Master Rational Numbers
Complete, step-by-step solutions for Exercises 1(A) to 1(E), with interactive answer panels, searchable questions and clean mathematical notation.
Exercise 1(A)
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Question 1(i)
A number which is not rational is called
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A number which is not rational is called an irrational number.
Hence, Option 3 is the correct option.
Question 1(ii)
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If x ≠ 0 then value of
\(\frac{0}{x}\)
is a rational number.
Hence, Option 1 is the correct option.
Question 1(iii)
The equation 5x + 7 = 0, gives the value of x which is
Show step-by-step answer
5x + 7 = 0
⇒ 5x = -7
\(\frac{- 7}{5}\)
As x is the form of
\(\frac{p}{q}\)
where p and q both are integers and q ≠ 0,
∴ x is a rational number.
Hence, Option 3 is the correct option.
Question 1(iv)
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Rational number
\(\frac{p}{q}\)
is in standard form, if p and q have no common factor and q ≠ 0
Hence, Option 3 is the correct option.
Question 1(v)
=
\(\frac{c}{d} + \frac{a}{b}\)
=
\(\frac{c}{d} - \frac{a}{b}\)
Show step-by-step answer
\(\frac{a}{b}\)
and
\(\frac{c}{d}\)
is commutative, if
\(\frac{a}{b} + \frac{c}{d}\)
=
\(\frac{c}{d} + \frac{a}{b}\)
Hence, Option 3 is the correct option.
Question 1(vi)
1
0
Show step-by-step answer
Hence, Option 2 is the correct option.
Question 2(i)
Add each pair of rational numbers, given below, and show that their addition (sum) is also a rational number:
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As
\(\frac{- 2}{8}\)
is in the form of
\(\frac{p}{q}\)
where p and q both are integers and q ≠ 0,
\(\frac{- 2}{8}\)
is a rational number.
Question 2(ii)
Add each pair of rational numbers, given below, and show that their addition (sum) is also a rational number:
Show step-by-step answer
As
\(\frac{- 12}{13}\)
is in the form of
\(\frac{p}{q}\)
where p and q both are integers and q ≠ 0,
\(\frac{- 12}{13}\)
is a rational number.
Question 2(iii)
Add each pair of rational numbers, given below, and show that their addition (sum) is also a rational number:
Show step-by-step answer
As
\(\frac{- 3}{11}\)
is in the form of
\(\frac{p}{q}\)
where p and q both are integers and q ≠ 0,
\(\frac{- 3}{11}\)
is a rational number.
Question 2(iv)
Add each pair of rational numbers, given below, and show that their addition (sum) is also a rational number:
Show step-by-step answer
As
\(\frac{1}{78}\)
is in the form of
\(\frac{p}{q}\)
where p and q both are integers and q ≠ 0,
\(\frac{1}{78}\)
is a rational number.
Question 2(v)
Add each pair of rational numbers, given below, and show that their addition (sum) is also a rational number:
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As
\(\frac{- 1}{6}\)
is in the form of
\(\frac{p}{q}\)
where p and q both are integers and q ≠ 0,
\(\frac{- 1}{6}\)
is a rational number.
Question 2(vi)
Add each pair of rational numbers, given below, and show that their addition (sum) is also a rational number:
Show step-by-step answer
As
\(\frac{- 8}{5}\)
is in the form of
\(\frac{p}{q}\)
where p and q both are integers and q ≠ 0,
\(\frac{- 8}{5}\)
is a rational number.
Question 2(vii)
Add each pair of rational numbers, given below, and show that their addition (sum) is also a rational number:
Show step-by-step answer
As
\(\frac{- 21}{8}\)
is in the form of
\(\frac{p}{q}\)
where p and q both are integers and q ≠ 0,
\(\frac{- 21}{8}\)
is a rational number.
Question 2(viii)
Add each pair of rational numbers, given below, and show that their addition (sum) is also a rational number:
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As
\(\frac{- 5}{54}\)
is in the form of
\(\frac{p}{q}\)
where p and q both are integers and q ≠ 0,
\(\frac{- 5}{54}\)
is a rational number.
Question 3(i)
Evaluate:
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\(\frac{5 \times 2}{9 \times 2} + \frac{- 7 \times 3}{6 \times 3} = \frac{10}{18} + \frac{- 21}{18} = \frac{10 + ( - 21 )}{18} = \frac{- 11}{18}\)
\(\frac{5}{9} + \frac{- 7}{6}\)
=
\(\frac{- 11}{18}\)
Question 3(ii)
Evaluate:
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\(4 + \frac{3}{- 5} = 3 \frac{2}{5}\)
Question 3(iii)
Evaluate:
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\(\frac{1}{- 15} + \frac{5}{- 12} = \frac{- 29}{60}\)
Question 3(iv)
Evaluate:
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\(\frac{5}{9} + \frac{3}{- 4}\)
=
\(\frac{- 7}{36}\)
Question 3(v)
Evaluate:
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\(\frac{- 8}{9}\)
+
\(\frac{- 5}{12}\)
=
\(\frac{- 47}{36}\)
Question 3(vi)
Evaluate:
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\(\frac{- 2}{7}\)
=
\(\frac{- 2}{7}\)
Question 3(vii)
Evaluate:
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\(\frac{5}{- 11}\)
+ 0 =
\(\frac{- 5}{11}\)
Question 3(viii)
Evaluate:
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\(\frac{- 3}{5}\)
=
\(\frac{7}{5}\)
Question 3(ix)
Evaluate:
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\(\frac{4}{- 9}\)
+ 1 =
\(\frac{5}{9}\)
Question 4(i)
Evaluate:
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\(\frac{3}{7} + \frac{- 4}{9} + \frac{- 11}{7} + \frac{7}{9} = \frac{- 51}{63}\)
Question 4(ii)
Evaluate:
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\(\frac{2}{3} + \frac{- 4}{5} + \frac{1}{3} + \frac{2}{5} = \frac{9}{15}\)
Question 4(iii)
Evaluate:
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\(\frac{4}{7} + 0 + \frac{- 8}{9} + \frac{- 13}{7} + \frac{17}{9} = \frac{- 18}{63}\)
Question 4(iv)
Evaluate:
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\(\frac{3}{8} + \frac{- 5}{12} + \frac{3}{7} + \frac{3}{12} + \frac{- 5}{8} + \frac{- 2}{7} = \frac{- 23}{84}\)
Question 5(i)
For each pair of rational numbers, verify commutative property of addition of rational numbers.
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∴ LHS = RHS
Hence,
\(\frac{- 8}{7} + \frac{5}{14} = \frac{5}{14} + \frac{- 8}{7}\)
So, the commutative property for the addition of the rational number is verified.
Question 5(ii)
For each pair of rational numbers, verify commutative property of addition of rational numbers.
Show step-by-step answer
Taking LHS:
\(\frac{5}{9} + \frac{5}{- 12} = \frac{5}{9} + \frac{- 5}{12}\)
Taking RHS:
\(\frac{5}{- 12} + \frac{5}{9} = \frac{- 5}{12} + \frac{5}{9}\)
∴ LHS = RHS
Hence,
\(\frac{5}{9} + \frac{5}{- 12} = \frac{5}{- 12} + \frac{5}{9}\)
So, the commutative property for the addition of the rational number is verified.
Question 5(iii)
For each pair of rational numbers, verify commutative property of addition of rational numbers.
Show step-by-step answer
Taking LHS:
\(\frac{- 4}{5} + \frac{- 13}{- 15} = \frac{- 4}{5} + \frac{13}{15}\)
Taking RHS:
\(\frac{- 13}{- 15} + \frac{- 4}{5} = \frac{13}{15} + \frac{- 4}{5}\)
∴ LHS = RHS
Hence,
\(\frac{- 4}{5} + \frac{- 13}{- 15} = \frac{- 13}{- 15} + \frac{- 4}{5}\)
So, the commutative property for the addition of the rational number is verified.
Question 5(iv)
For each pair of rational numbers, verify commutative property of addition of rational numbers.
Show step-by-step answer
Taking LHS:
\(\frac{2}{- 5} + \frac{11}{- 15} = \frac{- 2}{5} + \frac{- 11}{15}\)
Taking RHS:
\(\frac{11}{- 15} + \frac{2}{- 5} = \frac{- 11}{15} + \frac{- 2}{5}\)
∴ LHS = RHS
Hence,
\(\frac{2}{- 5} + \frac{11}{- 15} = \frac{11}{- 15} + \frac{2}{- 5}\)
So, the commutative property for the addition of the rational number is verified.
Question 5(v)
For each pair of rational numbers, verify commutative property of addition of rational numbers.
Show step-by-step answer
\(3 + \frac{- 2}{7} = \frac{- 2}{7} + 3\)
+3
Taking LHS:
\(3 + \frac{- 2}{7} = \frac{3}{1} + \frac{- 2}{7}\)
Taking RHS:
\(\frac{- 2}{7} + 3 = \frac{- 2}{7} + \frac{3}{1}\)
∴ LHS = RHS
Hence,
\(3 + \frac{- 2}{7} = \frac{- 2}{7} + 3\)
+3
So, the commutative property for the addition of the rational number is verified.
Question 5(vi)
For each pair of rational numbers, verify commutative property of addition of rational numbers.
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\(- 2 + \frac{3}{- 5} = \frac{3}{- 5} + - 2\)
+−2
Taking LHS:
\(- 2 + \frac{3}{- 5} = \frac{- 2}{1} + \frac{- 3}{5}\)
Taking RHS:
\(\frac{3}{- 5} + - 2 = \frac{- 3}{5} + \frac{- 2}{1}\)
∴ LHS = RHS
Hence,
\(- 2 + \frac{3}{- 5} = \frac{3}{- 5} + - 2\)
+−2
So, the commutative property for the addition of the rational number is verified.
Question 6(i)
For each set of rational numbers, given below, verify the associative property of addition of rational numbers:
Show step-by-step answer
\(( \frac{1}{2} + \frac{2}{3} ) + \frac{- 1}{6} = \frac{1}{2} + ( \frac{2}{3} + \frac{- 1}{6} )\)
)
=1
Taking RHS:
\(\frac{1}{2} + ( \frac{2}{3} + \frac{- 1}{6} )\)
)
\(\frac{1}{2} + ( \frac{2 \times 2}{3 \times 2} + \frac{- 1 \times 1}{6 \times 1} ) = \frac{1}{2} + ( \frac{4}{6} + \frac{- 1}{6} ) = \frac{1}{2} + ( \frac{4 + ( - 1 )}{6} ) = \frac{1}{2} + \frac{3}{6}\)
=1
∴ LHS = RHS
\(( \frac{1}{2} + \frac{2}{3} ) + \frac{- 1}{6} = \frac{1}{2} + ( \frac{2}{3} + \frac{- 1}{6} )\)
)
So, the associative property for the addition of the rational number is verified.
Question 6(ii)
For each set of rational numbers, given below, verify the associative property of addition of rational numbers:
Show step-by-step answer
\(( \frac{- 2}{5} + \frac{4}{15} ) + \frac{- 7}{10} = \frac{- 2}{5} + ( \frac{4}{15} + \frac{- 7}{10} )\)
)
Taking RHS:
\(\frac{- 2}{5} + ( \frac{4}{15} + \frac{- 7}{10} )\)
)
\(\frac{- 2}{5} + ( \frac{4 \times 2}{15 \times 2} + \frac{- 7 \times 3}{10 \times 3} ) = \frac{- 2}{5} + ( \frac{8}{30} + \frac{- 21}{30} ) = \frac{- 2}{5} + ( \frac{8 + ( - 21 )}{30} ) = \frac{- 2}{5} + \frac{- 13}{30}\)
∴ LHS = RHS
\(( \frac{- 2}{5} + \frac{4}{15} ) + \frac{- 7}{10} = \frac{- 2}{5} + ( \frac{4}{15} + \frac{- 7}{10} )\)
)
So, the associative property for the addition of the rational number is verified.
Question 6(iii)
For each set of rational numbers, given below, verify the associative property of addition of rational numbers:
Show step-by-step answer
To prove:
\(( \frac{- 7}{9} + \frac{2}{- 3} ) + \frac{- 5}{18} = \frac{- 7}{9} + ( \frac{2}{- 3} + \frac{- 5}{18} )\)
)
\(\frac{- 7}{9} + ( \frac{2}{- 3} + \frac{- 5}{18} ) = \frac{- 7}{9} + ( \frac{- 2}{3} + \frac{- 5}{18} )\)
)
∴ LHS = RHS
\(( \frac{- 7}{9} + \frac{2}{- 3} ) + \frac{- 5}{18} = \frac{- 7}{9} + ( \frac{2}{- 3} + \frac{- 5}{18} )\)
)
So, the associative property for the addition of the rational number is verified.
Question 6(iv)
For each set of rational numbers, given below, verify the associative property of addition of rational numbers:
Show step-by-step answer
To prove:
\(( - 1 + \frac{5}{6} ) + \frac{- 2}{3} = - 1 + ( \frac{5}{6} + \frac{- 2}{3} )\)
)
Taking RHS:
\(- 1 + ( \frac{5}{6} + \frac{- 2}{3} ) = \frac{- 1}{1} + ( \frac{5}{6} + \frac{- 2}{3} )\)
)
\(\frac{- 1}{1} + ( \frac{5 \times 1}{6 \times 1} + \frac{- 2 \times 2}{3 \times 2} ) = \frac{- 1}{1} + ( \frac{5}{6} + \frac{- 4}{6} ) = \frac{- 1}{1} + ( \frac{5 + ( - 4 )}{6} ) = \frac{- 1}{1} + \frac{1}{6}\)
∴ LHS = RHS
\(( - 1 + \frac{5}{6} ) + \frac{- 2}{3} = - 1 + ( \frac{5}{6} + \frac{- 2}{3} )\)
)
So, the associative property for the addition of the rational number is verified.
Question 7(i)
Write the additive inverse (negative) of:
Show step-by-step answer
\(\frac{- 3}{8}\)
=
\(- ( \frac{- 3}{8} )\)
)
\(\frac{3}{8}\)
Question 7(ii)
Write the additive inverse (negative) of:
Show step-by-step answer
=
\(\frac{- 4}{9}\)
\(\frac{- 4}{9}\)
=
\(- ( \frac{- 4}{9} )\)
)
\(\frac{4}{9}\)
Question 7(iii)
Write the additive inverse (negative) of:
Show step-by-step answer
=
\(\frac{4}{13}\)
\(\frac{4}{13}\)
=
\(- ( \frac{4}{13} )\)
)
\(- \frac{4}{13}\)
Question 7(iv)
Write the additive inverse (negative) of:
0
Show step-by-step answer
\(\frac{0}{1}\)
\(\frac{0}{1}\)
=
\(- ( \frac{0}{1} )\)
)
= 0
Question 7(v)
Write the additive inverse (negative) of:
-2
Show step-by-step answer
\(\frac{- 2}{1}\)
\(\frac{- 2}{1}\)
=
\(- ( \frac{- 2}{1} )\)
)
\(\frac{2}{1} = 2\)
=2
Question 7(vi)
Write the additive inverse (negative) of:
1
Show step-by-step answer
\(\frac{1}{1}\)
\(\frac{1}{1}\)
=
\(- ( \frac{1}{1} )\)
)
\(\frac{1}{1} = - 1\)
=−1
Question 8
Fill in the blanks:
\(\frac{- 5}{- 12}\)
= ............... .
\(\frac{- 5}{- 12}\)
+ its additive inverse = ............... .
\(\frac{a}{b}\)
is the additive inverse of
\(\frac{- c}{d}\)
, then
\(\frac{- c}{d}\)
is the additive inverse of ............... .
And, so
\(\frac{a}{b} + \frac{- c}{d} = \frac{- c}{d} + \frac{a}{b}\)
= ............... .
Show step-by-step answer
(i) Additive inverse of
\(\frac{- 5}{- 12} = - \frac{5}{12}\)
.
\(\frac{- 5}{- 12}\)
+ its additive inverse = 0
\(\frac{a}{b}\)
is the additive inverse of
\(\frac{- c}{d}\)
, then
\(\frac{- c}{d}\)
is the additive inverse of
\(\frac{a}{b}\)
.
And, so
\(\frac{a}{b} + \frac{- c}{d} = \frac{- c}{d} + \frac{a}{b} = 0\)
=0.
(i)
\(\frac{- 5}{- 12} = \frac{5}{12}\)
Additive inverse of
\(\frac{5}{12} = - \frac{5}{12}\)
(ii\() \frac{5}{12} + ( - \frac{5}{12}\)
)
=0
(iii) The sum of number and its additive inverse = Additive identity.
Question 9
State, true or false:
Show step-by-step answer
(i) False.
\(\frac{7 + 5}{9 + 5} = \frac{12}{14}\)
≠
\(\frac{7}{9}\)
(ii) False
\(\frac{7 - 5}{9 - 5} = \frac{2}{4}\)
≠
\(\frac{7}{9}\)
(iii) True
\(\frac{7\times5}{9\times5}=\frac79\)
(iv) True
(v) False
\(\frac{- 5}{- 12} = \frac{5}{12}\)
is a positive rational number.
(vi) False
We need to check if
\(\frac{- 13}{25}\)
is smaller than
\(\frac{- 25}{13}\)
And,
\(\frac{- 169}{325}\)
>
\(\frac{- 625}{325}\)
Hence,
\(\frac{- 13}{25}\)
>
\(\frac{- 25}{13}\)
Question 10
Show step-by-step answer
\(2\,\frac{1}{3}\)
kg
\(5\,\frac{5}{6}\)
kg
\(8\,\frac{3}{8}\)
kg
Total weight of basket with fruits = Weight of empty fruit basket + weight of grapes + weight of mangoes
\(= 2 \frac{1}{3} \mathrm{kg} + 5 \frac{5}{6} \mathrm{kg} + 8 \frac{3}{8} \mathrm{kg} = \frac{7}{3} \mathrm{kg} + \frac{35}{6} \mathrm{kg} + \frac{67}{8} \mathrm{kg}\)
kg
\(= \frac{7 \times 8}{3 \times 8} + \frac{35 \times 4}{6 \times 4} + \frac{67 \times 3}{8 \times 3} \mathrm{kg} = \frac{56}{24} + \frac{140}{24} + \frac{201}{24} \mathrm{kg} = \frac{56 + 140 + 201}{24} \mathrm{kg} = \frac{397}{24} \mathrm{kg} = 16 \frac{13}{24} \mathrm{kg}\)
kg
\(16\,\frac{13}{24}\)
kg.
Exercise 1(B)
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Question 1(i)
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Let
x be the other number.
\(2\,\frac{3}{4} + x = 8 \Rightarrow \frac{11}{4} + x = 8 \Rightarrow x = 8 - \frac{11}{4} \Rightarrow x = \frac{8}{1} - \frac{11}{4}\)
The sum of two rational numbers is 8, if one of them is
\(2\,\frac{3}{4}\)
, the other number is
\(5\,\frac{1}{4}\)
.
Hence, option 3 is correct option.
Question 1(ii)
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Hence, option 1 is correct option.
Question 1(iii)
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Let
x be the other number.
\(4\,\frac{1}{2} + x = - 6 \Rightarrow \frac{9}{2} + x = - 6 \Rightarrow x = - 6 - \frac{9}{2} \Rightarrow x = \frac{- 6}{1} - \frac{9}{2}\)
The sum of two rational numbers is -6, if one of them is
\(4\,\frac{1}{2}\)
, the other number is -
\(10\,\frac{1}{2}\)
.
Hence, option 4 is correct option.
Question 1(iv)
4
Show step-by-step answer
Let
x be subtracted from
\(5\,\frac{2}{3}\)
.
The number subtracted from
\(5\,\frac{2}{3}\)
to get -
\(1\,\frac{2}{3}\)
is
\(7\,\frac{1}{3}\)
.
Hence, option 4 is correct option.
Question 1(v)
4
Show step-by-step answer
Let
x be added to
\(5\,\frac{2}{3}\)
.
\(5\,\frac{2}{3} + x = - 1 \frac{2}{3} \Rightarrow \frac{17}{3} + x = - \frac{5}{3} \Rightarrow x = \frac{- 5}{3} - \frac{17}{3} \Rightarrow x = \frac{- 5 - 17}{3} \Rightarrow x = \frac{- 22}{3} \Rightarrow x = - 7 \frac{1}{3}\)
The number added to
\(5\,\frac{2}{3}\)
to get -1
\(\frac{2}{3}\)
is -
\(7\,\frac{1}{3}\)
.
Hence, option 2 is correct option.
Question 2(i)
Evaluate:
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Question 2(ii)
Evaluate:
Show step-by-step answer
Question 2(iii)
Evaluate:
Show step-by-step answer
Question 2(iv)
Evaluate:
Show step-by-step answer
Question 2(v)
Evaluate:
Show step-by-step answer
Question 2(vi)
Evaluate:
Show step-by-step answer
Question 3(i)
Subtract:
Show step-by-step answer
=−1
Question 3(ii)
Subtract:
Show step-by-step answer
Question 3(iii)
Subtract:
Show step-by-step answer
=−1
Question 3(iv)
Subtract:
Show step-by-step answer
Question 3(v)
Subtract:
Show step-by-step answer
Question 3(vi)
Subtract:
Show step-by-step answer
Question 4
Show step-by-step answer
Let
x be the other number.
\(\frac{2}{5} + x = \frac{9}{20} \Rightarrow x = \frac{9}{20} - \frac{2}{5}\)
The sum of two rational numbers is
\(\frac{9}{20}\)
, if one of them is
\(\frac{2}{5}\)
, the other number is
\(\frac{1}{20}\)
Question 5
Show step-by-step answer
Let
x be the other number.
\(\frac{- 8}{15} + x = \frac{- 2}{3} \Rightarrow x = \frac{- 2}{3} - \frac{- 8}{15}\)
The sum of two rational numbers is
\(\frac{- 2}{3}\)
, if one of them is
\(\frac{- 8}{15}\)
, the other number is
\(\frac{- 2}{15}\)
.
Question 6
Show step-by-step answer
Let
x be the other number.
\(\frac{- 8}{5} + x = - 6 \Rightarrow x = \frac{- 6}{1} - \frac{- 8}{5}\)
The sum of two rational numbers is -6, if one of them is
\(\frac{- 8}{5}\)
, the other number is
\(- 4 \frac{2}{5}\)
.
Question 7
Show step-by-step answer
Let
x be added to
\(\frac{- 7}{8}\)
.
The number added to
\(\frac{- 7}{8}\)
to get
\(\frac{5}{9}\)
is
\(1\,\frac{31}{72}\)
.
Question 8
Show step-by-step answer
Let
x be added to
\(\frac{- 5}{9}\)
.
The number added to
\(\frac{- 5}{9}\)
to get
\(\frac{- 2}{3}\)
is
\(\frac{- 1}{9}\)
.
Question 9
Show step-by-step answer
Let
x be subtracted from
\(\frac{- 5}{6}\)
.
The number subtracted from
\(\frac{- 5}{6}\)
to get
\(\frac{4}{9}\)
is -
\(1\,\frac{5}{18}\)
.
Question 10(i)
Show step-by-step answer
Let
x be subtracted from -2.
The number subtracted from -2 to get
\(\frac{3}{8}\)
is -
\(2\,\frac{3}{8}\)
.
Question 10(ii)
Show step-by-step answer
Let
x be added to -2.
The number added to -2 to get
\(\frac{3}{8}\)
is
\(2\,\frac{3}{8}\)
.
Question 11(i)
Evaluate:
Show step-by-step answer
\(( \frac{3}{7} - \frac{- 11}{7} ) + ( \frac{- 4}{9} - \frac{7}{9} ) = ( \frac{3 - ( - 11 )}{7} ) + ( \frac{- 4 - 7}{9} ) = ( \frac{14}{7} ) + ( \frac{- 11}{9} ) = ( \frac{2}{1} ) + ( \frac{- 11}{9} )\)
)
Question 11(ii)
Evaluate:
Show step-by-step answer
\(( \frac{2}{3} - \frac{1}{3} ) + ( \frac{- 4}{5} - \frac{2}{5} ) = ( \frac{2 - 1}{3} ) + ( \frac{- 4 - 2}{5} ) = ( \frac{1}{3} ) + ( \frac{- 6}{5} )\)
)
Question 11(iii)
Evaluate
Show step-by-step answer
\(( \frac{4}{7} - \frac{- 13}{7} ) + ( - \frac{- 8}{9} + \frac{17}{9} ) = ( \frac{4 - ( - 13 )}{7} ) + ( \frac{8 + 17}{9} ) = ( \frac{4 + 13}{7} ) + ( \frac{25}{9} ) = ( \frac{17}{7} ) + ( \frac{25}{9} )\)
)
Exercise 1(C)
Tap any answer panel to reveal the full working.
Question 1(i)
Show step-by-step answer
Let the number be
x.
\(5\,\frac{2}{3} \times x = - 1 \frac{2}{3} \Rightarrow \frac{17}{3} \times x = - \frac{5}{3} \Rightarrow x = - \frac{5}{3} \div \frac{17}{3} \Rightarrow x = - \frac{5}{3} \times \frac{3}{17} \Rightarrow x = - \frac{5 \times 3}{3 \times 17} \Rightarrow x = - \frac{15}{51} \Rightarrow x = - \frac{5}{17}\)
Hence, option 3 is the correct option.
Question 1(ii)
If a, b and c are three rational numbers, we have:
(a + b) x c = (a + c) x (b + c)
a x (b - c) = a x b - a x c
a x (b - c) = a x b - c
Show step-by-step answer
We know that, multiplication of rational numbers is distributive over their addition/subtraction.
∴ a x (b - c) = a x b - a x c
Hence, option 3 is the correct option.
Question 1(iii)
The product of a positive rational number and its reciprocal is:
Show step-by-step answer
Let a positive rational number be
\(\frac{a}{b}\)
.
\(\frac{a}{b} \times \frac{b}{a} = \frac{a \times b}{b \times a} = a b a b = 1\)\(\frac{ab}{ab}\)
=1
Hence, option 3 is the correct option.
Question 1(iv)
Show step-by-step answer
Let breadth be b
\(4\,\frac{2}{5}\)
cm =
\(\frac{22}{5}\)
cm.
\(7\,\frac{1}{3}\)
cm2 =
\(\frac{22}{3}\)
cm2.
\(\frac{22}{3} = \frac{22}{5} \times b \Rightarrow b = \frac{22}{3} \div \frac{22}{5} \Rightarrow b = \frac{22}{3} \times \frac{5}{22} \Rightarrow b = \frac{22 \times 5}{3 \times 22} \Rightarrow b = \frac{110}{66} \Rightarrow b = \frac{5}{3} \Rightarrow b = 1 \frac{2}{3}\)
Hence, option 2 is the correct option.
Question 1(v)
0
1
Show step-by-step answer
\(\frac{2}{7}\)
\(- \frac{2}{7}\)
\(\frac{2}{7} \times - \frac{2}{7} = - \frac{2 \times 2}{7 \times 7} = - \frac{4}{49} .\)
.
Hence, option 2 is the correct option.
Question 2(i)
Evaluate:
Show step-by-step answer
Hence,
\(\frac{- 14}{5} \times \frac{- 6}{7} = 2 \frac{2}{5}\)
Question 2(ii)
Evaluate:
Show step-by-step answer
Hence,
\(\frac{7}{6} \times \frac{- 18}{91} = \frac{- 3}{13}\)
Question 2(iii)
Evaluate:
Show step-by-step answer
Hence,
\(\frac{- 125}{72} \times \frac{9}{- 5} = 3 \frac{1}{8}\)
Question 2(iv)
Evaluate:
Show step-by-step answer
Hence,
\(\frac{- 11}{9} \times \frac{- 51}{- 44} = - 1 \frac{5}{12}\)
Question 2(v)
Evaluate:
Show step-by-step answer
=−8
\(- \frac{16}{5} \times \frac{20}{8} = - 8\)
=−8
Question 3(i)
Multiply
Show step-by-step answer
Hence,
\(\frac{5}{6} \times \frac{8}{9} = \frac{20}{27}\)
Question 3(ii)
Multiply:
Show step-by-step answer
Hence,
\(\frac{2}{7} \times \frac{- 14}{9} = \frac{- 4}{9}\)
Question 3(iii)
Multiply:
Show step-by-step answer
Hence,
\(\frac{- 7}{8} \times 4 = - 3 \frac{1}{2}\)
Question 3(iv)
Multiply:
Show step-by-step answer
Hence,
\(\frac{36}{- 7} \times \frac{- 9}{28} = 1 \frac{32}{49}\)
Question 3(v)
Multiply:
Show step-by-step answer
Hence,
\(\frac{- 7}{10} \times \frac{- 8}{15} = \frac{28}{75}\)
Question 4(i)
Evaluate:
Show step-by-step answer
)
=−1
Hence,
\(( \frac{2}{- 3} \times \frac{5}{4} ) + ( \frac{5}{9} \times \frac{3}{- 10} ) = ( \frac{- 6}{6} )\)
)
Question 4(ii)
Evaluate:
Show step-by-step answer
)
LCM of 2 and 5 is 2 x 5 = 10
\(= ( \frac{1 \times 5}{2 \times 5} ) - ( \frac{6 \times 2}{5 \times 2} ) = ( \frac{5}{10} ) - ( \frac{12}{10} ) = ( \frac{5 - 12}{10} ) = ( \frac{- 7}{10} )\)
)
Hence,
\(( 2 \times \frac{1}{4} ) - ( \frac{- 18}{7} \times \frac{- 7}{15} ) = ( \frac{- 7}{10} )\)
)
Question 4(iii)
Evaluate:
Show step-by-step answer
\(( \frac{- 5 \times 2}{1 \times 15} ) - ( \frac{- 6 \times 2}{1 \times 9} ) = ( \frac{- 10}{15} ) - ( \frac{- 12}{9} ) = ( \frac{- 2}{3} ) - ( \frac{- 4}{3} ) = ( \frac{- 2 - ( - 4 )}{3} ) = ( \frac{- 2 + 4}{3} ) = ( \frac{2}{3} )\)
)
Hence,
\(( - 5 \times \frac{5}{12} ) - ( - 6 \times \frac{2}{9} ) = ( \frac{2}{3} )\)
)
Question 4(iv)
Evaluate:
Show step-by-step answer
\(( \frac{8 \times - 3}{5 \times 2} ) + ( \frac{- 3 \times 9}{10 \times 16} ) = ( \frac{- 24}{10} ) + ( \frac{- 27}{160} ) = ( \frac{- 12}{5} ) + ( \frac{- 27}{160} )\)
)
Hence,
\(( \frac{8}{5} \times \frac{- 3}{2} ) + ( \frac{- 3}{10} \times \frac{9}{16} ) = - 2 \frac{91}{160}\)
Question 5(i)
Multiply each rational number, given below, by one (1):
Show step-by-step answer
Hence,
\(\frac{7}{- 5} \times 1 = \frac{7}{- 5}\)
Question 5(ii)
Multiply each rational number, given below, by one (1):
Show step-by-step answer
Hence,
\(\frac{- 3}{- 4} \times 1 = \frac{3}{4}\)
Question 5(iii)
Multiply each rational number, given below, by one (1):
0
Show step-by-step answer
×
1
=
0
0×1
=0
Hence, 0 x 1 = 0
Question 5(iv)
Multiply each rational number, given below, by one (1):
Show step-by-step answer
Hence,
\(\frac{- 8}{13} \times 1 = \frac{- 8}{13}\)
Question 5(v)
Multiply each rational number, given below, by one (1):
Show step-by-step answer
Hence,
\(\frac{- 6}{- 7} \times 1 = \frac{6}{7}\)
Question 6(i)
For each pair of rational numbers, given below, verify that the multiplication is commutative:
Show step-by-step answer
∴ LHS = RHS
Question 6(ii)
For each pair of rational numbers, given below, verify that the multiplication is commutative:
Show step-by-step answer
∴ LHS = RHS
Question 6(iii)
For each pair of rational numbers, given below, verify that the multiplication is commutative:
Show step-by-step answer
\(3 \times \frac{- 8}{9} = \frac{- 8}{9} \times 3\)
×3
∴ LHS = RHS
\(3 \times \frac{- 8}{9} = \frac{- 8}{9} \times 3\)
×3
Question 6(iv)
For each pair of rational numbers, given below, verify that the multiplication is commutative:
Show step-by-step answer
\(0 \times \frac{- 12}{17} = \frac{- 12}{17} \times 0\)
×0
=0
=0
∴ LHS = RHS
\(0 \times \frac{- 12}{17} = \frac{- 12}{17} \times 0\)
×0
Question 7(i)
Write the reciprocal (multiplicative inverse) of each rational number given below:
5
Show step-by-step answer
\(\frac{1}{5}\)
.
Question 7(ii)
Write the reciprocal (multiplicative inverse) of each rational number given below:
-3
Show step-by-step answer
\(- \frac{1}{3}\)
.
Question 7(iii)
Write the reciprocal (multiplicative inverse) of each rational number given below:
Show step-by-step answer
\(\frac{5}{11}\)
= reciprocal of
\(\frac{5}{11}\)
=
\(\frac{11}{5} = 2 \frac{1}{5}\)
.
Question 7(iv)
Write the reciprocal (multiplicative inverse) of each rational number given below:
Show step-by-step answer
\(\frac{- 7}{- 8}\)
= reciprocal of
\(\frac{7}{8}\)
=
\(\frac{8}{7} = 1 \frac{1}{7}\)
.
Question 7(v)
Write the reciprocal (multiplicative inverse) of each rational number given below:
Show step-by-step answer
\(\frac{- 8}{- 7}\)
= reciprocal of
\(\frac{8}{7}\)
=
\(\frac{7}{8}\)
.
Question 8(i)
Find the reciprocal (multiplicative inverse) of:
Show step-by-step answer
\(\frac{2}{5}\)
= reciprocal of
\(\frac{2}{5} = \frac{5}{2} = 2 \frac{1}{2}\)
.
Question 8(ii)
Find the reciprocal (multiplicative inverse) of:
Show step-by-step answer
\(\frac{104}{21}\)
= reciprocal of
\(\frac{104}{21}\)
=
\(\frac{21}{104}\)
.
Question 8(iii)
Find the reciprocal (multiplicative inverse) of:
Show step-by-step answer
\(\frac{3}{65}\)
= reciprocal of
\(\frac{3}{65}\)
=
\(\frac{65}{3} = 21 \frac{2}{3}\)
.
Question 9(i)
(
x
+
y
)
×
z
=
x
×
z
+
y
×
z
(x+y)×z=x×z+y×z, if
and
z
=
−
4
z=−4
Show step-by-step answer
x
+
y
)
×
z
=
x
×
z
+
y
×
z
(x+y)×z=x×z+y×z
\(( x + y ) \times z = ( \frac{4}{5} + \frac{- 2}{3} ) \times - 4\)
)×−4
\(= ( \frac{4 \times 3}{5 \times 3} + \frac{- 2 \times 5}{3 \times 5} ) \times - 4 = ( \frac{12}{15} + \frac{- 10}{15} ) \times - 4 = ( \frac{12 + ( - 10 )}{15} ) \times - 4 = ( \frac{2}{15} ) \times - 4 = ( \frac{2 \times - 4}{15 \times 1} ) = ( \frac{- 8}{15} )\)
)
∴ LHS = RHS
x
+
y
)
×
z
=
x
×
z
+
y
×
z
(x+y)×z=x×z+y×z
Question 9(ii)
(
x
+
y
)
×
z
=
x
×
z
+
y
×
z
(x+y)×z=x×z+y×z, if
Show step-by-step answer
x
+
y
)
×
z
=
x
×
z
+
y
×
z
(x+y)×z=x×z+y×z
\(= ( \frac{2 \times 5}{1 \times 5} + \frac{4 \times 1}{5 \times 1} ) \times \frac{3}{- 10} = ( \frac{10}{5} + \frac{4}{5} ) \times \frac{3}{- 10} = ( \frac{10 + 4}{5} ) \times \frac{3}{- 10} = ( \frac{14}{5} ) \times \frac{3}{- 10} = ( \frac{14 \times 3}{5 \times - 10} ) = ( \frac{42}{- 50} ) = ( \frac{- 21}{25} )\)
)
∴ LHS = RHS
x
+
y
)
×
z
=
x
×
z
+
y
×
z
(x+y)×z=x×z+y×z
Question 10(i)
x
×
(
y
−
z
)
=
x
×
y
−
x
×
z
x×(y−z)=x×y−x×z, if
and
z
=
3
z=3
Show step-by-step answer
×
(
y
−
z
)
=
x
×
y
−
x
×
z
x×(y−z)=x×y−x×z
\(x \times ( y - z ) = \frac{4}{5} \times ( \frac{- 7}{4} - 3 ) = \frac{4}{5} \times ( \frac{- 7}{4} - \frac{3}{1} )\)
)
\(= \frac{4}{5} \times ( \frac{- 7 \times 1}{4 \times 1} - \frac{3 \times 4}{1 \times 4} ) = \frac{4}{5} \times ( \frac{- 7}{4} - \frac{12}{4} ) = \frac{4}{5} \times ( \frac{- 7 - 12}{4} ) = \frac{4}{5} \times ( \frac{- 19}{4} ) = ( \frac{4 \times - 19}{5 \times 4} ) = ( \frac{- 76}{20} ) = ( \frac{- 19}{5} )\)
)
∴ LHS = RHS
×
(
y
−
z
)
=
x
×
y
−
x
×
z
x×(y−z)=x×y−x×z
Question 10(ii)
x
×
(
y
−
z
)
=
x
×
y
−
x
×
z
x×(y−z)=x×y−x×z, if
and
z
=
−
5
z=−5
Show step-by-step answer
×
(
y
−
z
)
=
x
×
y
−
x
×
z
x×(y−z)=x×y−x×z
\(x \times ( y - z ) = \frac{3}{4} \times ( \frac{8}{9} - ( - 5 ) ) = \frac{3}{4} \times ( \frac{8}{9} - \frac{- 5}{1} )\)
)
\(= \frac{3}{4} \times ( \frac{8 \times 1}{9 \times 1} - \frac{- 5 \times 9}{1 \times 9} ) = \frac{3}{4} \times ( \frac{8}{9} - \frac{- 45}{9} ) = \frac{3}{4} \times ( \frac{8 - ( - 45 )}{9} ) = \frac{3}{4} \times ( \frac{8 + 45}{9} ) = \frac{3}{4} \times ( \frac{53}{9} ) = ( \frac{3 \times 53}{4 \times 9} ) = ( \frac{159}{36} ) = ( \frac{53}{12} )\)
)
∴ LHS = RHS
×
(
y
−
z
)
=
x
×
y
−
x
×
z
x×(y−z)=x×y−x×z
Question 11
Name the multiplication property of rational numbers shown below:
\(\frac{- 7}{5} \times \frac{5}{- 7} = 1\)
=1
Show step-by-step answer
(i) Commutativity property
Reason
If
\(\frac{a}{b}\)
and
\(\frac{c}{d}\)
are any two rational numbers, then:
(ii) Associativity property
Reason
If
\(\frac{a}{b} , \frac{c}{d}\)
and
\(\frac{e}{f}\)
are any three rational numbers, then:
(iii) Distributivity property
Reason
If
\(\frac{a}{b} , \frac{c}{d}\)
and
\(\frac{e}{f}\)
are any three rational numbers, then:
\(\frac{a}{b} \times ( \frac{c}{d} + \frac{e}{f} ) = ( \frac{a}{b} \times \frac{c}{d} ) + ( \frac{a}{b} \times \frac{e}{f} )\)
)
(iv) Existence of inverse
Reason
\(\frac{a}{b}\)
= reciprocal of
\(\frac{a}{b} = \frac{b}{a}\)
.
(v) Existence of identity
Reason
For a rational number
\(\frac{a}{b}\)
,
\(1 \times \frac{a}{b} = \frac{a}{b} \times 1 = \frac{a}{b}\)
.
Question 12
Fill in the blanks:
(i) The product of two positive rational numbers is always ............... .
(ii) The product of two negative rational numbers is always ............... .
(iii) If two rational numbers have opposite signs then their product is always ............... .
(iv) The reciprocal of a positive rational number is ............... and the reciprocal of a negative rational number is ............... .
(v) Rational number 0 has ............... reciprocal.
(vi) The product of a non-zero rational number and its reciprocal is ............... .
(vii) The numbers ............... and ............... are their own reciprocals.
Show step-by-step answer
(i) The product of two positive rational numbers is always positive.
(ii) The product of two negative rational numbers is always positive.
(iii) If two rational numbers have opposite signs then their product is always negative.
(iv) The reciprocal of a positive rational number is positive and the reciprocal of a negative rational number is negative.
(v) Rational number 0 has no reciprocal.
(vi) The product of a non-zero rational number and its reciprocal is 1.
(vii) The numbers 1 and -1 are their own reciprocal.
(viii) If m is reciprocal of n, then the reciprocal of n is m.
(i) Let 2 positive rational numbers be
\(\frac{a}{b}\)
and
\(\frac{c}{d}\)
.
Hence,
\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} = a c b d\)\(\frac{ac}{bd}\)\(\frac{ac}{bd}\)
is also positive rational number.
(ii) Let 2 negative rational numbers be -
\(\frac{a}{b}\)
and -
\(\frac{c}{d}\)
.
Hence,
\(- \frac{a}{b} \times - \frac{c}{d} = \frac{- a \times - c}{b \times d} = a c b d\)\(\frac{ac}{bd}\)\(\frac{ac}{bd}\)
is positive rational number.
(iii) Let 2 rational numbers be
\(\frac{a}{b}\)
and -
\(\frac{c}{d}\)
.
Hence,
\(\frac{a}{b} \times - \frac{c}{d} = \frac{a \times - c}{b \times d} = - a c b d\)\(\frac{- ac}{bd}\)\(\frac{ac}{bd}\)
is negative rational number.
(iv) Let the positive rational number be
\(\frac{a}{b}\)
.
Reciprocal of
\(\frac{a}{b} = \frac{b}{a}\)
\(\frac{b}{a}\)
is a positive rational number.
Let the negative rational number be -
\(\frac{a}{b}\)
.
Reciprocal of -
\(\frac{a}{b} = - \frac{b}{a}\)
-
\(\frac{b}{a}\)
is a negative rational number.
(v) Reciprocal of
\(\frac{0}{1} = \frac{1}{0}\)
\(\frac{1}{0}\)
is not defined.
(vi) Let the positive rational number be
\(\frac{a}{b}\)
.
Reciprocal of
\(\frac{a}{b} = \frac{b}{a}\)
=1
\(\frac{1}{1} = \frac{1}{1} = 1\)
=1.
\(\frac{- 1}{1} = \frac{1}{- 1} = - 1\)
=−1.
(viii) If reciprocal of
\(\frac{m}{1} = \frac{n}{1}\)
Reciprocal of
\(\frac{n}{1} = \frac{m}{1}\)
Question 13
(i) its area
(ii) its perimeter
Show step-by-step answer
Length = 9 cm
\(10\,\frac{2}{3}\)
cm
Area = length x breadth
2
Area = 96 cm2
)
\(= 2 \times ( \frac{9 \times 3}{1 \times 3} + \frac{32 \times 1}{3 \times 1} ) = 2 \times ( \frac{27}{3} + \frac{32}{3} ) = 2 \times ( \frac{27 + 32}{3} ) = 2 \times ( \frac{59}{3} ) = ( \frac{59 \times 2}{3 \times 1} ) = ( \frac{118}{3} ) = 39 ( \frac{1}{3} )\)
)
\(39\,\frac{1}{3}\)
cm
Hence, area of the rectangular piece of paper is 96 cm2 and its perimeter is
\(39\,\frac{1}{3}\)
cm.
Question 14
Show step-by-step answer
\(7\,\frac{2}{5}\)
m =
\(\frac{37}{5}\)
m
\(4\,\frac{1}{6}\)
=
\(\frac{25}{6}\)
m
Area = length x breadth
\(= \frac{37}{5} \times \frac{25}{6} = \frac{37 \times 25}{5 \times 6} = \frac{925}{30} = \frac{185}{6} = 30 \frac{5}{6} m 2\)
m
2
)
\(= 2 \times ( \frac{37 \times 6}{5 \times 6} + \frac{25 \times 5}{6 \times 5} ) = 2 \times ( \frac{222}{30} + \frac{125}{30} ) = 2 \times ( \frac{222 + 125}{30} ) = 2 \times ( \frac{347}{30} ) = ( \frac{347 \times 2}{30 \times 1} ) = ( \frac{694}{30} ) = ( \frac{347}{15} ) = 23 ( \frac{2}{15} )\)
)
\(30\,\frac{5}{6}\)
m2 and perimeter =
\(23\,\frac{2}{15}\)
m
Exercise 1(D)
Tap any answer panel to reveal the full working.
Question 1(i)
Show step-by-step answer
\(- \frac{4}{9}\)
divided by
\(- \frac{2}{3}\)
gives
\(\frac{2}{3}\)
Hence, Option 1 is the correct option.
Question 1(ii)
Show step-by-step answer
Let
x be the number.
The rational number by which should
\(\frac{1}{2}\)
be divided to get
\(- \frac{2}{3}\)
is
\(- \frac{3}{4}\)
Hence, Option 2 is the correct option.
Question 1(iii)
For the three rational number a, b and c; which of the following is correct:
a x (b ÷ c) = (a ÷ b) x (a ÷ c)
a ÷ (b ÷ c) = (a ÷ b) ÷ (a ÷ c)
a ÷ (b ÷ c) ≠ a ÷ b ÷ c
Show step-by-step answer
We know that, division of rational numbers is not associative.
∴ a ÷ (b ÷ c) ≠ a ÷ b ÷ c
Hence, Option 4 is the correct option.
Question 1(iv)
Show step-by-step answer
Let the number be
x.
⇒x=−2
Hence, Option 4 is the correct option.
Question 1(v)
(8 ÷ 3) ÷ (3 ÷ 8) is equal to:
1
none of the above
Show step-by-step answer
Hence, Option 1 is the correct option.
Question 2(i)
Evaluate:
Show step-by-step answer
=3
\(1 \div \frac{1}{3} = 3\)
=3
Question 2(ii)
Evaluate:
Show step-by-step answer
=5
\(3 \div \frac{3}{5} = 5\)
=5
Question 2(iii)
Evaluate:
Show step-by-step answer
Hence,
\(- \frac{5}{12} \div \frac{1}{16} = - 6 \frac{2}{3}\)
Question 2(iv)
Evaluate:
Show step-by-step answer
Hence,
\(- \frac{21}{16} \div ( \frac{- 7}{8} ) = 1 \frac{1}{2}\)
Question 2(v)
Evaluate:
Show step-by-step answer
=0
\(0 \div ( \frac{- 4}{7} ) = 0\)
)=0
Question 2(vi)
Evaluate:
Show step-by-step answer
Hence,
\(\frac{8}{- 5} \div \frac{24}{25} = - 1 \frac{2}{3}\)
Question 2(vii)
Evaluate:
Show step-by-step answer
Hence,
\(- \frac{3}{4} \div ( - 9 ) = \frac{1}{12}\)
Question 2(viii)
Evaluate:
Show step-by-step answer
Hence,
\(\frac{3}{4} \div ( - \frac{5}{12} ) = - 1 \frac{4}{5}\)
Question 2(ix)
Evaluate:
Show step-by-step answer
Hence,
\(- 5 \div ( - \frac{10}{11} ) = 5 \frac{1}{2}\)
Question 2(x)
Evaluate:
Show step-by-step answer
Hence,
\(\frac{- 7}{11} \div ( \frac{- 3}{44} ) = 9 \frac{1}{3}\)
Question 3(i)
Divide:
Show step-by-step answer
=9
\(3 \div \frac{1}{3} = 9\)
=9
Question 3(ii)
Divide:
Show step-by-step answer
=4
\(- 2 \div - \frac{1}{2} = 4\)
=4
Question 3(iii)
Divide:
Show step-by-step answer
=0
\(0 \div \frac{7}{- 9} = 0\)
=0
Question 3(iv)
Divide:
Show step-by-step answer
Hence,
\(\frac{- 5}{8} \div \frac{1}{4} = - 2 \frac{1}{2}\)
Question 3(v)
Divide:
Show step-by-step answer
Hence,
\(- \frac{3}{4} \div - \frac{9}{16} = 1 \frac{1}{3}\)
Question 4
Show step-by-step answer
Let the number be
x.
The other number is
\(- 3 \frac{1}{2}\)
.
Question 5
Show step-by-step answer
Let the number be
x.
⇒x=6
The other number is 6.
Question 6(i)
Show step-by-step answer
if
\(m = \frac{5}{3}\)
, then
\(n = - 1 \frac{2}{3} .\)
.
Question 6(ii)
Show step-by-step answer
if
\(n = - \frac{10}{9}\)
, then
\(n = 2 \frac{1}{2} .\)
.
Question 7
Show step-by-step answer
Let the number be
x
\(- \frac{3}{4}\)
must be multiplied by
\(\frac{3}{4}\)
so that the product is
\(- \frac{9}{16}\)
.
Question 8
Show step-by-step answer
Let the number be
x
⇒x=−26
\(- \frac{8}{13}\)
must be multiplied by -26 so that the product is 16
Question 9
Show step-by-step answer
Let the cost of one litre of milk be ₹
x.
⇒x=14
The cost of one litre of milk = ₹14.
Question 10
Show step-by-step answer
Let the cost of 1 metre of cloth be ₹
x.
Hence, The cost of 1 meter of cloth is ₹
\(26\,\frac{1}{34}\)
.
Question 11
Show step-by-step answer
The sum of
\(\frac{3}{7}\)
and
\(\frac{- 5}{14}\)
Dividing the sum of
\(\frac{3}{7}\)
and
\(\frac{- 5}{14}\)
by
\(- \frac{1}{2}\)
On dividing the sum of
\(\frac{3}{7}\)
and
\(\frac{- 5}{14}\)
by
\(- \frac{1}{2}\)
we get
\(- \frac{1}{7}\)
.
Question 12(i)
Show step-by-step answer
\(( m + n ) \div ( m - n ) = ( \frac{2}{3} + \frac{3}{2} ) \div ( \frac{2}{3} - \frac{3}{2} )\)
)
m =
\(\frac{2}{3}\)
and
n =
\(\frac{3}{2}\)
then
\(( m + n ) \div ( m - n ) = - \frac{13}{5}\)
.
Question 12(ii)
Show step-by-step answer
\(( m + n ) \div ( m - n ) = ( \frac{3}{4} + \frac{4}{3} ) \div ( \frac{3}{4} - \frac{4}{3} )\)
)
m =
\(\frac{3}{4}\)
and
n =
\(\frac{4}{3}\)
then
\(( m + n ) \div ( m - n ) = - \frac{25}{7}\)
.
Question 12(iii)
Show step-by-step answer
\(( m + n ) \div ( m - n ) = [ \frac{4}{5} + ( - \frac{3}{10} ) ] \div [ \frac{4}{5} - ( - \frac{3}{10} ) ]\)
)]
m =
\(\frac{4}{5}\)
and
n = -
\(\frac{3}{10}\)
then
\(( m + n ) \div ( m - n ) = \frac{5}{11}\)
.
Question 13
Show step-by-step answer
Let the number be
x.
The other number is
\(10\,\frac{5}{7}\)
.
Question 14
Show step-by-step answer
The sum of
\(\frac{5}{8}\)
and
\(\frac{- 11}{12}\)
The difference of
\(\frac{3}{7}\)
and
\(\frac{5}{14}\)
Dividing the sum of
\(\frac{5}{8}\)
and
\(\frac{- 11}{12}\)
by the difference of
\(\frac{3}{7}\)
and
\(\frac{5}{14}\)
,
\(( \frac{5}{8} + \frac{- 11}{12} ) \div ( \frac{3}{7} - \frac{5}{14} ) = - 4 \frac{1}{12}\)
.
Question 15
Show step-by-step answer
\(5\,\frac{5}{7}\)
m2 =
\(\frac{40}{7}\)
m2
\(3\,\frac{3}{4}\)
m =
\(\frac{15}{4}\)
m
Let the breadth of the rectangular plate be b.
Area = length x breadth
)
\(1\,\frac{11}{21}\)
and perimeter =
\(10\,\frac{23}{42}\)
Question 16
Show step-by-step answer
\(7\,\frac{3}{26}\)
cm2 =
\(\frac{185}{26}\)
cm2
\(2\,\frac{9}{13}\)
m =
\(\frac{35}{13}\)
cm
Let the length of the piece of paper be l.
Area = length x breadth
)
\(2\,\frac{9}{14}\)
and perimeter =
\(10\,\frac{61}{91}\)
Exercise 1(E)
Tap any answer panel to reveal the full working.
Question 1(i)
In the following number line, points A and B represent:
In the following number line, points A and B represent: Rational Numbers, Concise Mathematics Solutions ICSE Class 8.
\(- 2 \frac{1}{5}\)
and
\(2\,\frac{3}{5}\)
Show step-by-step answer
In this number line, there are 5 small lines between every 2 consecutive integers, which means moving one step left from 0 gives
\(- \frac{1}{5}\)
.
\(- 1 \frac{4}{5}\)
\(2\,\frac{3}{5}\)
Hence, Option 3 is the correct option.
Question 1(ii)
Using the number line, given below; the length of line segment AB is:
\(\frac{13}{5}\)
=
\(2\,\frac{3}{5}\)
=
\(- 2 \frac{3}{5}\)
Show step-by-step answer
\(- 1 \frac{4}{5}\)
and B =
\(2\,\frac{3}{5}\)
.
\(\frac{9}{5} + \frac{13}{5}\)
Hence, Option 3 is the correct option.
Question 1(iii)
Show step-by-step answer
For any two rational numbers
\(\frac{a}{b}\)
and
\(\frac{c}{d}\)
,
\(( \frac{a + c}{b + d} )\)
) is also a rational number with its value lying between
\(\frac{a}{b}\)
and
\(\frac{c}{d}\)
.
Hence, Option 3 is the correct option.
Question 1(iv)
Show step-by-step answer
As we know that, for any two rational numbers
\(\frac{a}{b}\)
and
\(\frac{c}{d}\)
,
\(( \frac{a + c}{b + d} )\)
) is also a rational number with its value lying between
\(\frac{a}{b}\)
and
\(\frac{c}{d}\)
.
The rational number between
\(\frac{1}{3}\)
and
\(\frac{1}{2}\)
is
\(( \frac{1 + 1}{3 + 2} )\)
)
\(( \frac{2}{5} )\)
)
The rational number between
\(\frac{1}{3}\)
and
\(\frac{2}{5}\)
is
\(( \frac{1 + 2}{3 + 5} )\)
)
\(( \frac{3}{8} )\)
)
The rational number between
\(\frac{1}{2}\)
and
\(\frac{2}{5}\)
is
\(( \frac{1 + 2}{2 + 5} )\)
)
\(( \frac{3}{7} )\)
)
\(\frac{3}{7}\)
and
\(\frac{3}{8}\)
are two rational number between
\(\frac{1}{3}\)
and
\(\frac{1}{2}\)
Hence, option 1 is the correct option.
Question 1(v)
B and E respectively
C and D respectively
C and E respectively
B and F respectively
Show step-by-step answer
In this number line, there are 4 small lines between every 2 consecutive integers, which means moving one step towards left from 0 gives
\(- \frac{1}{4}\)
.
\(- 1 \frac{3}{4} = - \frac{7}{4}\)
\(\frac{3}{4}\)
\(- \frac{7}{4}\)
and
\(\frac{3}{4}\)
are represented by C and E, respectively.
Hence, option 3 is the correct option.
Question 2
Draw a number line and mark
Show step-by-step answer
Draw a number line as shown below:
Draw a number line and mark. Rational Numbers, Concise Mathematics Solutions ICSE Class 8.
In this number line
OA = AB = ...............= OA' = A'B' = 1 unit
Since the denominator of each given rational number is 4, divide each OA, AB, BC, OA', A'B',etc into four equal parts.
To represent
\(\frac{1}{4}\)
, move one step towards the right side of 0 to reach P as shown.
\(\frac{1}{4}\)
unit.
Hence, to represent
\(\frac{3}{4}\)
, move 3 steps towards the right side of 0 to reach point Q. So, Q represent
\(\frac{3}{4}\)
.
In the same way to represent
\(\frac{- 3}{4}\)
, move 3 steps towards the left side of 0 to reach point R. So, R represent
\(\frac{- 3}{4}\)
.
So, S represent
\(\frac{7}{4}\)
and T represent
\(\frac{- 7}{4}\)
.
Question 3
On a number line mark the points
Show step-by-step answer
Draw a number line as shown below:
On a number line mark the points. Rational Numbers, Concise Mathematics Solutions ICSE Class 8.
In this number line
OA = AB = ...............= OA' = A'B' = 1 unit
Since the denominator of each given rational number is 3, divide each OA, AB, BC, OA', A'B', etc into three equal parts.
To represent
\(\frac{1}{3}\)
, move one step towards the right side of 0 to reach P as shown.
\(\frac{1}{3}\)
unit.
Hence, to represent
\(\frac{2}{3}\)
, move 2 steps towards the right side of 0 to reach point Q. So, Q represents
\(\frac{2}{3}\)
.
In the same way to represent
\(\frac{- 2}{3}\)
, move 2 steps towards the left side of 0 to reach point R. So, R represents
\(\frac{- 2}{3}\)
.
Similarly, S represents
\(\frac{- 8}{3}\)
, T represents
\(\frac{7}{3}\)
and B' represents -2.
Question 4(i)
Insert one rational number between
Show step-by-step answer
As we know that, for any two rational numbers
\(\frac{a}{b}\)
and
\(\frac{c}{d}\)
,
\(( \frac{a + c}{b + d} )\)
) is also a rational number with its value lying between
\(\frac{a}{b}\)
and
\(\frac{c}{d}\)
.
The rational number between
\(\frac{3}{5}\)
and
\(\frac{5}{8}\)
is
\(( \frac{3 + 5}{5 + 8} )\)
)
\(( \frac{8}{13} )\)
)
Hence, one rational number between
\(\frac{3}{5}\)
and
\(\frac{5}{8}\)
is
\(\frac{8}{13}\)
.
Question 4(ii)
Insert one rational number between
Show step-by-step answer
As we know that, for any two rational numbers
\(\frac{a}{b}\)
and
\(\frac{c}{d}\)
,
\(( \frac{a + c}{b + d} )\)
) is also a rational number with its value lying between
\(\frac{a}{b}\)
and
\(\frac{c}{d}\)
.
The rational number between
\(\frac{1}{2}\)
and
\(\frac{2}{1}\)
is
\(( \frac{1 + 2}{2 + 1} )\)
)
\(( \frac{3}{3} )\)
)
=1
Hence, one rational number between
\(\frac{1}{2}\)
and 2 is 1.
Question 5
Insert two rational numbers between:
Show step-by-step answer
As we know that, for any two rational numbers
\(\frac{a}{b}\)
and
\(\frac{c}{d}\)
,
\(( \frac{a + c}{b + d} )\)
) is also a rational number with its value lying between
\(\frac{a}{b}\)
and
\(\frac{c}{d}\)
.
\(\frac{5}{7}\)
and
\(\frac{3}{8}\)
Hence, required rational numbers between
\(\frac{5}{7}\)
and
\(\frac{3}{8}\)
are :
\(\frac{13}{22}\)
and
\(\frac{8}{15}\)
Question 6
Insert three rational numbers between:
Show step-by-step answer
As we know that, for any two rational numbers
\(\frac{a}{b}\)
and
\(\frac{c}{d}\)
,
\(( \frac{a + c}{b + d} )\)
) is also a rational number with its value lying between
\(\frac{a}{b}\)
and
\(\frac{c}{d}\)
.
\(\frac{8}{11}\)
and
\(\frac{4}{9}\)
Hence, required rational numbers between
\(\frac{8}{11}\)
and
\(\frac{4}{9}\)
are :
\(\frac{11}{16} , \frac{3}{5}\)
and
\(\frac{1}{2}\)
Question 7
Show step-by-step answer
Make denominator of each given rational number equal to 15 (the LCM).
and
\(\frac{3}{5}\)
and
\(\frac{2}{3}\)
; multiply the numerator and the denominator of each rational number by 5 + 1 = 6.
\(\frac{9}{15} = \frac{9 \times 6}{15 \times 6} = \frac{54}{90} \therefore\)
and
\(\frac{3}{5}\)
and
\(\frac{2}{3}\)
are :
\(\frac{54}{90} , \frac{55}{90} , \frac{56}{90} , \frac{57}{90} , \frac{58}{90} , \frac{59}{90} , \frac{60}{90}\)
\(\frac{3}{5} , \frac{11}{18} , \frac{28}{45} , \frac{19}{30} , \frac{29}{45} , \frac{59}{90} , \frac{2}{3}\)
Hence,
\(\frac{11}{18} , \frac{28}{45} , \frac{19}{30} , \frac{29}{45}\)
and
\(\frac{59}{90}\)
lie between
\(\frac{3}{5}\)
and
\(\frac{2}{3}\)
.
Question 8
Show step-by-step answer
Make denominator of each given rational number equal to 18 (the LCM).
and
\(\frac{5}{6}\)
and
\(\frac{8}{9}\)
; multiply the numerator and the denominator of each rational number by 6 + 1 = 7.
and
Required rational numbers between
\(\frac{5}{6}\)
and
\(\frac{8}{9}\)
are :
\(\frac{105}{126} , \frac{106}{126} , \frac{107}{126} , \frac{108}{126} , \frac{109}{126} , \frac{110}{126} , \frac{111}{126} , \frac{112}{126}\)
\(\frac{5}{6} , \frac{53}{63} , \frac{107}{126} , \frac{6}{7} , \frac{109}{126} , \frac{55}{63} , \frac{37}{42} , \frac{8}{9}\)
Hence,
\(\frac{5}{6} , \frac{53}{63} , \frac{107}{126} , \frac{6}{7} , \frac{109}{126}\)
and
\(\frac{37}{42}\)
lie between
\(\frac{5}{6}\)
and
\(\frac{8}{9}\)
.
Question 9
Insert seven rational numbers between 2 and 3.
Show step-by-step answer
Write the endpoints with denominator 1:
To insert seven rational numbers, multiply both fractions by \(\frac88\).
The seven rational numbers between 2 and 3 are:
In simplified or mixed-number form:
Question 1(i)
an irrational number
a rational number
0
undefined
Show step-by-step answer
\(\frac{0}{0}\)
is undefined.
Hence, option 4 is the correct option.
Question 1(ii)
a and b are two rational numbers such that a + b = 0; then :
a = b
a and b are numerically equal
a and b are numerically equal but opposite in sign
none of the above
Show step-by-step answer
a + b = 0
a = 0 - b
a = -b
a and b are numerically equal but opposite in sign.
Hence, option 3 is the correct option
Question 1(iii)
1
0
Show step-by-step answer
\(\frac{3}{8}\)
=
\(- \frac{3}{8}\)
Hence, option 4 is the correct option.
Question 1(iv)
1
0
Show step-by-step answer
\(\frac{2}{3}\)
=
\(\frac{3}{2}\)
We need to find the sum of
\(\frac{2}{3}\)
and
\(\frac{3}{2}\)
Hence, option 2 is the correct option.
Question 1(v)
none of these
Show step-by-step answer
Let the number be
x.
Hence, option 3 is the correct option.
Question 1(vi)
Statement 2: Subtraction has no identity.
Which of the following options is correct?
Both the statement are true.
Both the statement are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.
Show step-by-step answer
For a rational number
\(\frac{7}{9} , \frac{7}{9} - 0 = \frac{7}{9} and 0 - \frac{7}{9} = - \frac{7}{9}\)
We know that,
For any number a,
a - 0 = a and 0 - a ≠ 0
Thus, subtraction only has right identity.
So, statement 1 is true.
Subtraction has 0 as right identity element.
So, statement 2 is false.
Hence, option 3 is the correct option.
Question 1(vii)
Reason (R) : For every non-zero rational number 'a', its additive inverse is '-a' such that a + (-a) = 0.
Both A and R are correct, and R is the correct explanation for A.
Both A and R are correct, and R is not the correct explanation for A.
A is true, but R is false.
A is false, but R is true.
Show step-by-step answer
The additive inverse of a number a is a number -a such that :
⇒ a + (-a) = 0.
So, reason (R) is true.
According to Assertion: Additive inverse of
\(\frac{2}{5}\)
is
\(- \frac{5}{2}\)
.
=0
So, assertion (A) is false.
Hence, option 4 is the correct option.
Question 1(viii)
Assertion (A): The multiplicative inverse of \(-\frac75\) is \(-\frac57\).
Reason (R): For every non-zero rational number a, there is a rational number \(\frac1a\) such that \(a\times\frac1a=1\).
Show step-by-step answer
We know that the multiplicative inverse of every non-zero rational number a is its reciprocal \(\frac1a\). Therefore, Reason (R) is true.
Hence, the multiplicative inverse of \(-\frac75\) is \(-\frac57\). Assertion (A) is also true, and Reason (R) correctly explains it.
Therefore, option 1 is correct.
Question 1(ix)
Both A and R are correct, and R is the correct explanation for A.
Both A and R are correct, and R is not the correct explanation for A.
A is true, but R is false.
A is false, but R is true.
Show step-by-step answer
According to Assertion:
A number is rational if it can be written in the form
\(\frac{p}{q}\)
, where p and q are integers.
Since,
\(\frac{5}{2}\)
is in the form of
\(\frac{p}{q}\)
as well as 5 and 2 are integers.
So, assertion (A) is true.
According to commutative property of addition: When two numbers are added together, then a change in their positions does not change the result.
When
\(\frac{p}{q} and \frac{r}{s}\)
are any two rational numbers then
\(\frac{p}{q} + \frac{r}{s} = \frac{r}{s} + \frac{p}{q}\)
, as addition of rational numbers is a commutative property.
So, reason (R) is true but it does not explain assertion.
Hence, option 2 is the correct option.
Question 1(x)
=0 then 0 ÷
\(\frac{11}{12} = 0\)
=0, a rational number.
Reason (R) : If a rational number is divided by some non - zero rational number, the result is always a rational number.
Both A and R are correct, and R is the correct explanation for A.
Both A and R are correct, and R is not the correct explanation for A.
A is true, but R is false.
A is false, but R is true.
Show step-by-step answer
When 0 is divided by any non-zero number, the result is 0.
\(\frac{11}{12} = 0\)
=0
\(\frac{0}{1}\)
is in the form of
\(\frac{p}{q}\)
.
So, assertion (A) is true.
The division of a rational number
\(\frac{a}{b}\)
by another non-zero rational number
\(\frac{c}{d}\)
is:
\(\Rightarrow \frac{a}{b} \div \frac{c}{d} \Rightarrow \frac{a}{b} \times \frac{d}{c} \Rightarrow a d b c\)\(\frac{ad}{bc}\)\(\frac{ad}{bc}\)
is in the form of
\(\frac{p}{q}\)
.
So, reason is true. But it does not explains about assertion.
Hence, option 2 is the correct option.
Question 2(i)
Write the rational number that does not have a reciprocal.
Show step-by-step answer
The rational number is
\(\frac{0}{1}\)
The reciprocal of
\(\frac{0}{1} = \frac{1}{0}\)
(not defined).
Hence, 0 is the rational number that does not have a reciprocal.
Question 2(ii)
Write the rational numbers that are equal to their reciprocal.
Show step-by-step answer
\(\frac{1}{1} = \frac{1}{1} = 1\)
=1.
\(\frac{- 1}{1} = \frac{1}{- 1} = - 1\)
=−1.
The numbers 1 and -1 are their own reciprocal.
Question 2(iii)
Show step-by-step answer
The reciprocal of
\(\frac{- 16}{17} = - \frac{17}{16}\)
.
Question 3
Show step-by-step answer
Make denominator of each given rational number equal to 6 (the LCM).
and
The rational number between
\(- \frac{9}{6}\)
and
\(\frac{10}{6}\)
are
\(- \frac{8}{6} , - \frac{7}{6} , - \frac{6}{6} , - \frac{5}{6} , - \frac{4}{6} , - \frac{3}{6} , - \frac{2}{6} , - \frac{1}{6} , \frac{0}{6} , \frac{1}{6} , \frac{2}{6} , \frac{3}{6} , \frac{4}{6} , \frac{5}{6} , \frac{6}{6} , \frac{7}{6} , \frac{8}{6} , \frac{9}{6}\)
From these rational numbers we can take any five rational number.
Hence, required rational numbers between
\(- \frac{3}{2}\)
and
\(\frac{5}{3}\)
are :
\(- \frac{4}{3} , - \frac{1}{1} , - \frac{2}{3} , - \frac{1}{3} , \frac{1}{3}\)
Hence,
\(- 1 \frac{1}{3} , - 1 , - \frac{2}{3} , - \frac{1}{3}\)
and
\(\frac{1}{3}\)
lies between
\(- \frac{3}{2}\)
and
\(\frac{5}{3}\)
.
Question 4
Write five rational numbers greater than -4.
Show step-by-step answer
There are infinite many rational number between -4 and ∞.
Hence, -3, -2, -1, 1, 2 are five rational number greater than -4.
Question 5
Show step-by-step answer
Let
x be added to
\(- 2 \frac{1}{2}\)
.
The number added to
\(- 2 \frac{1}{2}\)
to get
\(- 3 \frac{1}{3}\)
is
\(\frac{- 5}{6}\)
.
Question 6
Show step-by-step answer
Let
x be subtracted from
\(2\,\frac{1}{2}\)
.
The number subtracted from
\(2\,\frac{1}{2}\)
to get
\(- 3 \frac{1}{3}\)
is
\(5\,\frac{5}{6}\)
.
Question 7(i)
m - n ≠ n - m
Show step-by-step answer
m - n ≠ n - m
LHS:
RHS:
Hence, LHS ≠ RHS
m - n ≠ n - m
Question 7(ii)
-(m + n) = (-m) + (-n)
Show step-by-step answer
-(m + n) = (-m) + (-n)
LHS:
\(- ( m + n ) - ( - \frac{7}{9} + \frac{5}{6} )\)
)
\(= - ( - \frac{7 \times 2}{9 \times 2} + \frac{5 \times 3}{6 \times 3} ) = - ( - \frac{14}{18} + \frac{15}{18} ) = - ( \frac{- 14 + 15}{18} ) = - ( \frac{1}{18} )\)
)
RHS:
\(( - m ) + ( - n ) = - ( - \frac{7}{9} ) + ( - \frac{5}{6} ) = ( \frac{7}{9} ) + ( - \frac{5}{6} )\)
)
\(= ( \frac{7 \times 2}{9 \times 2} ) + ( - \frac{5 \times 3}{6 \times 3} ) = ( \frac{14}{18} ) + ( - \frac{15}{18} ) = ( \frac{14 + ( - 15 )}{18} ) = ( \frac{- 1}{18} )\)
)
Hence, LHS = RHS
−
(
m
+
n
)
=
(
−
m
)
+
(
−
n
)
∴−(m+n)=(−m)+(−n)
Question 8
Show step-by-step answer
The rational numbers are
\(- \frac{7}{3}\)
and
\(\frac{7}{4}\)
.
Hence, the rational number will be-
Draw a number line as shown below:
In this number line OA = AB = ........... = OA' = A'B' = 1 unit
Since, the denominator of each given rational numbers is 12, divide each of OA, AB,...,OA', A'B', etc. into twelve equal parts.
To represent
\(\frac{1}{12}\)
, moves one step towards the right side of O to reach point P as shown.
\(\frac{1}{12}\)
unit and so P represents
\(\frac{1}{12}\)
.
In the same way, to represent
\(- 2 \frac{4}{12}\)
, move 4 steps toward the left side of B' to reach point Q. Clearly, Q represents
\(- \frac{7}{3}\)
.
Similarly, to represent
\(1\,\frac{9}{12}\)
, move 9 steps toward the right side of A to reach point R. Clearly, R represents
\(\frac{7}{4}\)
.
Question 9(i)
Show step-by-step answer
\(- \frac{5}{6} + \frac{3}{8} = - \frac{11}{24}\)
Question 9(ii)
Show step-by-step answer
\(- \frac{5}{6} - \frac{3}{8} = - 1 \frac{5}{24}\)
Question 9(iii)
Show step-by-step answer
Question 9(iv)
Show step-by-step answer
Question 10(i)
Show step-by-step answer
Let the number be
x
\(- \frac{17}{21}\)
must be multiplied by
\(\frac{12}{17}\)
so that the product is
\(- \frac{4}{7}\)
.
Question 10(ii)
Show step-by-step answer
Let the number be
x
\(- 1 \frac{4}{17}\)
must be divided by
\(\frac{12}{17}\)
so that the product is
\(- \frac{4}{7}\)
.
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Frequently Asked Questions
Key ideas from the Rational Numbers chapter.
What is a rational number?
A rational number can be written in the form p/q, where p and q are integers and q is not zero.
How are two rational numbers added?
Use a common denominator, add the numerators, keep the common denominator and simplify the resulting fraction.
What is the additive inverse of a rational number?
The additive inverse of a number is the same number with the opposite sign. Their sum is zero.
Is addition of rational numbers commutative and associative?
Yes. For rational numbers a, b and c: a + b = b + a, and (a + b) + c = a + (b + c).
How can rational numbers be shown on a number line?
Divide the interval between consecutive integers into equal parts according to the denominator, then count the required parts from zero.