Rational Numbers

What a rational number is, why integers, fractions, mixed numbers and decimals that terminate or recur are all rational, how fractions and decimals are changed into each other, and how negative fractions and mixed numbers are written.

Before you start

The integers are $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ On the number line, the positive numbers are to the right of 0 and the negative numbers are to the left.

The numerical value of a number is its distance from 0 on the number line: 5 and $-5$ both have numerical value 5. Two numbers like these, with the same numerical value and different signs, are opposites.

In division, two numbers with the same sign give a positive answer and two numbers with different signs give a negative answer: $-12 \div 4 = -3$ , $12 \div (-4) = -3$ and $-12 \div (-4) = 3$ . Division by 0 is not defined.

A negative number that follows an operation sign is written in brackets, as in $12 \div (-4)$ . At the start of a calculation it is written without brackets, as in $-12 \div 4$ , which means the same as $(-12) \div 4$ .

A fraction is a division: $\frac{3}{4} = 3 \div 4$ . A mixed number is a whole number plus a fraction: $2\frac{1}{4} = 2 + \frac{1}{4} = \frac{9}{4}$ , because $2 \times 4 + 1 = 9$ .

What a Rational Number Is

When one integer is divided by another integer that is not 0, the answer is often not an integer: $3 \div 4 = \frac{3}{4}$ . The integers and all such fractions, positive and negative, are the rational numbers.

Definition

A rational number is a number that can be written in the form $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ .

The integers $a$ and $b$ may be positive or negative, and $a$ may be 0. But $b$ cannot be 0, because division by 0 is not defined.

All these kinds of numbers are rational, because each can be written in that form.

Kind of number	Example	Written as a fraction of two integers
Integer	$-4$ and $0$	$-4 = \frac{-4}{1}$ and $0 = \frac{0}{1}$
Fraction	$\frac{2}{3}$	It is already in this form.
Mixed number	$2\frac{1}{4}$	$2\frac{1}{4} = \frac{9}{4}$
Terminating decimal (it has a last digit)	$0.75$	$0.75 = \frac{75}{100} = \frac{3}{4}$
Recurring decimal (a digit or a group of digits repeats without end)	$0.333\ldots$	$0.333\ldots = \frac{1}{3}$ , because $1 \div 3 = 0.333\ldots$

A rational number can be written as a fraction in many ways: $0.75 = \frac{3}{4} = \frac{6}{8} = \frac{75}{100}$ . To show that a number is rational, one of them is enough.

Example 1: Showing that a number is rational

Show that each of these numbers is rational.
(a) $-15$
(b) $3\frac{2}{7}$
(c) $0.6$
(d) $\frac{0}{9}$
(e) Explain why $\frac{9}{0}$ is not a rational number.

Solution

(a) An integer is itself divided by 1: $-15 = \frac{-15}{1}$ .

(b) Change the mixed number to an improper fraction. $3 \times 7 + 2 = 23$ , so $3\frac{2}{7} = \frac{23}{7}$ .

(d) $\frac{0}{9}$ is already in the form $\frac{a}{b}$ , with $a = 0$ and $b = 9$ . Its value is $0 \div 9 = 0$ .

(e) Its denominator is 0, and division by 0 is not defined. So $\frac{9}{0}$ has no value: it is not a number at all.

(a) $\frac{-15}{1}$ (b) $\frac{23}{7}$ (c) $\frac{3}{5}$ (d) $\frac{0}{9}$ . In each one, $a$ and $b$ are integers and $b \neq 0$ , so the number is rational. (e) $\frac{9}{0}$ is not defined.

Rational Numbers as Decimals

Every rational number can also be written as a decimal. The decimal is one of two kinds.

Terminating and recurring decimals

A terminating decimal has a last digit, like $0.75$ .

A recurring decimal has a digit or a group of digits that repeats without end, like $0.333\ldots$ The three dots show that the digits go on. Without them, $0.333$ is $\frac{333}{1000}$ , which is not $\frac{1}{3}$ .

Changing a terminating decimal to a fraction

Count the decimal places. Write the digits after the decimal point over 10 for one decimal place, over 100 for two decimal places and over 1000 for three.
Keep a whole-number part in front, as the whole part of a mixed number.
Simplify the fraction.

For example, $0.75 = \frac{75}{100} = \frac{3}{4}$ and $2.3 = 2\frac{3}{10}$ .

Changing a fraction to a decimal

For a mixed number, keep the whole part and change only the fraction part: $2\frac{3}{5} = 2.6$ .
When the denominator is a factor of 10, 100 or 1000, there is a quick way: write an equivalent fraction with that denominator, as in $\frac{3}{5} = \frac{6}{10} = 0.6$ .
In every other case, and whenever the quick way is hard to see, divide the numerator by the denominator: division works for every fraction. Write a 0 after the remainder for each new decimal place, and go on until the remainder is 0, or until a remainder comes back that you have had before. From there the same digits repeat.

Watch the remainders, not the digits. A digit can come twice while the remainders are still different, and then the decimal has not started to repeat.

Example 2: Terminating decimals and fractions

(a) Write $0.48$ and $1.075$ as fractions in their simplest form.
(b) Write $\frac{13}{20}$ and $\frac{3}{8}$ as decimals.

Solution

(a) $0.48$ has two decimal places, so $0.48 = \frac{48}{100}$ . Divide 48 and 100 by 4: $\frac{48}{100} = \frac{12}{25}$ .

$1.075$ has three decimal places, so $1.075 = 1\frac{75}{1000}$ . Divide 75 and 1000 by 25: $1\frac{75}{1000} = 1\frac{3}{40}$ . As an improper fraction this is $\frac{43}{40}$ .

(b) 20 is a factor of 100, so $\frac{13}{20} = \frac{65}{100} = 0.65$ .

For $\frac{3}{8}$ , divide 3 by 8. 3 is less than 8, so the whole-number part is 0. Then 30 divided by 8 is 3 remainder 6, 60 divided by 8 is 7 remainder 4, and 40 divided by 8 is 5 with no remainder. The division stops: $3 \div 8 = 0.375$ .

(a) $0.48 = \frac{12}{25}$ and $1.075 = 1\frac{3}{40}$ , or $\frac{43}{40}$ (b) $\frac{13}{20} = 0.65$ and $\frac{3}{8} = 0.375$

For many fractions the division never stops. A remainder comes back, so the same digits come round again and again, and the decimal recurs. There is a short way to write such a decimal.

Writing a recurring decimal with a dot above the digit

When one digit repeats, put a dot above it: $0.333\ldots = 0.\dot{3}$ .

When a group of digits repeats, put a dot above the first and the last digit of the group only: $0.8181\ldots = 0.\dot{8}\dot{1}$ and $0.123123\ldots = 0.\dot{1}2\dot{3}$ .

Digits that do not repeat have no dot above them: $0.58333\ldots = 0.58\dot{3}$ .

Fraction	Decimal with three dots	With a dot above the digit	What repeats
$\frac{1}{3}$	$0.333\ldots$	$0.\dot{3}$	the digit 3
$\frac{7}{12}$	$0.58333\ldots$	$0.58\dot{3}$	the digit 3 only; the 5 and the 8 do not repeat
$\frac{9}{11}$	$0.8181\ldots$	$0.\dot{8}\dot{1}$	the two digits 81
$\frac{41}{333}$	$0.123123\ldots$	$0.\dot{1}2\dot{3}$	the three digits 123; the 2 in the middle has no dot above it

Example 3: Fractions as recurring decimals

Write each fraction as a decimal.
(a) $-\frac{4}{3}$
(b) $\frac{5}{6}$
(c) $\frac{1}{7}$

Solution

(a) Divide 4 by 3, then put the negative sign back. 4 divided by 3 is 1 remainder 1. Then 10 divided by 3 is 3 remainder 1, and the remainder is 1 every time after that. So the digit 3 repeats without end: $4 \div 3 = 1.333\ldots$

(b) Divide 5 by 6. 5 is less than 6, so the whole-number part is 0. Then 50 divided by 6 is 8 remainder 2. Next, 20 divided by 6 is 3 remainder 2, and the remainder is 2 every time after that. So the digit 3 repeats without end, but the 8 does not: $5 \div 6 = 0.8333\ldots$

(c) Divide 1 by 7. The whole-number part is 0. Then 10 divided by 7 is 1 remainder 3, 30 divided by 7 is 4 remainder 2, 20 divided by 7 is 2 remainder 6, 60 divided by 7 is 8 remainder 4, 40 divided by 7 is 5 remainder 5, and 50 divided by 7 is 7 remainder 1. The remainder 1 is the number the division started with, so the six digits 142857 repeat without end: $1 \div 7 = 0.142857142857\ldots$

(a) $-\frac{4}{3} = -1.333\ldots = -1.\dot{3}$ (b) $\frac{5}{6} = 0.8333\ldots = 0.8\dot{3}$ (c) $\frac{1}{7} = 0.142857142857\ldots = 0.\dot{1}4285\dot{7}$ . All three are recurring decimals.

Watch the remainders, not the digits

A digit can come twice before the decimal starts to repeat. Take $\frac{1}{44}$ and divide 1 by 44. 1 is less than 44, so the whole-number part is 0. Then 10 divided by 44 is 0 remainder 10, 100 divided by 44 is 2 remainder 12, and 120 divided by 44 is 2 remainder 32.

The digit 2 has come twice, but the remainders 12 and 32 are different, so nothing is repeating yet. Go on: 320 divided by 44 is 7 remainder 12. Now the remainder 12 has come back. The digits found since it first came up are 2 and 7, so the two digits 27 repeat without end: $\frac{1}{44} = 0.0227272\ldots = 0.02\dot{2}\dot{7}$ .

Stopping at the second 2 would give $0.0\dot{2} = 0.0222\ldots$ , which is a different number.

Why the decimal always terminates or recurs

In a division, every remainder is less than the divisor. When you divide by 7, the only remainders there can be are 0, 1, 2, 3, 4, 5 and 6. If a remainder is 0, the division stops and the decimal terminates. If no remainder is 0, a remainder must come back sooner or later, and from there the same digits repeat: the decimal recurs.

So when a fraction is changed to a decimal by division, the decimal always terminates or recurs.

It is true the other way round as well: every terminating decimal and every recurring decimal is a rational number. A terminating decimal is a fraction over 10, 100, 1000 and so on. A recurring decimal is also equal to a fraction of two integers. You do not need a method for finding that fraction, but these are worth knowing: $0.\dot{3} = \frac{1}{3}$ , $0.\dot{6} = \frac{2}{3}$ , $0.1\dot{6} = \frac{1}{6}$ and $0.\dot{1} = \frac{1}{9}$ .

A useful check: telling without dividing

This is a quick way to check an answer, not a rule to memorise: you can always find out by dividing. It uses the prime factors of a number, from the work on prime factorisation.

Write the fraction in its simplest form and find the prime factors of its denominator. If the denominator has no prime factor other than 2 and 5, the decimal terminates, because the fraction can then be written over 10, 100, 1000 and so on. If the denominator has any other prime factor, the decimal recurs.

$\frac{13}{20}$ terminates, because $20 = 2^{2} \times 5$ . $\frac{5}{6}$ recurs, because $6 = 2 \times 3$ has the prime factor 3. $\frac{9}{12}$ is not in its simplest form: $\frac{9}{12} = \frac{3}{4}$ and $4 = 2^{2}$ , so it terminates.

Negative Fractions and Mixed Numbers

A fraction is a division: $\frac{-3}{4}$ means $-3 \div 4$ , and $\frac{3}{-4}$ means $3 \div (-4)$ . In both, the signs are different, so both are negative and equal to $-\frac{3}{4}$ . In the same way, $\frac{-70}{14} = -5$ and $\frac{85}{-17} = -5$ .

Rule

$-\frac{3}{4} = \frac{-3}{4} = \frac{3}{-4}$ . The negative sign of a fraction can be written in front, in the numerator or in the denominator. We usually write it in front. Two negative signs give a positive fraction: $\frac{-3}{-4} = \frac{3}{4}$ .

$-\frac{3}{4}$ and $\frac{3}{4}$ are opposites. They have the same numerical value, $\frac{3}{4}$ , and lie on opposite sides of 0, at the same distance from it.

Negative mixed numbers

The negative sign belongs to the whole mixed number: $-2\frac{3}{5}$ means $-\left(2 + \frac{3}{5}\right) = -\frac{13}{5}$ . It does not mean $-2 + \frac{3}{5}$ . That number is $\frac{3}{5}$ of a unit to the right of $-2$ , which is $\frac{2}{5}$ of a unit to the left of $-1$ : it is $-1\frac{2}{5}$ .

To change a negative mixed number to an improper fraction, change the mixed number without its sign, then put the sign back: $2\frac{3}{5} = \frac{13}{5}$ , so $-2\frac{3}{5} = -\frac{13}{5}$ .

On the number line, $-2\frac{3}{5}$ is $2\frac{3}{5}$ units to the left of 0, so it lies between $-3$ and $-2$ .

The other point, $-1\frac{2}{5}$ , is 3 parts to the right of $-2$ . It is where $-2 + \frac{3}{5}$ is, and it is a different number.

Example 4: The sign of a fraction and of a mixed number

(a) Write $\frac{5}{-8}$ with the negative sign in front.
(b) Write $\frac{-12}{-18}$ in its simplest form.
(c) Write $-3\frac{2}{7}$ as an improper fraction.
(d) Write $-\frac{17}{5}$ as a mixed number.
(e) Write $-1.75$ as a mixed number, and then in the form $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ .

Solution

(a) $\frac{5}{-8}$ means $5 \div (-8)$ . The signs are different, so the fraction is negative: $\frac{5}{-8} = -\frac{5}{8}$ .

(b) $\frac{-12}{-18}$ means $-12 \div (-18)$ . The signs are the same, so the fraction is positive: $\frac{-12}{-18} = \frac{12}{18}$ . Divide 12 and 18 by 6: $\frac{12}{18} = \frac{2}{3}$ .

(c) Without the sign, $3 \times 7 + 2 = 23$ , so $3\frac{2}{7} = \frac{23}{7}$ . Put the sign back: $-3\frac{2}{7} = -\frac{23}{7}$ .

(d) Without the sign, 17 divided by 5 is 3 remainder 2, so $\frac{17}{5} = 3\frac{2}{5}$ . Put the sign back: $-\frac{17}{5} = -3\frac{2}{5}$ .

(e) Without the sign, $1.75 = 1\frac{75}{100} = 1\frac{3}{4} = \frac{7}{4}$ . Put the sign back: $-1.75 = -1\frac{3}{4} = -\frac{7}{4}$ . With the sign in the numerator, this is $\frac{-7}{4}$ .

(a) $-\frac{5}{8}$ (b) $\frac{2}{3}$ (c) $-\frac{23}{7}$ (d) $-3\frac{2}{5}$ (e) $-1\frac{3}{4}$ , which is $\frac{-7}{4}$

Example 5: Rational numbers in measurements

(a) Siti pours 2 litres of barley drink equally into 9 cups. Write the amount in each cup, in litres, as a fraction and as a decimal.
(b) The lowest floor of a car park is $7\frac{1}{4}$ m below ground level, so its height is written as $-7\frac{1}{4}$ m. Write $-7\frac{1}{4}$ as a decimal, and then in the form $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ .

Solution

(a) Each cup has $2 \div 9 = \frac{2}{9}$ litre. For the decimal, divide 2 by 9. 2 is less than 9, so the whole-number part is 0. Then 20 divided by 9 is 2 remainder 2, and the remainder is 2 every time after that. So the digit 2 repeats without end: $2 \div 9 = 0.222\ldots$

(b) Change the mixed number without its sign. Keep the whole part and change the fraction part: 4 is a factor of 100, so $\frac{1}{4} = \frac{25}{100} = 0.25$ and $7\frac{1}{4} = 7.25$ . Put the sign back: $-7\frac{1}{4} = -7.25$ .

For the form $\frac{a}{b}$ , $7 \times 4 + 1 = 29$ , so $7\frac{1}{4} = \frac{29}{4}$ . Put the sign back: $-7\frac{1}{4} = -\frac{29}{4}$ . With the sign in the numerator, this is $\frac{-29}{4}$ .

(a) Each cup has $\frac{2}{9}$ litre, which is $0.222\ldots = 0.\dot{2}$ litre. (b) $-7\frac{1}{4} = -7.25$ , which is $\frac{-29}{4}$ . Both measurements are rational numbers.

Common Mistakes

Leaving out the three dots, the dot above the digit, or the sign of a recurring decimal. $-\frac{5}{3} = -1.666\ldots = -1.\dot{6}$ . Written as $1.\dot{6}$ it has lost its negative sign, and written as $-1.666$ it is a different number.
Putting a dot above every digit that repeats. Only the first and the last digit of the group have a dot above them: $0.123123\ldots$ is $0.\dot{1}2\dot{3}$ .
Stopping a division as soon as a digit comes twice. Go on until a remainder comes back: $\frac{1}{44} = 0.02\dot{2}\dot{7}$ , not $0.0\dot{2}$ .
Reading $-2\frac{3}{5}$ as $-2 + \frac{3}{5}$ . The sign belongs to the whole mixed number: $-2\frac{3}{5} = -\left(2 + \frac{3}{5}\right) = -\frac{13}{5}$ .
Mixing up $\frac{0}{9}$ and $\frac{9}{0}$ . $\frac{0}{9} = 0$ , which is a rational number. $\frac{9}{0}$ is not a number, because division by 0 is not defined.

Practice

Work each question on paper before you open its solution.

Question 1

Show that each number is rational by writing it in the form $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ .
(a) $-9$
(b) $-3\frac{2}{5}$
(c) $-0.36$

ShowHide solution

(a) $-9 = \frac{-9}{1}$

(b) $-3\frac{2}{5} = -\frac{17}{5} = \frac{-17}{5}$

(a) $\frac{-9}{1}$ (b) $\frac{-17}{5}$ (c) $\frac{-9}{25}$ . Each is an integer over an integer that is not 0, so each number is rational.

Question 2

Write each decimal as a fraction in its simplest form.
(a) $0.85$
(b) $0.008$
(c) $-2.04$

ShowHide solution

(a) Two decimal places: $0.85 = \frac{85}{100}$ . Divide 85 and 100 by 5: $\frac{17}{20}$

(b) Three decimal places: $0.008 = \frac{8}{1000}$ . Divide 8 and 1000 by 8: $\frac{1}{125}$

(c) Without the sign, $2.04 = 2\frac{4}{100} = 2\frac{1}{25}$ . Put the sign back: $-2.04 = -2\frac{1}{25}$ . As an improper fraction, $2 \times 25 + 1 = 51$ , so this is $-\frac{51}{25}$ .

(a) $\frac{17}{20}$ (b) $\frac{1}{125}$ (c) $-2\frac{1}{25}$ , or $-\frac{51}{25}$

Question 3

Write each fraction as a decimal, and say whether the decimal terminates or recurs.
(a) $-\frac{7}{8}$
(b) $\frac{3}{11}$
(c) $\frac{2}{7}$

ShowHide solution

(a) Divide 7 by 8, then put the negative sign back. The whole-number part is 0. Then 70 divided by 8 is 8 remainder 6, 60 divided by 8 is 7 remainder 4, and 40 divided by 8 is 5 with no remainder. So $7 \div 8 = 0.875$ exactly.

(b) Divide 3 by 11. The whole-number part is 0. Then 30 divided by 11 is 2 remainder 8, then 80 divided by 11 is 7 remainder 3, and after that the remainders 8 and 3 keep coming back. So the digits 2 and 7 repeat without end: $3 \div 11 = 0.2727\ldots$

(c) Divide 2 by 7. The whole-number part is 0. Then 20 divided by 7 is 2 remainder 6, 60 divided by 7 is 8 remainder 4, 40 divided by 7 is 5 remainder 5, 50 divided by 7 is 7 remainder 1, 10 divided by 7 is 1 remainder 3, and 30 divided by 7 is 4 remainder 2. The remainder 2 is the number the division started with, so the six digits 285714 repeat without end: $2 \div 7 = 0.285714285714\ldots$

(a) $-\frac{7}{8} = -0.875$ , a terminating decimal. (b) $\frac{3}{11} = 0.2727\ldots = 0.\dot{2}\dot{7}$ , a recurring decimal. (c) $\frac{2}{7} = 0.285714285714\ldots = 0.\dot{2}8571\dot{4}$ , a recurring decimal.

Question 4

(a) Write each recurring decimal in the short way, putting a dot above each digit that needs one: $0.444\ldots$ , $2.53636\ldots$ and $0.702702\ldots$
(b) Write $0.\dot{6}\dot{1}$ and $1.\dot{2}3\dot{4}$ with three dots in place of the dots above the digits, showing the first eight decimal places.
(c) Use the check with prime factors to say whether each fraction gives a terminating or a recurring decimal: $\frac{9}{40}$ , $\frac{7}{12}$ and $\frac{21}{28}$ . Then confirm each answer by dividing.

ShowHide solution

(a) In $0.444\ldots$ the digit 4 repeats. In $2.53636\ldots$ the two digits 36 repeat and the 5 does not. In $0.702702\ldots$ the three digits 702 repeat, so a dot goes above the 7 and above the 2 only.

(b) In $0.\dot{6}\dot{1}$ the digits 61 repeat. In $1.\dot{2}3\dot{4}$ the dots are above the first and the last digit of the group 234, so 234 repeats.

(c) $\frac{9}{40}$ and $\frac{7}{12}$ are already in their simplest form. $40 = 2^{3} \times 5$ , which has no prime factor other than 2 and 5, so $\frac{9}{40}$ terminates. $12 = 2^{2} \times 3$ , which has the prime factor 3, so $\frac{7}{12}$ recurs. $\frac{21}{28}$ is not in its simplest form: divide 21 and 28 by 7 to get $\frac{3}{4}$ . Since $4 = 2^{2}$ , it terminates.

By division, for $\frac{9}{40}$ : the whole-number part is 0. Then 90 divided by 40 is 2 remainder 10, 100 divided by 40 is 2 remainder 20, and 200 divided by 40 is 5 with no remainder. The division stops: $9 \div 40 = 0.225$ . The digit 2 came twice, but the remainders 10 and 20 were different, so nothing repeated.

For $\frac{7}{12}$ : the whole-number part is 0. Then 70 divided by 12 is 5 remainder 10, 100 divided by 12 is 8 remainder 4, and 40 divided by 12 is 3 remainder 4. The remainder 4 has come back, so the digit 3 repeats without end: $7 \div 12 = 0.58333\ldots$

For $\frac{21}{28} = \frac{3}{4}$ : the whole-number part is 0. Then 30 divided by 4 is 7 remainder 2, and 20 divided by 4 is 5 with no remainder. The division stops: $3 \div 4 = 0.75$ .

(a) $0.\dot{4}$ , $2.5\dot{3}\dot{6}$ and $0.\dot{7}0\dot{2}$ (b) $0.61616161\ldots$ and $1.23423423\ldots$ (c) $\frac{9}{40}$ terminates, $\frac{7}{12}$ recurs and $\frac{21}{28}$ terminates. By division, $\frac{9}{40} = 0.225$ , $\frac{7}{12} = 0.58\dot{3}$ and $\frac{21}{28} = 0.75$ .

Question 5

(a) Write $\frac{7}{-9}$ with the negative sign in front.
(b) Write $\frac{-15}{-20}$ in its simplest form.
(c) Write $-4\frac{5}{6}$ as an improper fraction.
(d) Write $-\frac{31}{8}$ as a mixed number.
(e) Mei Ling says that $-1\frac{1}{2}$ is the same number as $-1 + \frac{1}{2}$ . Is she right?

ShowHide solution

(a) $\frac{7}{-9}$ means $7 \div (-9)$ . The signs are different, so the fraction is negative.

(b) The signs are the same, so the fraction is positive: $\frac{-15}{-20} = \frac{15}{20}$ . Divide 15 and 20 by 5.

(d) Without the sign, 31 divided by 8 is 3 remainder 7, so $\frac{31}{8} = 3\frac{7}{8}$ . Put the sign back.

(e) $-1\frac{1}{2}$ means $-\left(1 + \frac{1}{2}\right) = -\frac{3}{2}$ , which lies between $-2$ and $-1$ . But $-1 + \frac{1}{2}$ is half a unit to the right of $-1$ , which is $-\frac{1}{2}$ .

(a) $-\frac{7}{9}$ (b) $\frac{3}{4}$ (c) $-\frac{29}{6}$ (d) $-3\frac{7}{8}$ (e) No. $-1\frac{1}{2} = -\frac{3}{2}$ , but $-1 + \frac{1}{2} = -\frac{1}{2}$ .

Question 6

Say whether each statement is true or false, and give a reason.
(a) Every integer is a rational number.
(b) $\frac{0}{-6}$ is a rational number.
(c) $\frac{-6}{0}$ is a rational number.
(d) $0.777$ and $0.\dot{7}$ are the same number.

ShowHide solution

(a) Any integer can be written as itself divided by 1, for example $-8 = \frac{-8}{1}$ . This is an integer over an integer that is not 0.

(b) It is in the form $\frac{a}{b}$ with $a = 0$ and $b = -6$ . Both are integers and $b \neq 0$ . Its value is $0 \div (-6) = 0$ .

(d) $0.777$ is a terminating decimal: it stops after three decimal places and is equal to $\frac{777}{1000}$ . But $0.\dot{7}$ is the recurring decimal $0.7777\ldots$ , which has a 7 in the fourth decimal place. In $0.777$ the fourth decimal place is 0. So $0.\dot{7}$ is a little greater than $0.777$ .

(a) True (b) True (c) False (d) False

Question 7

A shop sells nuts by mass. Three bags of nuts have masses of $\frac{4}{5}$ kg, $\frac{7}{25}$ kg and $\frac{5}{16}$ kg.
(a) Write each mass as a decimal.
(b) A fourth bag has a mass of $0.275$ kg. Write this mass as a fraction of a kilogram in its simplest form.

ShowHide solution

(a) 5 is a factor of 10, so $\frac{4}{5} = \frac{8}{10} = 0.8$ .

25 is a factor of 100, so $\frac{7}{25} = \frac{28}{100} = 0.28$ .

For $\frac{5}{16}$ , divide 5 by 16. The whole-number part is 0. Then 50 divided by 16 is 3 remainder 2, 20 divided by 16 is 1 remainder 4, 40 divided by 16 is 2 remainder 8, and 80 divided by 16 is 5 with no remainder. The division stops: $5 \div 16 = 0.3125$ .

(b) $0.275$ has three decimal places, so $0.275 = \frac{275}{1000}$ . Divide 275 and 1000 by 25: $\frac{275}{1000} = \frac{11}{40}$ .

(a) The masses are $0.8$ kg, $0.28$ kg and $0.3125$ kg. (b) The mass of the fourth bag is $\frac{11}{40}$ kg.

Question 8

After a dry month, the change in the water level of a reservoir is $-1\frac{7}{20}$ m.
(a) What does the negative sign tell you about the water level?
(b) Write $-1\frac{7}{20}$ as an improper fraction.
(c) Write $-1\frac{7}{20}$ as a decimal. Does the decimal terminate or recur?

ShowHide solution

(a) A negative change is a fall. The water level went down by $1\frac{7}{20}$ m.

(b) Without the sign, $1 \times 20 + 7 = 27$ , so $1\frac{7}{20} = \frac{27}{20}$ . Put the sign back.

(c) Keep the whole part and change the fraction part. 20 is a factor of 100, so $\frac{7}{20} = \frac{35}{100} = 0.35$ and $1\frac{7}{20} = 1.35$ . Put the sign back. The decimal has a last digit, so it terminates.

(a) The water level fell. (b) $-\frac{27}{20}$ (c) $-1\frac{7}{20} = -1.35$ , a terminating decimal. The change in the water level is $-1.35$ m.

Question 9

A stall holder pours 5 litres of sugar cane juice equally into 12 cups.
(a) Write the amount in each cup as a fraction of a litre.
(b) Write this amount as a decimal, with a dot above the digit that repeats.
(c) His helper writes the amount in each cup as $0.416$ litre. Is this exactly the amount in each cup? Explain.

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(a) Each cup has $5 \div 12 = \frac{5}{12}$ litre.

(b) Divide 5 by 12. The whole-number part is 0. Then 50 divided by 12 is 4 remainder 2, 20 divided by 12 is 1 remainder 8, and 80 divided by 12 is 6 remainder 8. The remainder is 8 every time after that. So the digit 6 repeats without end, but the 4 and the 1 do not: $5 \div 12 = 0.41666\ldots$

(c) $0.416$ is a terminating decimal, equal to $\frac{416}{1000}$ . The amount in each cup is $0.41666\ldots$ , which has a 6 in the fourth decimal place. In $0.416$ the fourth decimal place is 0. The two numbers are close, but they are not equal.

(a) $\frac{5}{12}$ litre (b) $0.41\dot{6}$ litre (c) No. $0.416$ is a little less than $\frac{5}{12} = 0.41\dot{6}$ .

Question 10

Hui Min changes $\frac{3}{13}$ to a decimal by dividing 3 by 13.
(a) Which remainders are possible when a whole number is divided by 13?
(b) Explain why the group of digits that repeats cannot have more than 12 digits.
(c) Do the division, and write $\frac{3}{13}$ as a decimal, putting a dot above each digit that needs one.

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(a) Every remainder is less than the divisor, so the possible remainders are the whole numbers from 0 to 12.

(b) If a remainder is 0, the division stops. If no remainder is 0, every remainder is one of the 12 numbers from 1 to 12. So within 12 steps of the division, a remainder that has come up before must come back, and from there the same digits repeat.

(c) The whole-number part is 0. Then 30 divided by 13 is 2 remainder 4, 40 divided by 13 is 3 remainder 1, 10 divided by 13 is 0 remainder 10, 100 divided by 13 is 7 remainder 9, 90 divided by 13 is 6 remainder 12, and 120 divided by 13 is 9 remainder 3. The remainder 3 is the number the division started with, so the six digits 230769 repeat without end: $3 \div 13 = 0.230769230769\ldots$

(a) 0, 1, 2 and so on, up to 12 (b) There are only 12 remainders other than 0, so a remainder must come back within 12 steps. (c) $\frac{3}{13} = 0.230769230769\ldots = 0.\dot{2}3076\dot{9}$ , a recurring decimal with a group of six digits.

Key Points

A rational number can be written as $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ . Integers, fractions, mixed numbers, terminating decimals and recurring decimals are all rational.
A terminating decimal is a fraction over 10, 100, 1000 and so on: $0.48 = \frac{48}{100} = \frac{12}{25}$ . A fraction is changed to a decimal by division. Go on until the remainder is 0 or a remainder comes back: the decimal always terminates or recurs.
A recurring decimal is written with a dot above the digit that repeats, or above the first and the last digit of the group that repeats: $\frac{1}{3} = 0.\dot{3}$ , $\frac{9}{11} = 0.\dot{8}\dot{1}$ and $\frac{1}{7} = 0.\dot{1}4285\dot{7}$ .
$-\frac{3}{4} = \frac{-3}{4} = \frac{3}{-4}$ , and $-2\frac{3}{5}$ means $-\left(2 + \frac{3}{5}\right) = -\frac{13}{5}$ .
To change a negative mixed number or a negative decimal, change it without its sign, then put the sign back: $-1.75 = -1\frac{3}{4} = -\frac{7}{4}$ .