Fractions in Maths

A fraction is a number that can represent part of a whole, a point on a number line, a ratio, or a division. In

\frac34,

The numerator is 3 and the denominator is 4. Dividing one whole into four equal parts makes each part a quarter. The numerator tells us that three of those quarters are being considered.

The parts must be equal in size. Three selected pieces represent $3/4$ only when they are three of four equal shares of the same whole.

In $a/b$ , the denominator must be non-zero: $b\ne0$ .

On this page (9 sections)

Fractions on a number line

A fraction has one exact position on the number line. The fraction $3/4$ sits between 0 and 1. An improper fraction can sit beyond 1. For example,

\frac74=1\frac34.

The What Works Clearinghouse recommends teaching fractions as numbers and using number lines as a central representation. A number line gives fractions a common scale on which to compare their values.

Locating seven quarters on a number line

0¼½¾11¼1½1¾7/42

Every interval is one quarter. Seven quarters and one and three quarters name the same point.

Diagram explained

The positions, in order, are 0, one quarter, one half, three quarters, 1, one and one quarter, one and one half, one and three quarters, and 2. All eight intervals have equal length. The point at one and three quarters is also seven quarters.

Equivalent fractions

Multiplying the numerator and denominator by the same non-zero number changes the notation but not the value:

\frac34=\frac{3\times2}{4\times2}=\frac68.

This principle supports simplifying, comparing, and adding fractions. A fraction is in simplest form when the numerator and denominator have no common factor greater than 1.

Equivalent fractions use the same whole

3/4

6/8

Splitting each quarter in two gives eighths. The selected amount is unchanged.

Diagram explained

Both bars have the same total length. The upper bar has four equal parts with three selected. The lower bar has eight equal parts with six selected. Each selected quarter corresponds to two selected eighths, so three quarters equals six eighths.

Comparing fractions

A larger denominator does not automatically mean a larger fraction. Compare $5/8$ and $2/3$ using a common denominator:

\frac58=\frac{15}{24},\qquad \frac23=\frac{16}{24}.

Therefore,

\frac23>\frac58.

Benchmarks such as $0$ , $1/2$ , and $1$ can also make comparisons faster.

Adding and subtracting fractions

To add or subtract fractions, express their parts using the same denominator:

\frac23+\frac14 =\frac8{12}+\frac3{12} =\frac{11}{12}.

The denominator describes the unit being counted. Twelfths can be added to twelfths in the same way that metres can be added to metres.

Add parts of the same size

2/3 = 8/12

1/4 = 3/12

11/12

The whole stays the same. Express both fractions in twelfths before adding.

Diagram explained

All three bars represent a whole of the same length and are divided into twelve equal parts. Eight selected twelfths represent two thirds. Three selected twelfths represent one quarter. Their sum selects eleven of the twelve equal parts.

Multiplying and dividing fractions

To multiply fractions, multiply the numerators and denominators, then simplify:

\frac35\times\frac{10}{9} =\frac{30}{45} =\frac23.

Dividing by a fraction asks how many groups of that fractional size fit in the first quantity. The reciprocal method is a compact way to calculate:

\frac34\div\frac25 =\frac34\times\frac52 =\frac{15}{8} =1\frac78.

The reciprocal turns the divisor into 1. Multiplying both entries in the division by that reciprocal preserves the quotient.

Reference: choose the operation before choosing a procedure

Choose the operation before choosing a procedure
Operation	What the structure requires
Add or subtract	Use equal-sized units, usually by finding a common denominator.
Multiply	Multiply numerators and multiply denominators, then simplify.
Divide by a non-zero fraction	Multiply by the reciprocal of the divisor.

Fractions, decimals and percentages

A fraction bar represents division, so a fraction can be converted to a decimal by dividing numerator by denominator:

\frac34=3\div4=0.75.

A percentage expresses a quantity per hundred:

\frac34=0.75=75\%.

These are three representations of exactly the same number.

Checking fraction calculations

Keep the whole and the size of each part clear. Use equivalent fractions to compare values or add like-sized parts, and use a benchmark or number line to check the answer.

CTS’s guide to personalized mathematics instruction discusses support for specific difficulties, such as identifying the whole or comparing fraction sizes.

Practice questions

Try the questions, explain your reasoning, then check the answers.

Simplify $6/8$ .
Calculate $1/3+1/6$ .
Which is larger: $5/8$ or $2/3$ ? Use a common denominator.

Show answers

$6/8=3/4$ . Divide the numerator and denominator by 2.
$1/3+1/6=2/6+1/6=3/6=1/2$ . Express both fractions in sixths before adding.
$2/3=16/24>15/24=5/8$ . Sixteen twenty-fourths is greater than fifteen twenty-fourths.

Sources and further reading

U.S. Institute of Education Sciences, What Works Clearinghouse. Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades. The guide supports systematic instruction, precise mathematical language, well-chosen representations, and number lines.
National Research Council. Adding It Up: Helping Children Learn Mathematics. The report describes mathematical proficiency as an integration of conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition.
OpenStax. Prealgebra 2e. Open educational reference for number systems, fractions, decimals, percents, arithmetic, and related prealgebra concepts.
U.S. Institute of Education Sciences, What Works Clearinghouse. Developing Effective Fractions Instruction for Kindergarten Through 8th Grade. The guide recommends treating fractions as numbers, using number lines, and connecting procedures to conceptual meaning.