SPI Mathematics · Number System

Fractions and Rational Numbers

A rational number is one integer divided by another non-zero integer; you compare, add and take fractions of them with a common denominator.

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Why this matters

Ten questions in the bank, six EASY and four MODERATE. Three test what a rational number is and which one lies between two others, four are fraction sums and word problems, and three turn a recurring decimal into a fraction.

Concept 1 of 3: Rational numbers and comparing them

A rational number is any number you can write as a fraction of two integers. To compare two of them, give them the same denominator or turn both into decimals, then read them on the number line.

Definition

A rational number is $\frac{p}{q}$ where $p$ and $q$ are integers and $q \ne 0$ .
The numerator may be 0: $\frac{0}{5} = 0$ is rational. The denominator may be negative: $\frac{3}{-4} = -\frac{3}{4}$ . Only a zero denominator is barred.
Every integer is rational: $7 = \frac{7}{1}$ .
On the negative side, a bigger size means a smaller number: $-\frac{1}{2} < -\frac{1}{4}$ .
Between any two rationals lies another, for example their average.

Rational number

\frac{p}{q}, \quad p, q \text{ integers}, \quad q \ne 0 \qquad \text{a number between } x \text{ and } y: \frac{x + y}{2}

Worked example

Which of

-\frac{3}{4}

-\frac{2}{5}

and

-\frac{1}{2}

lies between

-\frac{7}{10}

and

-\frac{9}{20}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Number System · Fractions and Rational Numbers

Choose the rational number which does not lie between

-\frac{2}{3}

and

-\frac{1}{5}

Reading negative fractions by size

-\frac{2}{3}

is less than

-\frac{1}{5}

, because it lies further left on the number line. Every number between two negatives is negative, so a positive option can never lie between them.

Thinking the numerator cannot be 0

\frac{0}{q}

is a rational number (it is 0) for any non-zero integer q. The rule that forbids 0 is for the denominator.

Drill 2 more on rational numbers and comparing them

Concept 2 of 3: Adding fractions and taking a fraction of a number

Fractions add only when they count the same size of piece, so first give them a common denominator. 'A fraction of' something means multiply. In a word problem, write each group as a fraction of the whole before you use any number.

Definition

Add or subtract over the LCM of the denominators: $\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$ .
$\frac{a}{b}$ of $N$ means $\frac{a}{b} \times N$ .
A fraction of the rest multiplies the fraction that is left: $\frac{1}{2}$ of the remaining $\frac{3}{5}$ is $\frac{3}{10}$ of the whole.
'What must be subtracted from $a$ to get $b$ ?' The answer is $a - b$ . 'What must be added to $a$ to get $b$ ?' The answer is $b - a$ .

Fraction sums and parts

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \qquad \frac{a}{b} \text{ of } N = \frac{a}{b} \times N

Worked example

In a class,

\frac{1}{3}

of the students play cricket and

\frac{1}{4}

play football. Half of the rest play chess, and the other 15 play nothing. How many students are in the class?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Number System · Fractions and Rational Numbers

\frac{2}{5}

of total number of students of a school come by car while

\frac{1}{4}

of students come by bus to school. All the other students walk to school of which

\frac{1}{3}

walk on their own and the rest are escorted by their parents. If 224 students come to school walking on their own, how many students study in that school?

Taking a fraction of the whole instead of the rest

If half of the remaining people stay home, multiply

\frac{1}{2}

by the fraction that remains. Taking

\frac{1}{2}

of the whole group gives a wrong total.

Adding the denominators

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

\frac{5}{6}

, not

\frac{2}{5}

. Only the numerators add, and only after the denominators are made the same.

Drill 3 more on adding fractions and taking a fraction of a number

Concept 3 of 3: Recurring decimals as fractions

A decimal that repeats forever is still a fraction. Multiply by 10 for every digit in the repeating block, subtract the original, and the endless tail cancels.

Definition

$0.\overline{6}$ means $0.666\ldots$ and $0.\overline{45}$ means $0.4545\ldots$
A one-digit block over 9, a two-digit block over 99: $0.\overline{7} = \frac{7}{9}$ , $0.\overline{12} = \frac{12}{99}$ .
Keep the whole-number part outside: $4.\overline{2} = 4 + \frac{2}{9} = \frac{38}{9}$ .
A fraction in lowest terms gives a decimal that ends when its denominator has no prime factors but 2 and 5; otherwise the decimal recurs.
Every terminating or recurring decimal is rational.

Pure recurring decimals

0.\overline{a} = \frac{a}{9} \qquad 0.\overline{ab} = \frac{ab}{99}

abthe two-digit block that repeats, read as one number

Worked example

Write

0.\overline{27}

as a fraction in lowest terms.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Number System · Fractions and Rational Numbers

The value of

2.\overline{45} + 0.\overline{36}

Putting the whole part into the block

3.\overline{18}

3 + \frac{18}{99} = \frac{35}{11}

, not

\frac{318}{99}

. Only the repeating digits go over the nines.

Mixing up where the bar starts

0.\overline{45}

0.4545\ldots

, but

0.4\overline{5}

0.4555\ldots

. They are different numbers, and both can appear among the options.

Drill 2 more on recurring decimals as fractions

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Rational numbers and comparing them
Rational number
$\frac{p}{q}, \quad p, q \text{ integers}, \quad q \ne 0 \qquad \text{a number between } x \text{ and } y: \frac{x + y}{2}$
Adding fractions and taking a fraction of a number
Fraction sums and parts
$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \qquad \frac{a}{b} \text{ of } N = \frac{a}{b} \times N$
Recurring decimals as fractions
Pure recurring decimals
$0.\overline{a} = \frac{a}{9} \qquad 0.\overline{ab} = \frac{ab}{99}$

Watch out for (6)

Reading negative fractions by size
→ Rational numbers and comparing them
Thinking the numerator cannot be 0
→ Rational numbers and comparing them
Taking a fraction of the whole instead of the rest
→ Adding fractions and taking a fraction of a number
Adding the denominators
→ Adding fractions and taking a fraction of a number
Putting the whole part into the block
→ Recurring decimals as fractions
Mixing up where the bar starts
→ Recurring decimals as fractions

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