Simple and Decimal Fractions

In daily life — half a kilo, a quarter of a litre, ₹12.50 — we use fractions and decimals all the time. In exams they appear on their own and inside almost every other chapter (percentage, interest, average). Here you will learn what a fraction is, how to turn it into a decimal, what to do with recurring decimals, and how to do the four operations fast.

1

Kinds of fractions

Simple and decimal

All the names for simple and decimal fractions, and which fractions give a terminating decimal.

Simple fractions

Numerator, denominator and the kinds

The parts of a fraction, proper, improper and mixed fractions, equivalent fractions and reciprocals.

When something is cut into equal parts and we take some of them, we write $\frac{numerator}{denominator}$ . The denominator says into how many equal parts the whole was cut; the numerator says how many of them are taken. $\frac{3}{8}$ means: the whole was cut into 8 equal parts and 3 are taken.

Kind	How to spot it	Example
Proper fraction	Numerator < denominator (value below 1)	$\frac{5}{9}$
Improper fraction	Numerator ≥ denominator (value 1 or more)	$\frac{17}{5}$
Mixed number	A whole number plus a proper fraction	$3 \frac{2}{5}$
Like fractions	Same denominators	$\frac{2}{7}, \frac{5}{7}$
Unlike fractions	Different denominators	$\frac{2}{3}, \frac{5}{7}$
Equivalent fractions	Multiply or divide top and bottom by the same number	$\frac{2}{3} = \frac{4}{6} = \frac{10}{15}$

Mixed → improper: $3 \frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{17}{5}$ (whole × denominator + numerator).
Improper → mixed: $\frac{17}{5}$ : 17 ÷ 5 = 3 remainder 2 → $3 \frac{2}{5}$ .
Reciprocal: the reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$ ; a fraction × its reciprocal = 1.

Example 1 · Basic

Convert

4 \frac{3}{7}

to an improper fraction and

\frac{59}{8}

to a mixed number.

Solution, step by step

1First: (4 × 7 + 3) ÷ 7 = 31/7.
2Second: 59 ÷ 8 = 7 remainder 3 → $7 \frac{3}{8}$ .

Answer $\frac{31}{7}$ and $7 \frac{3}{8}$

BasicTry it yourself first

Write

\frac{38}{7}

as a mixed number.

BasicTry it yourself first

Write

9 \frac{4}{6}

as an improper fraction.

StandardTry it yourself first

Write

\frac{85}{12}

as a mixed number.

StandardTry it yourself first

Write

7 \frac{5}{9}

as an improper fraction.

Decimal fractions: terminating and recurring

Which fraction stops when

What a decimal fraction is, and how to tell in advance whether a fraction gives a terminating or a recurring decimal.

A fraction whose denominator is 10, 100, 1000 … is a decimal fraction, written with a point: $\frac{35}{100} = 0.35$ . After the point the first place is tenths (1/10), the second hundredths (1/100), the third thousandths (1/1000) …

Kind	Meaning	Example
Terminating decimal	Ends after a few digits	$\frac{3}{8} = 0.375$
Pure recurring	Digits repeat right after the point	$\frac{1}{3} = 0. \overline{3}$ , $\frac{4}{11} = 0. \overline{36}$
Mixed recurring	A few digits once, then repetition	$\frac{1}{6} = 0.1 \overline{6}$

RememberThe test: put the fraction in lowest terms. If the denominator has only 2s and/or 5s as prime factors the decimal terminates; any other prime (3, 7, 11 …) makes it recurring.

Example 2 · Standard

Which of

\frac{7}{40}

and

\frac{5}{12}

gives a terminating decimal? Write it.

Solution, step by step

140 = 2³ × 5 → only 2 and 5 → terminating.
212 = 2² × 3 → has a 3 → recurring.
3 $\frac{7}{40} = 0.175$ (dividing 7 by 40).

Answer $\frac{7}{40} = 0.175$

BasicTry it yourself first

Which of these fractions has a terminating decimal?

BasicTry it yourself first

Which of these fractions has a terminating decimal?

StandardTry it yourself first

Which of these fractions has a terminating decimal?

StandardTry it yourself first

Which of these fractions has a terminating decimal?

StandardTry it yourself first

Which of these fractions has a terminating decimal?

2

Changing the form

Fraction ↔ decimal ↔ recurring

Turning fractions into decimals, decimals into fractions, and recurring decimals into simple fractions.

Fractions and terminating decimals

Numerator ÷ denominator, and over

1 0^{k}

Fraction to decimal (by dividing) and terminating decimal to fraction (using a power of 10).

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Recurring decimals to fractions

A denominator of nines and zeros

The rule that gives the fraction straight from the repeating digits.

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3

The four operations

Add · subtract · multiply · divide

The four operations on simple fractions and decimals, and the easiest route in each.

Operations on simple fractions

Common denominators, cross-cancelling, flip and multiply

Equalise denominators to add or subtract; multiply numerators and denominators directly; flip the divisor to divide.

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Operations on decimals

Count the decimal places

Line up the points to add or subtract, count the total decimal places to multiply, make the divisor whole to divide.

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4

Comparing fractions

Which is bigger

Four ways to order fractions and how to pick the fastest.

Four ways to compare

Cross-multiply · equal denominators · equal numerators · decimals

Which method is fastest in each situation.

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Putting in order

Ascending and descending

Arranging three or four fractions from small to big (or the reverse).

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5

Word problems

Fractions and decimals in use

The word “of” means multiply, “the rest” means subtract — one pattern for fraction word problems.

A part of a quantity and the rest

“Of” = multiply, “the rest” = subtract

Taking a fraction of a quantity, the reverse question (part to whole) and fractions applied one after another.

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Decimal word problems

Prices, measures and pieces

Decimal questions about rupees, kilos and metres.

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6

Choosing the method, traps and fast tricks

Clues · traps · tricks

Which method for which situation, the commonest slips and some fast tricks.

The commonest mistakes

Six traps

Mistakes with fractions and decimals are mostly carelessness.

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Fast tricks

Telescoping and neat identities

A few patterns that shrink a long calculation.

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7

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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8

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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Kinds of fractions

Simple fractions

Decimal fractions: terminating and recurring

Changing the form

Fractions and terminating decimals

Recurring decimals to fractions

The four operations

Operations on simple fractions

Operations on decimals

Comparing fractions

Four ways to compare

Putting in order

Word problems

A part of a quantity and the rest

Decimal word problems

Choosing the method, traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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