $\frac{8÷4}{12÷4}=\frac{2}{3}$. In real life, if you cut a cake into 6 pieces and eat 3, that's $\frac{3}{6}$, which is the same as $\frac{1}{2}$ of the cake.

A common mistake is forgetting to divide by the HCF, which means the fraction is not in its simplest form.

To compare and order fractions, look at their numerators and denominators: if denominators are the same, compare numerators; if numerators are the same, compare denominators; for unlike denominators, use equivalent fractions or benchmarks like $\frac{1}{2}$.

When fractions have the same denominator, the one with the larger numerator is greater. For example, $\frac{3}{7}>\frac{2}{7}$. When fractions have the same numerator, the one with the smaller denominator is greater, because each part is bigger. For example, $\frac{2}{5}>\frac{2}{7}$. For unlike denominators, convert them to equivalent fractions with a common denominator. For example, to compare $\frac{3}{4}$ and $\frac{5}{8}$, convert $\frac{3}{4}$ to $\frac{6}{8}$, so $\frac{6}{8}>\frac{5}{8}$.

A misconception is thinking a bigger denominator always means a bigger fraction, but actually, it means the parts are smaller.

To add or subtract fractions, first make their denominators the same (find the LCM if needed), then add or subtract the numerators and keep the denominator. For mixed fractions, convert them to improper fractions before calculating.

For like denominators, simply add or subtract the numerators: $\frac{2}{9}+\frac{4}{9}=\frac{6}{9}$, which simplifies to $\frac{2}{3}$. For unlike denominators, find the LCM of the denominators, convert each fraction to an equivalent one with the LCM as denominator, then add or subtract. For example, $\frac{1}{3}+\frac{1}{4}$: LCM of 3 and 4 is 12, so $\frac{1}{3}=\frac{4}{12}$ and $\frac{1}{4}=\frac{3}{12}$, so $\frac{4}{12}+\frac{3}{12}=\frac{7}{12}$. For mixed fractions, $1\frac{1}{2}+2\frac{1}{3}$: convert to improper fractions ($\frac{3}{2}$ and $\frac{7}{3}$), find LCM (6), convert ($\frac{9}{6}$ and $\frac{14}{6}$), add ($\frac{23}{6}$), and if needed, write as a mixed fraction ($3\frac{5}{6}$).

A common error is forgetting to convert mixed fractions or not simplifying the final answer.

To solve word problems with fractions, identify the operation (addition or subtraction), convert all numbers to like denominators if needed, perform the calculation, and simplify the answer.

Read the problem carefully to decide if you need to add or subtract. For example, if Rani ate $\frac{1}{4}$ of a pizza and her brother ate $\frac{2}{4}$, together they ate $\frac{1}{4}+\frac{2}{4}=\frac{3}{4}$ of the pizza. If a water tank is $\frac{5}{8}$ full and $\frac{1}{4}$ is used, the remaining water is $\frac{5}{8}-\frac{1}{4}$. Convert $\frac{1}{4}$ to $\frac{2}{8}$, so $\frac{5}{8}-\frac{2}{8}=\frac{3}{8}$. Always check if your answer can be simplified.

A common mistake is not converting to like denominators before adding or subtracting, which leads to wrong answers.