MathematicsClass 6ICSE

Why You Flip the Second Fraction When Dividing

Learn the types of fractions, how to order unlike fractions using the LCM, and how to add, subtract, multiply and divide them and always simplify the answer.

What are the types of fractions?

A fraction shows a part of a whole, written

\frac{numerator}{denominator}

— the denominator says how many equal parts the whole is cut into, and the numerator says how many of those parts you have.

This page covers everything in the ICSE Class 6 Mathematics chapter on fractions: the types and how to convert between them, ordering unlike fractions, and all four operations with answers in simplest form.

How do you convert an improper fraction into a mixed number?

First the names. A proper fraction has a numerator smaller than its denominator, like

\frac{3}{5}

. An improper fraction has a numerator equal to or larger than its denominator, like

\frac{11}{4}

. A mixed number pairs a whole number with a proper fraction, like

2 \frac{3}{4}

. Like fractions share a denominator; unlike fractions do not. A unit fraction has numerator 1. Equivalent fractions look different but represent the same amount, such as

\frac{1}{2}

and

\frac{4}{8}

.

To convert improper to mixed, divide and keep the remainder over the divisor:

\frac{11}{4} = 2 remainder 3 = 2 \frac{3}{4}

To convert back, multiply the whole number by the denominator and add the numerator:

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

For example, three and a half chapatis is

3 \frac{1}{2}

as a mixed number and

\frac{7}{2}

as an improper fraction — the same food, written two ways.

How do you compare and order unlike fractions?

First, reduce to lowest terms by dividing numerator and denominator by their HCF:

\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}

To order unlike fractions, convert them to like fractions using the LCM of the denominators, then compare numerators.

Arrange

\frac{2}{3}

\frac{3}{4}

and

\frac{5}{6}

in ascending order. The LCM of 3, 4 and 6 is 12:

\frac{2}{3} = \frac{8}{12} \frac{3}{4} = \frac{9}{12} \frac{5}{6} = \frac{10}{12}

So the order is

\frac{2}{3} < \frac{3}{4} < \frac{5}{6}

.

There is a useful shortcut for fractions with the same numerator: the one with the smaller denominator is larger, because the whole is cut into fewer, bigger pieces. So

\frac{1}{3} > \frac{1}{5}

. This is where intuition misleads students — the bigger number underneath makes a smaller share.

How do you add and subtract fractions?

For like fractions, add or subtract the numerators and keep the denominator:

\frac{3}{7} + \frac{2}{7} = \frac{5}{7}

For unlike fractions, first make the denominators the same using their LCM:

\frac{1}{4} + \frac{2}{3} = \frac{3}{12} + \frac{8}{12} = \frac{11}{12}

With mixed numbers, convert to improper fractions first:

2 \frac{1}{2} + 1 \frac{1}{3} = \frac{5}{2} + \frac{4}{3} = \frac{15}{6} + \frac{8}{6} = \frac{23}{6} = 3 \frac{5}{6}

In real life, adding

\frac{1}{4}

kg of sugar to

\frac{2}{3}

kg needs exactly this: a common denominator of 12 gives

\frac{11}{12}

kg.

Never add the denominators.

\frac{1}{4} + \frac{2}{3}

is not

\frac{3}{7}

— the denominator names the size of the piece, and adding pieces does not change their size.

Formula

How do you multiply and divide fractions?

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

\frac{2}{3} \times \frac{3}{5} = \frac{6}{15} = \frac{2}{5}

To divide, multiply by the reciprocal of the divisor — the reciprocal of

\frac{a}{b}

\frac{b}{a}

\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1 \frac{7}{8}

Why flipping works: dividing by

\frac{2}{5}

asks how many two-fifths fit into the amount, and since each fifth fits five times per whole, multiplying by

\frac{5}{2}

counts them. Because

\frac{2}{5} \times \frac{5}{2} = 1

, the reciprocal undoes the divisor exactly.

For a fractional part of a quantity, multiply. Three-fifths of 200 g is

\frac{3}{5} \times 200 = 120 g

Note that dividing by a fraction smaller than 1 makes the answer larger, which is the opposite of dividing by a whole number.

Exam tip

Exam tip: leaving the answer unsimplified

Marks disappear at the last line. An answer of

\frac{6}{15}

is correct arithmetic but incomplete — it must be reduced to

\frac{2}{5}

.

Build a two-part habit for every fraction answer. First, reduce to lowest terms by dividing by the HCF. Second, if the result is improper, convert it to a mixed number unless the question asked otherwise.

So

\frac{15}{8}

should be written as

1 \frac{7}{8}

, and

\frac{20}{4}

should be written simply as 5. Read your final line back and ask whether anything still divides.

Did you know

Why is a bigger denominator a smaller share?

The denominator counts how many equal pieces the whole is cut into. Cut one roti into three and each piece is large; cut the same roti into eight and each piece is small.

So

\frac{1}{3}

is more food than

\frac{1}{8}

, even though 8 is the bigger number — the number below the line is counting cuts, not quantity.

Key takeaways

Fractions in 30 seconds

- Fractions are proper, improper or mixed, and like or unlike depending on their denominators; convert improper to mixed by dividing and keeping the remainder.
- Reduce to lowest terms using the HCF, and order unlike fractions by converting them to like fractions with the LCM.
- Add and subtract only after the denominators match; never add the denominators themselves.
- Multiply numerators and denominators straight across; divide by multiplying by the reciprocal, since a fraction times its reciprocal is 1.
- Always finish by simplifying, and convert an improper answer to a mixed number.

You will remember all of this far better after answering five questions on it than after reading it twice.