Rational Numbers ICSE Class 8: Properties, Number Line and Word Problems

What Rational Numbers is about

Weightage: CISCE publishes no question paper, marking scheme or chapter weightage for Class VIII Mathematics: your school sets and marks the exam. Rational Numbers sits in the Number System theme of the CISCE Curriculum for Upper Primary Classes (VI-VIII).Source: CISCE Curriculum for Upper Primary Classes (VI-VIII), RDCD 2016

Key concepts

Rational numbers and standard form

A rational number $\frac{p}{q}$ is in standard form when:
1. the denominator $q$ is positive, and
2. $p$ and $q$ have no common factor other than 1.

To get standard form, move any minus sign to the numerator and divide top and bottom by their HCF. So $\frac{3}{-4}=\frac{-3}{4}$ and $\frac{-18}{-24}=\frac{3}{4}$.

Equivalent rational numbers have the same standard form: $\frac{2}{3}=\frac{4}{6}=\frac{-10}{-15}$.

To compare two rational numbers, write both with the same positive denominator and compare numerators. On the number line, the one further right is greater.

Worked example

Write $\frac{36}{-48}$ in standard form and compare it with $\frac{-5}{6}$.

Try it first, then show the solutionSolution

1. Move the sign up: $\frac{36}{-48}=\frac{-36}{48}$.
2. HCF of 36 and 48 is 12, so $\frac{-36}{48}=\frac{-3}{4}$.
3. Common denominator 12: $\frac{-3}{4}=\frac{-9}{12}$ and $\frac{-5}{6}=\frac{-10}{12}$.
4. $-9>-10$, so $\frac{-3}{4}>\frac{-5}{6}$.

Answer: $\frac{-3}{4}$, which is greater than $\frac{-5}{6}$.

Closure, commutative and associative properties

For rational numbers $\frac{a}{b}$, $\frac{c}{d}$ and $\frac{e}{f}$:

• Closure: the sum, difference and product of two rational numbers are rational. Division is closed only if you never divide by 0.
• Commutative: $\frac{a}{b}+\frac{c}{d}=\frac{c}{d}+\frac{a}{b}$ and $\frac{a}{b}\times \frac{c}{d}=\frac{c}{d}\times \frac{a}{b}$. Subtraction and division are not commutative: $\frac{1}{2}-\frac{1}{3}\ne \frac{1}{3}-\frac{1}{2}$.
• Associative: $\left(\frac{a}{b}+\frac{c}{d}\right)+\frac{e}{f}=\frac{a}{b}+\left(\frac{c}{d}+\frac{e}{f}\right)$, and the same for multiplication. Not for subtraction or division.

These properties let you regroup a long sum so that the awkward fractions cancel first.

Worked example

Find $\frac{2}{5}+\frac{-3}{7}+\frac{3}{7}$ in the quickest way.

Try it first, then show the solutionSolution

1. By the associative property, group the last two: $\frac{2}{5}+\left(\frac{-3}{7}+\frac{3}{7}\right)$.
2. The bracket is 0.
3. So the sum is $\frac{2}{5}$.

Answer: $\frac{2}{5}$

Identities, inverses and the distributive property

• Additive identity: $\frac{a}{b}+0=\frac{a}{b}$. Zero changes nothing when added.
• Multiplicative identity: $\frac{a}{b}\times 1=\frac{a}{b}$.
• Additive inverse (negative) of $\frac{a}{b}$ is $\frac{-a}{b}$, because their sum is 0.
• Multiplicative inverse (reciprocal) of $\frac{a}{b}$ is $\frac{b}{a}$ (for $a\ne 0$), because their product is 1. Zero has no reciprocal.
• Distributive: $\frac{a}{b}\times \left(\frac{c}{d}+\frac{e}{f}\right)=\frac{a}{b}\times \frac{c}{d}+\frac{a}{b}\times \frac{e}{f}$.

When two products share a factor, use the distributive property backwards to take the common factor out. It saves a lot of working.

Worked example

Simplify $\frac{-3}{5}\times \frac{2}{7}+\frac{-3}{5}\times \frac{5}{7}$.

Try it first, then show the solutionSolution

1. Both terms have the factor $\frac{-3}{5}$.
2. Distributive property: $\frac{-3}{5}\times \left(\frac{2}{7}+\frac{5}{7}\right)$.
3. The bracket is $\frac{7}{7}=1$, so the answer is $\frac{-3}{5}\times 1$.

Answer: $\frac{-3}{5}$

Rational numbers on the number line

To mark $\frac{p}{q}$ on the number line:
1. Write it as a mixed number if it is improper, so you know which two integers it lies between.
2. Divide each unit length into $q$ equal parts.
3. Count $p$ parts from 0: to the right if positive, to the left if negative.

For example, $\frac{-7}{4}=-1\frac{3}{4}$ lies between $-1$ and $-2$: divide that unit into 4 parts and mark the third mark to the left of $-1$.

Draw-and-label checklist: a straight line with arrows at both ends; 0 and the integers on either side marked evenly; the unit split into equal parts; the point marked with a dot and labelled with its value.

Worked example

Between which two integers does $\frac{-7}{4}$ lie, and how is it marked?

Try it first, then show the solutionSolution

1. $\frac{-7}{4}=-1\frac{3}{4}$, so it lies between $-2$ and $-1$.
2. Split the unit from $-1$ to $-2$ into 4 equal parts.
3. Starting at 0, count 7 quarter-parts to the left; the point is 3 parts past $-1$.

Answer: Between $-2$ and $-1$, three quarters of the way from $-1$ towards $-2$.

Rational numbers between two rational numbers

Between any two different rational numbers there is always another one, so there are infinitely many between them. Two methods:

Mean method: $\frac{1}{2}(x+y)$ lies exactly halfway between $x$ and $y$. Repeat to get more.

Common-denominator method: write both with the same denominator; if there is no room between the numerators, multiply top and bottom by 10 (or any number) to make room. For example, between $\frac{1}{3}$ and $\frac{1}{2}$: $\frac{2}{6}$ and $\frac{3}{6}$ have nothing between, but $\frac{20}{60}$ and $\frac{30}{60}$ have $\frac{21}{60},\frac{22}{60},\dots ,\frac{29}{60}$.

This is a big difference from integers: there is no integer between 3 and 4.

Worked example

Find the rational number halfway between $\frac{-1}{3}$ and $\frac{1}{2}$.

Try it first, then show the solutionSolution

1. Sum: $\frac{-1}{3}+\frac{1}{2}=\frac{-2}{6}+\frac{3}{6}=\frac{1}{6}$.
2. Half of it: $\frac{1}{6}÷2=\frac{1}{12}$.

Answer: $\frac{1}{12}$

Word problems with rational numbers

Most word problems are about a part of a part. Follow three rules:
1. Let the unknown whole be $x$ (or 1 whole).
2. Translate "of" as multiply, and "the remainder" as what is left after the earlier step.
3. Write one equation for what is finally left (or used) and solve.

Always check by putting your answer back into the story.

Worked example

Aman spends $\frac{2}{5}$ of his savings on a bag and $\frac{1}{3}$ of the remainder on shoes. He has ₹600 left. How much were his savings?

Try it first, then show the solutionSolution

1. Let the savings be $x$. After the bag, $\frac{3}{5}x$ is left.
2. Shoes $=\frac{1}{3}\times \frac{3}{5}x=\frac{1}{5}x$.
3. Left $=\frac{3}{5}x-\frac{1}{5}x=\frac{2}{5}x=600$, so $x=1500$.
4. Check: bag ₹600, shoes ₹300, left ₹600 ✓.

Answer: ₹1,500

Maths in real life: Rational Numbers

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Properties of Rational Numbers
• Rational Numbers Between Two Rational Numbers

Rational Numbers formula sheet

Addition

$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$

Or use the LCM of the denominators.

Multiplication

$\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}$

Cancel common factors first.

Division

$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}$

c ≠ 0: multiply by the reciprocal.

Distributive property

$x(y+z)=xy+xz$

Works for any rational x, y, z.

Mean of two rationals

$m=\frac{1}{2}(x+y)$

m lies exactly halfway between x and y.

Inverses

$\frac{a}{b}+\frac{-a}{b}=0;\frac{a}{b}\times \frac{b}{a}=1$

Additive inverse and reciprocal (a ≠ 0).

How the school exam asks Rational Numbers

The question patterns to expect, and the method that works for each.

Tricks to remember Rational Numbers

Short memory aids. Say each one aloud once, then use it on a question the same day.

• CCAD: the four big properties

  Closure, Commutative, Associative, Distributive. Subtraction and division fail the middle two, so remember 'only + and × can swap and regroup'.

  Use it when: Naming properties and spotting the false ones.

• Zero adds, one multiplies

  0 is the identity for adding, 1 is the identity for multiplying: the number that does nothing.

  Use it when: Identity questions.

• Flip for reciprocal, flip the sign for negative

  Reciprocal: turn the fraction upside down. Additive inverse: change only the sign.

  Use it when: Inverse questions; never mix the two.

• Stuck? Multiply by 10

  If two fractions with the same denominator have no room between them, multiply top and bottom by 10 and the gap opens up.

  Use it when: Inserting rational numbers between two numbers.

Rational Numbers: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Rational Numbers

The slips examiners see most often, and what to write instead.

• ✗Mistake: Leaving the minus sign in the denominator

  Answers like $\frac{3}{-4}$ are not in standard form.

  Wrong working

  Answer: $\frac{3}{-4}$

  Right working

  Answer: $\frac{-3}{4}$

  ✓Do this: Always finish with a positive denominator and the fraction fully reduced.

  Usually costs: 1 mark for the final answer.

• ✗Mistake: Mixing up reciprocal and additive inverse

  The reciprocal of $\frac{-2}{3}$ is $\frac{-3}{2}$, not $\frac{3}{2}$ or $\frac{2}{3}$.

  Wrong working

  Reciprocal of $\frac{-2}{3}$ is $\frac{2}{3}$

  Right working

  Reciprocal of $\frac{-2}{3}$ is $\frac{-3}{2}$ (product 1)

  ✓Do this: Check: an inverse must give 0 when added, a reciprocal must give 1 when multiplied.

  Usually costs: The whole 1-mark question.

• ✗Mistake: Assuming subtraction is commutative or associative

  Students regroup $a-b-c$ as $a-(b-c)$, which changes the answer.

  ✓Do this: Only regroup sums and products. Turn subtraction into adding the negative first: $a+(-b)+(-c)$.

  Usually costs: The value and later steps.

• ✗Mistake: Taking a fraction of the whole instead of the remainder

  '$\frac{1}{4}$ of the remainder' is a part of what is left, not of the original amount.

  Wrong working

  Food $=\frac{1}{4}x$

  Right working

  Food $=\frac{1}{4}\times \frac{2}{3}x=\frac{1}{6}x$

  ✓Do this: Write the remainder after each step before taking the next fraction.

  Usually costs: Usually 2 to 3 marks in a word problem.

• ✗Mistake: Unequal parts on the number line

  A sketch with uneven divisions does not show the point correctly.

  ✓Do this: Use a ruler, mark equal unit lengths and split each unit into exactly q equal parts.

  Usually costs: 1 mark for the diagram.

Practice questions on Rational Numbers

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   The additive inverse of $\frac{-7}{-9}$ is:
   (a) $\frac{7}{9}$
   (b) $\frac{-7}{9}$
   (c) $\frac{9}{7}$
   (d) $\frac{-9}{7}$

   Hint

   First simplify the signs of $\frac{-7}{-9}$.

   Answer

   (b): $\frac{-7}{-9}=\frac{7}{9}$, whose additive inverse is $\frac{-7}{9}$.

2. Q2 · 2 marks · Short answerExam level

   Express $\frac{56}{-84}$ in standard form.

   Hint

   HCF of 56 and 84 is 28.

   Answer

   $\frac{-56}{84}=\frac{-2}{3}$.

3. Q3 · 3 marks · Short answerExam level

   Represent $\frac{-5}{3}$ and $\frac{4}{3}$ on the same number line.

   Hint

   Write them as mixed numbers and split each unit into 3 parts.

   Answer

   $\frac{-5}{3}=-1\frac{2}{3}$: two parts to the left of $-1$ (between $-2$ and $-1$). $\frac{4}{3}=1\frac{1}{3}$: one part to the right of 1 (between 1 and 2). Each unit divided into 3 equal parts, both points labelled.

4. Q4 · 3 marks · Short answerHarder exam level

   Simplify $\left(\frac{-5}{8}+\frac{3}{4}\right)÷\left(\frac{-7}{16}+\frac{1}{2}\right)$.

   Hint

   Work out each bracket over a common denominator.

   Answer

   First bracket $=\frac{1}{8}$, second $=\frac{1}{16}$. $\frac{1}{8}÷\frac{1}{16}=2$.

5. Q5 · 1 mark · Assertion-reasonHarder exam level

   Assertion (A): Division of rational numbers is closed.
   Reason (R): 0 is a rational number.
   (a) (A) is true and (R) is false.
   (b) (A) is false and (R) is true.
   (c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
   (d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).

   Hint

   Can you divide by 0?

   Answer

   (b). Dividing by the rational number 0 is not defined, so division is not closed; (R) is true since $0=\frac{0}{1}$.

6. Q6 · 2 marks · Short answerHarder exam level

   What number should be added to $\frac{-5}{8}$ to get $\frac{1}{6}$?

   Hint

   Subtract: required − given.

   Answer

   $\frac{1}{6}-\frac{-5}{8}=\frac{4}{24}+\frac{15}{24}=\frac{19}{24}$.

7. Q7 · 2 marks · Short answerHarder exam level

   By what number should $\frac{-33}{16}$ be divided to get $\frac{-11}{4}$?

   Hint

   Divisor = dividend ÷ quotient.

   Answer

   $\frac{-33}{16}÷\frac{-11}{4}=\frac{33}{16}\times \frac{4}{11}=\frac{3}{4}$.

8. Q8 · 2 marks · Short answerHarder exam level

   The product of two rational numbers is $\frac{-14}{27}$. If one of them is $\frac{7}{9}$, find the other.

   Hint

   Divide the product by the known number.

   Answer

   $\frac{-14}{27}÷\frac{7}{9}=\frac{-14}{27}\times \frac{9}{7}=\frac{-2}{3}$.

9. Q9 · 4 marks · Long answerHarder exam level

   Using the mean method, insert three rational numbers between $\frac{-2}{5}$ and $\frac{-1}{3}$, and write them in ascending order.

   Hint

   Find the mean, then the mean of each half.

   Answer

   Mean $=\frac{1}{2}\left(\frac{-6}{15}+\frac{-5}{15}\right)=\frac{-11}{30}$. Between $\frac{-2}{5}$ and $\frac{-11}{30}$: $\frac{-23}{60}$. Between $\frac{-11}{30}$ and $\frac{-1}{3}$: $\frac{-21}{60}=\frac{-7}{20}$. Order: $\frac{-23}{60}<\frac{-11}{30}<\frac{-7}{20}$.

10. Q10 · 3 marks · Short answerHarder exam level

    A rod $15\frac{3}{4}$ m long is cut into equal pieces, each $2\frac{5}{8}$ m long. How many pieces are there?

    Hint

    Change both to improper fractions and divide.

    Answer

    $\frac{63}{4}÷\frac{21}{8}=\frac{63}{4}\times \frac{8}{21}=6$ pieces.

11. Q11 · 4 marks · Long answerHOTS / competency

    When 12 litres of oil are poured out of a full can, it is $\frac{5}{8}$ full. Find the capacity of the can and how much oil is left.

    Hint

    12 litres is the fraction that was poured out.

    Answer

    Poured out $=1-\frac{5}{8}=\frac{3}{8}$ of the can $=12$ L, so capacity $=12÷\frac{3}{8}=32$ L. Oil left $=\frac{5}{8}\times 32=20$ L.

12. Q12 · 4 marks · Long answerHOTS / competency

    Using general form, show that if $x$ and $y$ are rational numbers with $x<y$, then $\frac{1}{2}(x+y)$ is rational and lies between $x$ and $y$.

    Hint

    Find $m-x$ and $y-m$.

    Answer

    Let $x=\frac{a}{b}$, $y=\frac{c}{d}$. Then $m=\frac{ad+bc}{2bd}$, an integer over a non-zero integer, so $m$ is rational. $m-x=\frac{y-x}{2}>0$ and $y-m=\frac{y-x}{2}>0$, so $x<m<y$.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.Define a rational number.Show answer

   A number of the form p/q where p and q are integers and q ≠ 0.

2. 2.What is the additive identity for rational numbers?Show answer

   0.

3. 3.What is the multiplicative identity?Show answer

   1.

4. 4.Which rational number has no reciprocal?Show answer

   0.

5. 5.Is subtraction of rational numbers commutative?Show answer

   No.

6. 6.Write −15/−25 in standard form.Show answer

   3/5.

7. 7.How many rational numbers lie between 1/3 and 1/2?Show answer

   Infinitely many.

8. 8.What is the rational number halfway between −1/2 and 5/4?Show answer

   3/8 (the mean of the two numbers).

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you write any rational number in standard form?
• ☐Can you name the property used in each step of a simplification?
• ☐Do you know that 0 has no reciprocal?
• ☐Can you use the distributive property backwards to take out a common factor?
• ☐Can you mark −7/4 on a number line with equal parts?
• ☐Can you insert five rational numbers between two given ones?
• ☐Do you write the remainder before taking a fraction of it in word problems?

Rational Numbers: frequently asked questions