Fractions & Ratios
26 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 number representing a part of a whole, written as one integer over another non-zero integer.
Equivalent Fractions
Two or more fractions that look different but represent exactly the same amount or value.
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 comparison of two quantities that shows their relative sizes, written as $a:b$ or $\frac{a}{b}$.
Proportions
An equation stating that two ratios are equal, used to find an unknown when three of the four values are known.
Rates
A ratio that compares two quantities measured in different units, often expressed as 'per' one unit of the second.
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
Arranging three or more fractions from least to greatest (or greatest to least).
Mixed Numbers
A number consisting of a whole number and a proper fraction combined, such as $2\frac{3}{4}$.
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
Multiplying two fractions by multiplying the numerators together and the denominators together.
Dividing Fractions
Dividing by a fraction by multiplying by its reciprocal (inverting the divisor and multiplying).
Fraction of a Number
Finding a fractional part of a whole number by multiplying the fraction by that number.
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
Adding, subtracting, multiplying, and dividing numbers that contain decimal points.
Percent of a Number
Calculating a given percentage of a quantity by converting the percent to a decimal (or fraction) and multiplying.
Percent Change
The ratio of the change in a quantity to the original value, expressed as a percentage.
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.