Important Formulas & Points to Remember: Working with Fractions | Mathematics Olympiad Class 7 PDF Download

 Page 1


1 M u l t i p l i c a t i o n o f F r a c t i o n s
• Multiplying a Whole Number b y a Fr action :
– T o multiply a whole number b y a fr action, divide the whole number b y the denomina-
tor of the fr action and multiply b y the numer ator .
– Example: 3×
1
4
=
3
4
, since3÷4=
3
4
.
– Example:
2
5
×3=
2×3
5
=
6
5
=1
1
5
.
• Multiplying Two Fr actions :
– Br ahmagupta’ s formula (628 CE):
a
b
×
c
d
=
a×c
b×d
.
– Example:
3
4
×
2
5
=
3×2
4×5
=
6
20
=
3
10
.
– Example:
5
4
×
7
18
=
5×7
4×18
=
35
72
.
• Simplifying Before Multiplication :
– Cancel common f actors between numer ators and denominators before multiplying to
simplify the result.
– Example:
12
7
×
5
24
=
12×5
7×24
=
1×5
7×2
=
5
14
.
– Example:
15
14
×
2
45
=
15×2
14×45
=
1×2
14×3
=
2
3×14
=
2
3×3
=
2
9
.
• Geometric Interpretation :
– The product of two fr actions represents the area of a rectangle with sides equal to the
fr actions, using a unit square as the whole.
– Example:
1
2
×
1
4
=
1
8
, as a rectangle with sides
1
2
and
1
4
has area
1
8
of the unit square.
• Relationship Between Product and Numbers Mul tiplied :
– If both numbers are greater than 1, the product is greater than both: e.g.,
4
3
×4=
16
3
>
4,
4
3
.
– If both numbers are between 0 and 1, the product is less than both: e.g.,
3
4
×
2
5
=
3
10
<
3
4
,
2
5
.
– If one number is between 0 and 1 and the other is greater than 1, the product is less
than the larger number and greater than the smaller: e.g.,
3
4
×5=
15
4
, where
3
4
<
15
4
< 5 .
• Order of Multiplication :
– The order of multiplication does not affect the result:
a
b
×
c
d
=
c
d
×
a
b
.
– Example:
1
2
×
1
4
=
1
4
×
1
2
=
1
8
.
2 D i v i s i o n o f F r a c t i o n s
• Division as Multiplication b y Reciprocal :
– Br ahmagupta’ s formula (628 CE):
a
b
÷
c
d
=
a
b
×
d
c
=
a×d
b×c
.
– The reciprocal of a fr action
c
d
is
d
c
, where
c
d
×
d
c
=1 .
– Example:
2
3
÷
3
5
=
2
3
×
5
3
=
2×5
3×3
=
10
9
=1
1
9
.
– Example: 3÷
2
3
=3×
3
2
=
9
2
=4
1
2
.
• Relationship Between Quotient and Dividend/Divis or :
– If the divisor is between 0 and 1, the quotient is greater than the dividend: e.g.,6÷
1
4
=
6×4= 24> 6 .
– If the divisor is greater than 1, the quotient is less than the dividend: e.g.,6÷3= 2< 6 .
– The quotient is the reciprocal of the divisor multiplied b y the dividend.
1
Page 2


1 M u l t i p l i c a t i o n o f F r a c t i o n s
• Multiplying a Whole Number b y a Fr action :
– T o multiply a whole number b y a fr action, divide the whole number b y the denomina-
tor of the fr action and multiply b y the numer ator .
– Example: 3×
1
4
=
3
4
, since3÷4=
3
4
.
– Example:
2
5
×3=
2×3
5
=
6
5
=1
1
5
.
• Multiplying Two Fr actions :
– Br ahmagupta’ s formula (628 CE):
a
b
×
c
d
=
a×c
b×d
.
– Example:
3
4
×
2
5
=
3×2
4×5
=
6
20
=
3
10
.
– Example:
5
4
×
7
18
=
5×7
4×18
=
35
72
.
• Simplifying Before Multiplication :
– Cancel common f actors between numer ators and denominators before multiplying to
simplify the result.
– Example:
12
7
×
5
24
=
12×5
7×24
=
1×5
7×2
=
5
14
.
– Example:
15
14
×
2
45
=
15×2
14×45
=
1×2
14×3
=
2
3×14
=
2
3×3
=
2
9
.
• Geometric Interpretation :
– The product of two fr actions represents the area of a rectangle with sides equal to the
fr actions, using a unit square as the whole.
– Example:
1
2
×
1
4
=
1
8
, as a rectangle with sides
1
2
and
1
4
has area
1
8
of the unit square.
• Relationship Between Product and Numbers Mul tiplied :
– If both numbers are greater than 1, the product is greater than both: e.g.,
4
3
×4=
16
3
>
4,
4
3
.
– If both numbers are between 0 and 1, the product is less than both: e.g.,
3
4
×
2
5
=
3
10
<
3
4
,
2
5
.
– If one number is between 0 and 1 and the other is greater than 1, the product is less
than the larger number and greater than the smaller: e.g.,
3
4
×5=
15
4
, where
3
4
<
15
4
< 5 .
• Order of Multiplication :
– The order of multiplication does not affect the result:
a
b
×
c
d
=
c
d
×
a
b
.
– Example:
1
2
×
1
4
=
1
4
×
1
2
=
1
8
.
2 D i v i s i o n o f F r a c t i o n s
• Division as Multiplication b y Reciprocal :
– Br ahmagupta’ s formula (628 CE):
a
b
÷
c
d
=
a
b
×
d
c
=
a×d
b×c
.
– The reciprocal of a fr action
c
d
is
d
c
, where
c
d
×
d
c
=1 .
– Example:
2
3
÷
3
5
=
2
3
×
5
3
=
2×5
3×3
=
10
9
=1
1
9
.
– Example: 3÷
2
3
=3×
3
2
=
9
2
=4
1
2
.
• Relationship Between Quotient and Dividend/Divis or :
– If the divisor is between 0 and 1, the quotient is greater than the dividend: e.g.,6÷
1
4
=
6×4= 24> 6 .
– If the divisor is greater than 1, the quotient is less than the dividend: e.g.,6÷3= 2< 6 .
– The quotient is the reciprocal of the divisor multiplied b y the dividend.
1
3 S o m e P r o b l e m s I n v o l v i n g F r a c t i o n s
• Sharing and Dividing Quantities :
– Example: F or 5 cups of tea using
1
4
litre of milk, milk per cup =
1
4
÷5=
1
4
×
1
5
=
1
20
litre.
– Example: T o cover 7
1
2
=
15
2
square units with bricks of area
1
25
, number of bricks =
15
2
÷
1
25
=
15
2
×25=
375
2
=187
1
2
.
• Fr actional Rates :
– Example: F our fountains fill a cistern in 1,
1
2
,
1
4
, and
1
5
da ys. Rates per da y: 1,2,4,5 .
T otal r ate =1+2+4+5 =12 . Time to fill =
1
12
da ys.
• Area of Shaded Regions :
– Example: F or a square with area 1, a smaller square is
1
4
, a triangle within it is
1
2
×
1
4
=
1
8
,
and the shaded region is
3
4
×
1
8
=
3
32
.
• Historical Problems :
– Bhaskar acharya’ s problem: A miser gives
1
2
×
2
3
×
3
4
×
1
5
×
1
16
×
1
4
=
6
7680
=
1
1280
dr amma
= 1 cowrie shell (1 dr amma = 1280 cowrie shells).
• Coin Conversions :
– Given: 1 gold dinar = 12 silver dr ammas, 1 dr amma = 4 copper panas, 1 pana = 30
cowrie shells.
– 1 copper pana =
1
12
×
1
4
=
1
48
gold dinar .
– 1 cowrie shell =
1
30
copper pana =
1
30
×
1
48
=
1
1440
gold dinar .
K ey P oints to Remember
• Multiplication of Fr actions :
– Use Br ahmagupta’ s formula:
a
b
×
c
d
=
a×c
b×d
.
– Simplify b y canceling common factors before multiplying.
– The product’ s magnitude depends on whether the fr actions are greater than 1 or be-
tween 0 and 1.
– Multiplication is commutative: order does not affect the result.
• Division of Fr actions :
– Use Br ahmagupta’ s formula:
a
b
÷
c
d
=
a
b
×
d
c
=
a×d
b×c
.
– The reciprocal of
c
d
is
d
c
.
– The quotient is greater than the dividend if the divisor is between 0 and 1, and less if
the divisor is greater than 1.
• Pr actical Applications :
– Fr actions are used in sharing quantities (e.g., dividing milk among cups), calculating
areas, and solving r ate problems (e.g., fountains filling a cistern).
– Historical context: Indian mathematicians lik e Br ahmagupta (628 CE) and Bhaskar a II
(1150 CE) formalized fr action arithmetic, influencing global mathematics.
• Geometric Interpretation :
– Multiplication of fr actions can be visualized as the area of a rectangle with fr actional
sides.
– Division can be interpreted as finding how man y times one fr action fits into another .
• Historical Significance :
2
Page 3


1 M u l t i p l i c a t i o n o f F r a c t i o n s
• Multiplying a Whole Number b y a Fr action :
– T o multiply a whole number b y a fr action, divide the whole number b y the denomina-
tor of the fr action and multiply b y the numer ator .
– Example: 3×
1
4
=
3
4
, since3÷4=
3
4
.
– Example:
2
5
×3=
2×3
5
=
6
5
=1
1
5
.
• Multiplying Two Fr actions :
– Br ahmagupta’ s formula (628 CE):
a
b
×
c
d
=
a×c
b×d
.
– Example:
3
4
×
2
5
=
3×2
4×5
=
6
20
=
3
10
.
– Example:
5
4
×
7
18
=
5×7
4×18
=
35
72
.
• Simplifying Before Multiplication :
– Cancel common f actors between numer ators and denominators before multiplying to
simplify the result.
– Example:
12
7
×
5
24
=
12×5
7×24
=
1×5
7×2
=
5
14
.
– Example:
15
14
×
2
45
=
15×2
14×45
=
1×2
14×3
=
2
3×14
=
2
3×3
=
2
9
.
• Geometric Interpretation :
– The product of two fr actions represents the area of a rectangle with sides equal to the
fr actions, using a unit square as the whole.
– Example:
1
2
×
1
4
=
1
8
, as a rectangle with sides
1
2
and
1
4
has area
1
8
of the unit square.
• Relationship Between Product and Numbers Mul tiplied :
– If both numbers are greater than 1, the product is greater than both: e.g.,
4
3
×4=
16
3
>
4,
4
3
.
– If both numbers are between 0 and 1, the product is less than both: e.g.,
3
4
×
2
5
=
3
10
<
3
4
,
2
5
.
– If one number is between 0 and 1 and the other is greater than 1, the product is less
than the larger number and greater than the smaller: e.g.,
3
4
×5=
15
4
, where
3
4
<
15
4
< 5 .
• Order of Multiplication :
– The order of multiplication does not affect the result:
a
b
×
c
d
=
c
d
×
a
b
.
– Example:
1
2
×
1
4
=
1
4
×
1
2
=
1
8
.
2 D i v i s i o n o f F r a c t i o n s
• Division as Multiplication b y Reciprocal :
– Br ahmagupta’ s formula (628 CE):
a
b
÷
c
d
=
a
b
×
d
c
=
a×d
b×c
.
– The reciprocal of a fr action
c
d
is
d
c
, where
c
d
×
d
c
=1 .
– Example:
2
3
÷
3
5
=
2
3
×
5
3
=
2×5
3×3
=
10
9
=1
1
9
.
– Example: 3÷
2
3
=3×
3
2
=
9
2
=4
1
2
.
• Relationship Between Quotient and Dividend/Divis or :
– If the divisor is between 0 and 1, the quotient is greater than the dividend: e.g.,6÷
1
4
=
6×4= 24> 6 .
– If the divisor is greater than 1, the quotient is less than the dividend: e.g.,6÷3= 2< 6 .
– The quotient is the reciprocal of the divisor multiplied b y the dividend.
1
3 S o m e P r o b l e m s I n v o l v i n g F r a c t i o n s
• Sharing and Dividing Quantities :
– Example: F or 5 cups of tea using
1
4
litre of milk, milk per cup =
1
4
÷5=
1
4
×
1
5
=
1
20
litre.
– Example: T o cover 7
1
2
=
15
2
square units with bricks of area
1
25
, number of bricks =
15
2
÷
1
25
=
15
2
×25=
375
2
=187
1
2
.
• Fr actional Rates :
– Example: F our fountains fill a cistern in 1,
1
2
,
1
4
, and
1
5
da ys. Rates per da y: 1,2,4,5 .
T otal r ate =1+2+4+5 =12 . Time to fill =
1
12
da ys.
• Area of Shaded Regions :
– Example: F or a square with area 1, a smaller square is
1
4
, a triangle within it is
1
2
×
1
4
=
1
8
,
and the shaded region is
3
4
×
1
8
=
3
32
.
• Historical Problems :
– Bhaskar acharya’ s problem: A miser gives
1
2
×
2
3
×
3
4
×
1
5
×
1
16
×
1
4
=
6
7680
=
1
1280
dr amma
= 1 cowrie shell (1 dr amma = 1280 cowrie shells).
• Coin Conversions :
– Given: 1 gold dinar = 12 silver dr ammas, 1 dr amma = 4 copper panas, 1 pana = 30
cowrie shells.
– 1 copper pana =
1
12
×
1
4
=
1
48
gold dinar .
– 1 cowrie shell =
1
30
copper pana =
1
30
×
1
48
=
1
1440
gold dinar .
K ey P oints to Remember
• Multiplication of Fr actions :
– Use Br ahmagupta’ s formula:
a
b
×
c
d
=
a×c
b×d
.
– Simplify b y canceling common factors before multiplying.
– The product’ s magnitude depends on whether the fr actions are greater than 1 or be-
tween 0 and 1.
– Multiplication is commutative: order does not affect the result.
• Division of Fr actions :
– Use Br ahmagupta’ s formula:
a
b
÷
c
d
=
a
b
×
d
c
=
a×d
b×c
.
– The reciprocal of
c
d
is
d
c
.
– The quotient is greater than the dividend if the divisor is between 0 and 1, and less if
the divisor is greater than 1.
• Pr actical Applications :
– Fr actions are used in sharing quantities (e.g., dividing milk among cups), calculating
areas, and solving r ate problems (e.g., fountains filling a cistern).
– Historical context: Indian mathematicians lik e Br ahmagupta (628 CE) and Bhaskar a II
(1150 CE) formalized fr action arithmetic, influencing global mathematics.
• Geometric Interpretation :
– Multiplication of fr actions can be visualized as the area of a rectangle with fr actional
sides.
– Division can be interpreted as finding how man y times one fr action fits into another .
• Historical Significance :
2
– Fr actions were used in a ncient Indian texts lik e the Shulbasutr a (c. 800 BCE) for geom-
etry and rituals.
– The concept of reducing fr actions to lowest terms (apavartana) was noted b y Umasvati
(c. 150 CE).
– Indian fr action arithmetic was tr ansmitted to Ar ab and European mathematicians, be-
coming standard b y the 17th century .
3