Fractions & Ratios
27 concepts in Math
Fractions and ratios describe parts of wholes and comparisons between quantities — ideas that students encounter constantly in everyday life, from splitting a recipe in half to understanding percentages on a test score. Mastering fractions means understanding what they represent, not just memorizing procedures for adding or multiplying them. Students learn to find equivalent fractions, convert between fractions and decimals, compare magnitudes, and perform arithmetic with unlike denominators. Ratios extend these ideas to compare two separate quantities, leading naturally into proportional reasoning, unit rates, and percentages. Research consistently shows that fraction understanding in elementary school is one of the strongest predictors of success in higher mathematics, making this topic especially important to get right.
Suggested learning path: Start with understanding what fractions represent using visual models, then practice equivalent fractions and comparisons before moving to fraction arithmetic and ratio reasoning.
Fractions
A fraction is a number of the form $\frac{a}{b}$ where $a$ (the numerator) counts how many equal parts you have and $b$ (the denominator, which must not be zero) tells how many equal parts the whole is divided into.
Equivalent Fractions
Two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent if they represent the same value, which happens exactly when $a \times d = b \times c$ (cross-multiplication gives equal products).
Decimals
Numbers written with a decimal point where each position to the right represents tenths, hundredths, thousandths, etc.
Percentages
A way of expressing a quantity as a fraction of 100, written with the symbol % to mean 'per hundred.'
Ratios
A ratio compares two or more quantities by showing how many times one contains the other, written as $a:b$ or $\frac{a}{b}$. Unlike fractions, ratios can compare parts to parts, not just parts to wholes.
Proportions
An equation stating that two ratios are equal, used to find an unknown when three of the four values are known.
Rates
A rate is a ratio that compares two quantities measured in different units, expressing how much of one quantity corresponds to a given amount of another. It is often written as 'per' one unit of the second quantity, such as miles per hour or dollars per pound.
Fraction on a Number Line
Locating and representing a fraction as a precise point on a number line by dividing the unit interval into equal parts.
Comparing Fractions
Determining which of two fractions is greater, less, or equal using common denominators, benchmarks, or cross-multiplication.
Ordering Fractions
Ordering fractions means arranging a set of fractions from least to greatest (or greatest to least) by converting them to a common denominator or to decimals so their sizes can be directly compared.
Mixed Numbers
A mixed number combines a whole number and a proper fraction, such as $3\frac{1}{4}$, representing the sum of the whole part and fractional part: $3 + \frac{1}{4} = \frac{13}{4}$.
Improper Fractions
A fraction where the numerator is greater than or equal to the denominator, representing a value of one or more.
Mixed-Improper Conversion
The process of converting between mixed-number form and improper-fraction form, which represent the same value.
Adding Fractions with Like Denominators
Adding fractions that share the same denominator by adding the numerators and keeping the denominator.
Subtracting Fractions with Like Denominators
Subtracting fractions that share the same denominator by subtracting the numerators and keeping the denominator.
Adding Fractions with Unlike Denominators
Adding fractions with different denominators by first rewriting them with a common denominator (usually the LCD), then adding numerators.
Subtracting Fractions with Unlike Denominators
Subtracting fractions with different denominators by first rewriting them with a common denominator, then subtracting numerators.
Multiplying Fractions
To multiply fractions, multiply the numerators together and the denominators together: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$. Simplify the result by cancelling common factors.
Dividing Fractions
Dividing by a fraction means multiplying by its reciprocal: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$. This works because division asks 'how many groups of this size fit?'
Fraction of a Number
Finding a fraction of a number means multiplying that number by the fraction: $\frac{a}{b}$ of $n$ equals $\frac{a}{b} \times n = \frac{a \times n}{b}$. It answers 'what is this part of the whole amount?'
Decimal-Fraction Conversion
Converting between fraction form and decimal form of a number: divide numerator by denominator for fraction-to-decimal, and use place value to go the other way.
Decimal Operations
Decimal operations — addition, subtraction, multiplication, and division — follow the same rules as whole-number arithmetic but require careful attention to decimal point placement and alignment.
Percent of a Number
Calculating a given percentage of a quantity by converting the percent to a decimal (or fraction) and multiplying.
Percent Change
Percent change measures how much a quantity has increased or decreased relative to its original value, calculated as $\frac{\text{new} - \text{original}}{\text{original}} \times 100\%$.
Percent Applications
Using percentages to solve real-world problems involving tax, tip, discount, markup, and simple interest.
Adding Fractions
Adding fractions combines parts of a whole by rewriting both with a common denominator and then adding the numerators.
Fraction Line Plots
A fraction line plot displays fractional data by placing marks above a number line scaled in fractional units (halves, quarters, eighths, etc.).