A Fraction Can Have Another Fraction Hiding Inside It

It can. In $\frac{\frac{2}{3}}{\frac{4}{5}}$ both the numerator and the denominator are themselves fractions — and that is called a complex fraction.

It looks alarming and simplifies in one step. This page covers everything in the ICSE Class 7 Mathematics chapter's first part: classifying fractions, converting between improper and mixed forms, like and unlike fractions, and comparing and inserting fractions.

The same fraction is classified in three separate ways at once, so a question may want more than one label.

By denominator:

- Decimal fraction — the denominator is 10, 100, 1000 and so on: $\frac{7}{10}$, $\frac{23}{100}$.
- Vulgar fraction — the denominator is any other number: $\frac{3}{4}$, $\frac{5}{7}$.

By size:

- Proper fraction — numerator less than denominator, so its value is less than 1: $\frac{3}{5}$.
- Improper fraction — numerator greater than or equal to the denominator, so its value is 1 or more: $\frac{7}{4}$.
- Mixed number — a whole number with a proper fraction: $1\frac{3}{4}$.

By form:

- Simple fraction — numerator and denominator are both whole numbers: $\frac{3}{4}$.
- Complex fraction — the numerator or the denominator, or both, is itself a fraction: $\frac{\frac{2}{3}}{4}$ or $\frac{\frac{2}{3}}{\frac{4}{5}}$.

So $\frac{7}{4}$ is vulgar, improper and simple, all three at the same time.

A measuring jug marked in tenths shows decimal fractions, while a recipe asking for $\frac{3}{4}$ cup gives a vulgar one.

The classification students collapse is the first one. A decimal fraction is not the same thing as a decimal number — $\frac{7}{10}$ is a decimal fraction written in fraction form, and 0.7 is the same value written in decimal form.

Improper to mixed: divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator, and the denominator stays.

$$\frac{17}{5}:17÷5=3\text{ remainder }2\Rightarrow 3\frac{2}{5}$$

Mixed to improper: multiply the whole number by the denominator, add the numerator, and keep the denominator.

$$3\frac{2}{5}=\frac{3\times 5+2}{5}=\frac{17}{5}$$

To reduce a fraction to its lowest terms, divide the numerator and denominator by their HCF:

$$\frac{24}{36}=\frac{24÷12}{36÷12}=\frac{2}{3}$$

A fraction in lowest terms has no common factor left except 1.

Buying $\frac{5}{2}$ kg of sugar is the same as asking for $2\frac{1}{2}$ kg, and the shopkeeper will understand the second form more easily.

The check worth running is a size check. An improper fraction must convert to a mixed number greater than 1, so $\frac{17}{5}$ becoming $3\frac{2}{5}$ is believable while $\frac{17}{5}$ becoming $\frac{3}{5}$ is not. And the fractional part of a mixed number must always be proper — $2\frac{7}{5}$ is not a finished answer, since it should be $3\frac{2}{5}$.

Like fractions have the same denominator: $\frac{2}{7}$, $\frac{5}{7}$, $\frac{9}{7}$.

Unlike fractions have different denominators: $\frac{3}{4}$ and $\frac{5}{6}$.

Equivalent fractions have different numerators and denominators but the same value: $\frac{1}{2}$, $\frac{2}{4}$, $\frac{3}{6}$, $\frac{50}{100}$ are all one half. They are obtained by multiplying or dividing both parts by the same non-zero number.

To make unlike fractions like, there are two routes.

Common denominator. Use the LCM of the denominators. For $\frac{3}{4}$ and $\frac{5}{6}$, the LCM of 4 and 6 is 12:

$$\frac{3}{4}=\frac{9}{12},\frac{5}{6}=\frac{10}{12}$$

Common numerator. Use the LCM of the numerators instead. For $\frac{3}{4}$ and $\frac{5}{6}$, the LCM of 3 and 5 is 15:

$$\frac{3}{4}=\frac{15}{20},\frac{5}{6}=\frac{15}{18}$$

That second method is less familiar and has its own comparison rule, which is the reason it is taught. With equal numerators, the fraction with the smaller denominator is the larger fraction — because the whole is cut into fewer pieces, so each piece is bigger. So $\frac{15}{18}>\frac{15}{20}$, which tells us $\frac{5}{6}>\frac{3}{4}$ without any addition at all.

Sharing one roti between 3 people rather than 4 is the same idea — fewer shares, larger portions.

Take the mean of the two — add them and divide by 2. The result always lies between them.

Worked example. Insert a fraction between $\frac{1}{2}$ and $\frac{3}{5}$:

$$\frac{1}{2}+\frac{3}{5}=\frac{5}{10}+\frac{6}{10}=\frac{11}{10}$$

$$\text{mean}=\frac{11}{10}÷2=\frac{11}{10}\times \frac{1}{2}=\frac{11}{20}$$

Checking with a common denominator of 20: $\frac{1}{2}=\frac{10}{20}$ and $\frac{3}{5}=\frac{12}{20}$, so $\frac{10}{20}<\frac{11}{20}<\frac{12}{20}$. Correct.

A quicker route for a single insertion is to write both with a larger common denominator and look for a gap. With denominator 20, $\frac{10}{20}$ and $\frac{12}{20}$ leave $\frac{11}{20}$ sitting between them.

Ordering a set. Arrange $\frac{2}{3},\frac{3}{4},\frac{5}{8}$ in ascending order. The LCM of 3, 4 and 8 is 24:

$$\frac{16}{24},\frac{18}{24},\frac{15}{24}$$

So the ascending order is $\frac{5}{8},\frac{2}{3},\frac{3}{4}$.

The fact worth carrying away is that this never runs out. Between any two different fractions there is always another one — take the mean of $\frac{10}{20}$ and $\frac{11}{20}$ and you get $\frac{21}{40}$, and the process can be repeated for ever.