RS Aggarwal Solutions: Fractions | Mathematics (Maths) Class 6 PDF Download

 Page 1


Points to Remember :
1. Fraction. The number of the forms 
a
b
where p and q are integers and q  0, is
called a fraction. Here p is called the num-
erator and ‘q’ is called the denominator.
Fraction can also be represent on a
number line. e.g. 
2
3
5
7
11
19
, ,
 are called
fractions.
2
3
 means two parts from 3 and is read as
two-third.
5
7
 means, five parts from 7 and is read
five-seventh.
11
19
 means 11 parts from 19 and is read
as eleven-nineteenth.
2. (A) Equivalent fractions. Two or more
fractions representing the same part of a
whole are called equivalent fractions.
(B) Rule to form equivalent fractions.
To get a fractions equivalent to a given
fractions, we multiply or divide the
numerator and denominator of the given
fraction by the same non-zero number.
(C) To test of two equivalent fractions.
Let 
a
b
 and 
c
d
 are two equivalent fractions,
these    
a
b
c
d
   ad = bc.
In other words, we can say that
if ad = bc then, fractions 
a
b
 and 
c
d
 are
equal.
3. Like and unlike fractions :
(A) Like fractions. Fractions having
same denominators are called like
fractions e.g. 
1
7
3
7
5
7
4
7
, , , .
(B) Unlike fractions. Fractions with
different denominators are called unlike
fractions e.g. 
2
3
5
7
4
9
, , .
(C) To convert unlike fractions to like
fractions. We can convert unlike
fractions into like fractions by equalising
their denominator with the help of using
their L.C.M.
4. Fractions in simplest form or in lowest
terms. A fraction is said to be in the
simplest form if the HCF of its numerator
and denominator is 1.
5. Proper, improper and mixed
fractions :
(A) Proper fractions. A fraction whose
numerator is less than its denominator is
called a proper fraction e.g. 
1
2
5
7
4
9
, , etc.
(B) Improper fractions : A fraction
whose numerator is greater than its
denominator, is called improper fraction
e.g. 
5
4
7
3
9
2
, , etc.
(C) Mixed fractions. A combination of
a whole number and a proper fraction is
Page 2


Points to Remember :
1. Fraction. The number of the forms 
a
b
where p and q are integers and q  0, is
called a fraction. Here p is called the num-
erator and ‘q’ is called the denominator.
Fraction can also be represent on a
number line. e.g. 
2
3
5
7
11
19
, ,
 are called
fractions.
2
3
 means two parts from 3 and is read as
two-third.
5
7
 means, five parts from 7 and is read
five-seventh.
11
19
 means 11 parts from 19 and is read
as eleven-nineteenth.
2. (A) Equivalent fractions. Two or more
fractions representing the same part of a
whole are called equivalent fractions.
(B) Rule to form equivalent fractions.
To get a fractions equivalent to a given
fractions, we multiply or divide the
numerator and denominator of the given
fraction by the same non-zero number.
(C) To test of two equivalent fractions.
Let 
a
b
 and 
c
d
 are two equivalent fractions,
these    
a
b
c
d
   ad = bc.
In other words, we can say that
if ad = bc then, fractions 
a
b
 and 
c
d
 are
equal.
3. Like and unlike fractions :
(A) Like fractions. Fractions having
same denominators are called like
fractions e.g. 
1
7
3
7
5
7
4
7
, , , .
(B) Unlike fractions. Fractions with
different denominators are called unlike
fractions e.g. 
2
3
5
7
4
9
, , .
(C) To convert unlike fractions to like
fractions. We can convert unlike
fractions into like fractions by equalising
their denominator with the help of using
their L.C.M.
4. Fractions in simplest form or in lowest
terms. A fraction is said to be in the
simplest form if the HCF of its numerator
and denominator is 1.
5. Proper, improper and mixed
fractions :
(A) Proper fractions. A fraction whose
numerator is less than its denominator is
called a proper fraction e.g. 
1
2
5
7
4
9
, , etc.
(B) Improper fractions : A fraction
whose numerator is greater than its
denominator, is called improper fraction
e.g. 
5
4
7
3
9
2
, , etc.
(C) Mixed fractions. A combination of
a whole number and a proper fraction is
called a mixed fraction e.g. 1
5
7
2
3
4
5
1
8
, ,
etc.
(D) To convert a mixed fraction into
an improper fraction. We know that a
mixed fraction = A whole number + a
fraction
e.g. 
1
5
7
1
5
7
1 7 5
7
 
7 5
7
12
7
.
Method. Multiply the whole number with
the denominator of the fraction and add
the numerator of the fraction, then the
new numerator is the numerator of the
improper fraction with same
denominator.
(E) To convert an improper fraction
with a mixed fraction. On dividing the
numerator by denominator, we get
quotient i.e. whole number. Then, whole
number plus 
Remainder
Denominator
 is the required
mixed fraction e.g. 
15
4
3
3
4
3
3
4
.
6. Comparison of fractions :
(A) Comparing fractions with same
denominator.
Rule. Among two fractions with the
same denominator, the greater
numerator is greater fraction.
(B) Comparing fractions with same
numerator.
Rule. Among two fraction with same
numerator the one with smaller
denominator is the greater fraction.
(C) General method for comparison :
(i) By means of cross multiplication.
(ii) By converting the given fractions into
like fractions.
7. Addition of fractions :
(A) Addition of like fractions
Sum of their numerators
Denominator
(B) Addition of unlike fractions.
Change the given fractions into equivalent
like fractions and then add them as given
in (A).
8. Subtraction of fractions. We use similar
methods as in addition for subtraction of
fractions.
Exercise 5A
Q.1. Write the fraction representing the shaded
portion :
(i) (ii)
(iii) (iv)
(v) (vi)
Sol. (i)
4
3
(ii)
4
1
(iii)
3
2
(iv)
10
3
(v)
9
4
(vi)
8
3
Q.2. Shade 
9
4
 of the given figure.
Page 3


Points to Remember :
1. Fraction. The number of the forms 
a
b
where p and q are integers and q  0, is
called a fraction. Here p is called the num-
erator and ‘q’ is called the denominator.
Fraction can also be represent on a
number line. e.g. 
2
3
5
7
11
19
, ,
 are called
fractions.
2
3
 means two parts from 3 and is read as
two-third.
5
7
 means, five parts from 7 and is read
five-seventh.
11
19
 means 11 parts from 19 and is read
as eleven-nineteenth.
2. (A) Equivalent fractions. Two or more
fractions representing the same part of a
whole are called equivalent fractions.
(B) Rule to form equivalent fractions.
To get a fractions equivalent to a given
fractions, we multiply or divide the
numerator and denominator of the given
fraction by the same non-zero number.
(C) To test of two equivalent fractions.
Let 
a
b
 and 
c
d
 are two equivalent fractions,
these    
a
b
c
d
   ad = bc.
In other words, we can say that
if ad = bc then, fractions 
a
b
 and 
c
d
 are
equal.
3. Like and unlike fractions :
(A) Like fractions. Fractions having
same denominators are called like
fractions e.g. 
1
7
3
7
5
7
4
7
, , , .
(B) Unlike fractions. Fractions with
different denominators are called unlike
fractions e.g. 
2
3
5
7
4
9
, , .
(C) To convert unlike fractions to like
fractions. We can convert unlike
fractions into like fractions by equalising
their denominator with the help of using
their L.C.M.
4. Fractions in simplest form or in lowest
terms. A fraction is said to be in the
simplest form if the HCF of its numerator
and denominator is 1.
5. Proper, improper and mixed
fractions :
(A) Proper fractions. A fraction whose
numerator is less than its denominator is
called a proper fraction e.g. 
1
2
5
7
4
9
, , etc.
(B) Improper fractions : A fraction
whose numerator is greater than its
denominator, is called improper fraction
e.g. 
5
4
7
3
9
2
, , etc.
(C) Mixed fractions. A combination of
a whole number and a proper fraction is
called a mixed fraction e.g. 1
5
7
2
3
4
5
1
8
, ,
etc.
(D) To convert a mixed fraction into
an improper fraction. We know that a
mixed fraction = A whole number + a
fraction
e.g. 
1
5
7
1
5
7
1 7 5
7
 
7 5
7
12
7
.
Method. Multiply the whole number with
the denominator of the fraction and add
the numerator of the fraction, then the
new numerator is the numerator of the
improper fraction with same
denominator.
(E) To convert an improper fraction
with a mixed fraction. On dividing the
numerator by denominator, we get
quotient i.e. whole number. Then, whole
number plus 
Remainder
Denominator
 is the required
mixed fraction e.g. 
15
4
3
3
4
3
3
4
.
6. Comparison of fractions :
(A) Comparing fractions with same
denominator.
Rule. Among two fractions with the
same denominator, the greater
numerator is greater fraction.
(B) Comparing fractions with same
numerator.
Rule. Among two fraction with same
numerator the one with smaller
denominator is the greater fraction.
(C) General method for comparison :
(i) By means of cross multiplication.
(ii) By converting the given fractions into
like fractions.
7. Addition of fractions :
(A) Addition of like fractions
Sum of their numerators
Denominator
(B) Addition of unlike fractions.
Change the given fractions into equivalent
like fractions and then add them as given
in (A).
8. Subtraction of fractions. We use similar
methods as in addition for subtraction of
fractions.
Exercise 5A
Q.1. Write the fraction representing the shaded
portion :
(i) (ii)
(iii) (iv)
(v) (vi)
Sol. (i)
4
3
(ii)
4
1
(iii)
3
2
(iv)
10
3
(v)
9
4
(vi)
8
3
Q.2. Shade 
9
4
 of the given figure.
Sol. In the figure, 
9
4
 is shaded
3. In the given figure, if we say that the
shaded region is 
4
1
, then identify the error
in it.
Sol. In the figure, whole rectangle is not
divided into four equal parts.
Q. 4. Write a fraction for each of the following :
(i) three-fourths (ii) four-sevenths
(iii) two-fifths (iv) three-tenths
(v) one-eighth (vi) five-sixths
(vii) eight-ninths (viii) seven-twelfths
Sol. (i) Three-fourths 
3
4
.
(ii) Four-sevenths 
4
7
.
(iii) Two-fifths 
2
5
.
(iv) Three-tenths 
3
10
.
(v) One-eighth 
1
8
.
(vi) Five-sixths 
5
6
.
(vii) Eight-ninths 
8
9
.
(viii) Seven-twelfths 
7
12
.    Ans.
Q. 5. Write down the numerator and denomi-
nator in each of the following fractions :
(i)
4
9
(ii)
11
6
(iii)
15
8
(iv)
12
17
(v)
1
5
Sol. (i) In 
4
9
, numerator is 4 and denominator
is 9.
(ii) In 
11
6
, numerator is 6 and denominator
is 11.
(iii) In 
15
8
, numerator is 8 and denominator
is 15.
(iv) In 
12
17
, numerator is 12 and denominator
is 17.
(v)
1
5
, numerator is 5 and denominator is 1.
Q. 6. Write down the fraction in which
(i) numerator = 3, denominator = 8
(ii) numerator = 5, denominator = 12
(iii) numerator = 7, denominator = 16
(iv) numerator = 8, denominator = 15
Sol. (i) Numerator = 3, Denominator = 8, then
fraction 
3
8
.
(ii) Numerator = 5, Denominator = 12, then
fraction = 
12
5
.
(iii) Numerator = 7, Denominator = 16, then
fraction 
7
16
.
(iv) Numerator = 8, Denominator = 15, then
fraction 
8
15
.   Ans.
Q.7. Write down the fractional number for
each of the following :
Page 4


Points to Remember :
1. Fraction. The number of the forms 
a
b
where p and q are integers and q  0, is
called a fraction. Here p is called the num-
erator and ‘q’ is called the denominator.
Fraction can also be represent on a
number line. e.g. 
2
3
5
7
11
19
, ,
 are called
fractions.
2
3
 means two parts from 3 and is read as
two-third.
5
7
 means, five parts from 7 and is read
five-seventh.
11
19
 means 11 parts from 19 and is read
as eleven-nineteenth.
2. (A) Equivalent fractions. Two or more
fractions representing the same part of a
whole are called equivalent fractions.
(B) Rule to form equivalent fractions.
To get a fractions equivalent to a given
fractions, we multiply or divide the
numerator and denominator of the given
fraction by the same non-zero number.
(C) To test of two equivalent fractions.
Let 
a
b
 and 
c
d
 are two equivalent fractions,
these    
a
b
c
d
   ad = bc.
In other words, we can say that
if ad = bc then, fractions 
a
b
 and 
c
d
 are
equal.
3. Like and unlike fractions :
(A) Like fractions. Fractions having
same denominators are called like
fractions e.g. 
1
7
3
7
5
7
4
7
, , , .
(B) Unlike fractions. Fractions with
different denominators are called unlike
fractions e.g. 
2
3
5
7
4
9
, , .
(C) To convert unlike fractions to like
fractions. We can convert unlike
fractions into like fractions by equalising
their denominator with the help of using
their L.C.M.
4. Fractions in simplest form or in lowest
terms. A fraction is said to be in the
simplest form if the HCF of its numerator
and denominator is 1.
5. Proper, improper and mixed
fractions :
(A) Proper fractions. A fraction whose
numerator is less than its denominator is
called a proper fraction e.g. 
1
2
5
7
4
9
, , etc.
(B) Improper fractions : A fraction
whose numerator is greater than its
denominator, is called improper fraction
e.g. 
5
4
7
3
9
2
, , etc.
(C) Mixed fractions. A combination of
a whole number and a proper fraction is
called a mixed fraction e.g. 1
5
7
2
3
4
5
1
8
, ,
etc.
(D) To convert a mixed fraction into
an improper fraction. We know that a
mixed fraction = A whole number + a
fraction
e.g. 
1
5
7
1
5
7
1 7 5
7
 
7 5
7
12
7
.
Method. Multiply the whole number with
the denominator of the fraction and add
the numerator of the fraction, then the
new numerator is the numerator of the
improper fraction with same
denominator.
(E) To convert an improper fraction
with a mixed fraction. On dividing the
numerator by denominator, we get
quotient i.e. whole number. Then, whole
number plus 
Remainder
Denominator
 is the required
mixed fraction e.g. 
15
4
3
3
4
3
3
4
.
6. Comparison of fractions :
(A) Comparing fractions with same
denominator.
Rule. Among two fractions with the
same denominator, the greater
numerator is greater fraction.
(B) Comparing fractions with same
numerator.
Rule. Among two fraction with same
numerator the one with smaller
denominator is the greater fraction.
(C) General method for comparison :
(i) By means of cross multiplication.
(ii) By converting the given fractions into
like fractions.
7. Addition of fractions :
(A) Addition of like fractions
Sum of their numerators
Denominator
(B) Addition of unlike fractions.
Change the given fractions into equivalent
like fractions and then add them as given
in (A).
8. Subtraction of fractions. We use similar
methods as in addition for subtraction of
fractions.
Exercise 5A
Q.1. Write the fraction representing the shaded
portion :
(i) (ii)
(iii) (iv)
(v) (vi)
Sol. (i)
4
3
(ii)
4
1
(iii)
3
2
(iv)
10
3
(v)
9
4
(vi)
8
3
Q.2. Shade 
9
4
 of the given figure.
Sol. In the figure, 
9
4
 is shaded
3. In the given figure, if we say that the
shaded region is 
4
1
, then identify the error
in it.
Sol. In the figure, whole rectangle is not
divided into four equal parts.
Q. 4. Write a fraction for each of the following :
(i) three-fourths (ii) four-sevenths
(iii) two-fifths (iv) three-tenths
(v) one-eighth (vi) five-sixths
(vii) eight-ninths (viii) seven-twelfths
Sol. (i) Three-fourths 
3
4
.
(ii) Four-sevenths 
4
7
.
(iii) Two-fifths 
2
5
.
(iv) Three-tenths 
3
10
.
(v) One-eighth 
1
8
.
(vi) Five-sixths 
5
6
.
(vii) Eight-ninths 
8
9
.
(viii) Seven-twelfths 
7
12
.    Ans.
Q. 5. Write down the numerator and denomi-
nator in each of the following fractions :
(i)
4
9
(ii)
11
6
(iii)
15
8
(iv)
12
17
(v)
1
5
Sol. (i) In 
4
9
, numerator is 4 and denominator
is 9.
(ii) In 
11
6
, numerator is 6 and denominator
is 11.
(iii) In 
15
8
, numerator is 8 and denominator
is 15.
(iv) In 
12
17
, numerator is 12 and denominator
is 17.
(v)
1
5
, numerator is 5 and denominator is 1.
Q. 6. Write down the fraction in which
(i) numerator = 3, denominator = 8
(ii) numerator = 5, denominator = 12
(iii) numerator = 7, denominator = 16
(iv) numerator = 8, denominator = 15
Sol. (i) Numerator = 3, Denominator = 8, then
fraction 
3
8
.
(ii) Numerator = 5, Denominator = 12, then
fraction = 
12
5
.
(iii) Numerator = 7, Denominator = 16, then
fraction 
7
16
.
(iv) Numerator = 8, Denominator = 15, then
fraction 
8
15
.   Ans.
Q.7. Write down the fractional number for
each of the following :
(i)
3
2
(ii)
9
4
(iii)
5
2
(iv)
10
7
(v)
3
1
(vi)
4
3
(vii)
8
3
(viii)
14
9
(ix)
11
5
(x)
15
6
Sol. (i)
3
2
 = two-thirds (ii)
9
4
 = four-ninths
(iii)
5
2
 = two-fifths
(iv)
10
7
 = seven-tenths
(v)
3
1
 = one-thirds
(vi)
4
3
 = three-fourth
(vii)
8
3
 = three-eighths
(viii)
14
9
 = nine-fourteenths
(ix)
11
5
 = five-elevanths
(x)
15
6
 = six-fifteenths
Q.8. What fraction of an hour is 24 minutes ?
Sol. 24 minutes is the fraction of 1 hour i.e.,
60 minutes = 
60
24
Q.9. How many natural numbers are there from
2 to 10 ? What fraction of them are prime
numbers ?
Sol. Natural number between 2 to 10 are
2, 3, 4, 5, 6, 7, 8, 9, 10 = 9
Out of these prime number are 2, 3, 5, 7 = 4
Fraction = 
9
4
Q. 10. Determine 
2
3
 of a collection of
(i)
3
2
 of 15 pens (ii) 
3
2
 of 27 balls
(iii)
3
2
 of 36 balloons
Sol. (i) 
2
3
 of 15 pens 
2
3
15= 2 × 5
= 10 pens.
(ii)
2
3
 of 27 balls 
2
3
27 2 9 18 balls .
(iii)
2
3
 of 36 balloons 
2
3
36 = 2 × 12
= 24 balloons.   Ans.
Q. 11. Determine 
3
4
 of a collection of
(i) 16 cups (ii) 28 rackets
(iii) 32 books
Sol. (i) 
3
4
 of 16 cups 
3
4
16 = 3 × 4
= 12 cups.
(ii)
3
4
 of 28 rackets 
3
4
28 = 3 × 7
= 21 rackets.
(iii)
3
4
 of 32 books 
3
4
32 = 3 × 8
           = 24 books.   Ans.
Page 5


Points to Remember :
1. Fraction. The number of the forms 
a
b
where p and q are integers and q  0, is
called a fraction. Here p is called the num-
erator and ‘q’ is called the denominator.
Fraction can also be represent on a
number line. e.g. 
2
3
5
7
11
19
, ,
 are called
fractions.
2
3
 means two parts from 3 and is read as
two-third.
5
7
 means, five parts from 7 and is read
five-seventh.
11
19
 means 11 parts from 19 and is read
as eleven-nineteenth.
2. (A) Equivalent fractions. Two or more
fractions representing the same part of a
whole are called equivalent fractions.
(B) Rule to form equivalent fractions.
To get a fractions equivalent to a given
fractions, we multiply or divide the
numerator and denominator of the given
fraction by the same non-zero number.
(C) To test of two equivalent fractions.
Let 
a
b
 and 
c
d
 are two equivalent fractions,
these    
a
b
c
d
   ad = bc.
In other words, we can say that
if ad = bc then, fractions 
a
b
 and 
c
d
 are
equal.
3. Like and unlike fractions :
(A) Like fractions. Fractions having
same denominators are called like
fractions e.g. 
1
7
3
7
5
7
4
7
, , , .
(B) Unlike fractions. Fractions with
different denominators are called unlike
fractions e.g. 
2
3
5
7
4
9
, , .
(C) To convert unlike fractions to like
fractions. We can convert unlike
fractions into like fractions by equalising
their denominator with the help of using
their L.C.M.
4. Fractions in simplest form or in lowest
terms. A fraction is said to be in the
simplest form if the HCF of its numerator
and denominator is 1.
5. Proper, improper and mixed
fractions :
(A) Proper fractions. A fraction whose
numerator is less than its denominator is
called a proper fraction e.g. 
1
2
5
7
4
9
, , etc.
(B) Improper fractions : A fraction
whose numerator is greater than its
denominator, is called improper fraction
e.g. 
5
4
7
3
9
2
, , etc.
(C) Mixed fractions. A combination of
a whole number and a proper fraction is
called a mixed fraction e.g. 1
5
7
2
3
4
5
1
8
, ,
etc.
(D) To convert a mixed fraction into
an improper fraction. We know that a
mixed fraction = A whole number + a
fraction
e.g. 
1
5
7
1
5
7
1 7 5
7
 
7 5
7
12
7
.
Method. Multiply the whole number with
the denominator of the fraction and add
the numerator of the fraction, then the
new numerator is the numerator of the
improper fraction with same
denominator.
(E) To convert an improper fraction
with a mixed fraction. On dividing the
numerator by denominator, we get
quotient i.e. whole number. Then, whole
number plus 
Remainder
Denominator
 is the required
mixed fraction e.g. 
15
4
3
3
4
3
3
4
.
6. Comparison of fractions :
(A) Comparing fractions with same
denominator.
Rule. Among two fractions with the
same denominator, the greater
numerator is greater fraction.
(B) Comparing fractions with same
numerator.
Rule. Among two fraction with same
numerator the one with smaller
denominator is the greater fraction.
(C) General method for comparison :
(i) By means of cross multiplication.
(ii) By converting the given fractions into
like fractions.
7. Addition of fractions :
(A) Addition of like fractions
Sum of their numerators
Denominator
(B) Addition of unlike fractions.
Change the given fractions into equivalent
like fractions and then add them as given
in (A).
8. Subtraction of fractions. We use similar
methods as in addition for subtraction of
fractions.
Exercise 5A
Q.1. Write the fraction representing the shaded
portion :
(i) (ii)
(iii) (iv)
(v) (vi)
Sol. (i)
4
3
(ii)
4
1
(iii)
3
2
(iv)
10
3
(v)
9
4
(vi)
8
3
Q.2. Shade 
9
4
 of the given figure.
Sol. In the figure, 
9
4
 is shaded
3. In the given figure, if we say that the
shaded region is 
4
1
, then identify the error
in it.
Sol. In the figure, whole rectangle is not
divided into four equal parts.
Q. 4. Write a fraction for each of the following :
(i) three-fourths (ii) four-sevenths
(iii) two-fifths (iv) three-tenths
(v) one-eighth (vi) five-sixths
(vii) eight-ninths (viii) seven-twelfths
Sol. (i) Three-fourths 
3
4
.
(ii) Four-sevenths 
4
7
.
(iii) Two-fifths 
2
5
.
(iv) Three-tenths 
3
10
.
(v) One-eighth 
1
8
.
(vi) Five-sixths 
5
6
.
(vii) Eight-ninths 
8
9
.
(viii) Seven-twelfths 
7
12
.    Ans.
Q. 5. Write down the numerator and denomi-
nator in each of the following fractions :
(i)
4
9
(ii)
11
6
(iii)
15
8
(iv)
12
17
(v)
1
5
Sol. (i) In 
4
9
, numerator is 4 and denominator
is 9.
(ii) In 
11
6
, numerator is 6 and denominator
is 11.
(iii) In 
15
8
, numerator is 8 and denominator
is 15.
(iv) In 
12
17
, numerator is 12 and denominator
is 17.
(v)
1
5
, numerator is 5 and denominator is 1.
Q. 6. Write down the fraction in which
(i) numerator = 3, denominator = 8
(ii) numerator = 5, denominator = 12
(iii) numerator = 7, denominator = 16
(iv) numerator = 8, denominator = 15
Sol. (i) Numerator = 3, Denominator = 8, then
fraction 
3
8
.
(ii) Numerator = 5, Denominator = 12, then
fraction = 
12
5
.
(iii) Numerator = 7, Denominator = 16, then
fraction 
7
16
.
(iv) Numerator = 8, Denominator = 15, then
fraction 
8
15
.   Ans.
Q.7. Write down the fractional number for
each of the following :
(i)
3
2
(ii)
9
4
(iii)
5
2
(iv)
10
7
(v)
3
1
(vi)
4
3
(vii)
8
3
(viii)
14
9
(ix)
11
5
(x)
15
6
Sol. (i)
3
2
 = two-thirds (ii)
9
4
 = four-ninths
(iii)
5
2
 = two-fifths
(iv)
10
7
 = seven-tenths
(v)
3
1
 = one-thirds
(vi)
4
3
 = three-fourth
(vii)
8
3
 = three-eighths
(viii)
14
9
 = nine-fourteenths
(ix)
11
5
 = five-elevanths
(x)
15
6
 = six-fifteenths
Q.8. What fraction of an hour is 24 minutes ?
Sol. 24 minutes is the fraction of 1 hour i.e.,
60 minutes = 
60
24
Q.9. How many natural numbers are there from
2 to 10 ? What fraction of them are prime
numbers ?
Sol. Natural number between 2 to 10 are
2, 3, 4, 5, 6, 7, 8, 9, 10 = 9
Out of these prime number are 2, 3, 5, 7 = 4
Fraction = 
9
4
Q. 10. Determine 
2
3
 of a collection of
(i)
3
2
 of 15 pens (ii) 
3
2
 of 27 balls
(iii)
3
2
 of 36 balloons
Sol. (i) 
2
3
 of 15 pens 
2
3
15= 2 × 5
= 10 pens.
(ii)
2
3
 of 27 balls 
2
3
27 2 9 18 balls .
(iii)
2
3
 of 36 balloons 
2
3
36 = 2 × 12
= 24 balloons.   Ans.
Q. 11. Determine 
3
4
 of a collection of
(i) 16 cups (ii) 28 rackets
(iii) 32 books
Sol. (i) 
3
4
 of 16 cups 
3
4
16 = 3 × 4
= 12 cups.
(ii)
3
4
 of 28 rackets 
3
4
28 = 3 × 7
= 21 rackets.
(iii)
3
4
 of 32 books 
3
4
32 = 3 × 8
           = 24 books.   Ans.
Q.12. Neelam has 25 pencils. She gives 
5
4
of them to Meena. How many pencils
does Meena get ? How many pencils
are left with Neelam ?
Sol. Total number of pencils Neelam has = 25
No. of pencils given to Meena
= 
5
4
 of 25
= 
5
4
 × 25 = 20
No. of pencils left with Neelam
= 25 – 20 = 5
Q. 13. Represent each of the following fractions
on a number line :
(i)
3
8
(ii)
5
9
(iii)
4
7
(iv)
2
5
(v)
4
1
Sol. (i) 
3
8
Take a line segment OA = one unit of
length
Divide it into 8 equal parts and take 3
parts at P, then P represents 
3
8
.
(ii)
5
9
(a) Take a line segment OA = one unit of
length.
(b) Divide it into nine equal parts and take 5
parts at P, then P represents 
5
9
.
(iii)
4
7
(a) Take a line segment OA = one unit of
length.
(b) Divide it into 7 equal parts and take 4
parts at P then P represents 
4
7
.
(iv)
2
5
(a) Take a line segment OA = 1 unit of length.
(b) Divide it with 5 equal parts and take 2
parts and P then P represents 
2
5
.
(v)
4
1
(a) Take a line segment OA = 1 unit of length.
(b) Divide it with 4 equal parts and take 1
parts and P then P represents 
4
1
.
0 1 2 3 4
1
4
P
Exercise 5B
Q. 1. Which of the following are proper
fractions ?
2
1
, 
5
3
, 
10
6
, 
4
7
, 2, 
8
15
, 
16
16
, 
11
10
, 
10
23
Sol. We know that, a fraction is proper if its
denominator is greater than its numerator.
Therefore,
2
1
, 
5
3
 and 
10
11
 are proper fractions.  Ans.
Q. 2. Which of the following are improper