WE DISTRIBUTE,
6 YET THINGS
MULTIPLY
We have seen how algebra makes use of letter symbols to write general
statements about patterns and relations in a compact manner. Algebra
can also be used to justify or prove claims and conjectures (like the many
properties you saw in the previous chapter) and to solve problems of
various kinds.
Distributivity is a property relating multiplication and addition that
is captured concisely using algebra. In this chapter, we explore different
types of multiplication patterns and show how they can be described in
the language of algebra by making use of distributivity.
6.1 Some Properties of Multiplication
Increments in Products
Consider the multiplication of two numbers, say, 23 × 27.
1. By how much does the product increase if the first number (23) is
increased by 1?
2. What if the second number (27) is increased by 1?
3. How about when both numbers are increased by 1?
Do you see a pattern that could help generalise our observations to the
product of any two numbers?
Let us first consider a simpler problem — find the increase in the
product when 27 is increased by 1. From the definition of multiplication
(and the commutative property), it is clear that the product increases by
23. This can be seen from the distributive property of multiplication as
well. If a, b and c are three numbers, then —
Reprint 2026-27
, We Distribute, Yet Things Multiply
a (b + c) = ab + ac
This property can be visualised nicely using a diagram:
b columns c columns
..... ......
..... ......
..... ......
. . . . . . . .
a rows . . . . . . . .
. . . ab
. . . . ac .
. . . . . . . .
..... ......
..... ......
a (b + c)
This is called the distributive property of multiplication over
addition. Using the identity a (b + c) = ab + ac with a = 23, b = 27, and
c = 1, we have
23 ( 27 + 1) = 23 × 27 + 23
Increase
Remember that here, a (b + c) and 23 (27 + 1) mean a × (b + c), and
23 × (27 + 1), respectively. We usually skip writing the ‘×’ symbol before
or after brackets, just as in the case of expressions like 5a, xy, etc.
We can also similarly expand (a + b) c using the distributive property
as follows —
(a + b) c = c (a + b) (commutativity of multiplication)
= ca + cb (distributivity)
= ac + bc (commutativity of multiplication)
We can use the distributive property to find, in general, how much a
product increases if one or both the numbers in the product are increased
by 1. Suppose the initial two numbers are a and b. If one of the numbers,
say b, is increased by 1, then we have —
a ( b + 1) = ab × a
Increase
Now let us see what happens if both numbers in a product are
increased by 1. If in a product ab, both a and b are increased by 1, then
we obtain (a + 1) (b + 1).
137
Reprint 2026-27
, Ganita Prakash | Grade 8
How do we expand this?
Let us consider (a + 1) as a single term. Then, by the distributive
property, we have
(a + 1) (b + 1) = (a + 1) b + (a + 1) 1
If a = 23, and, b = 27, we get
Again applying the distributive
property, we obtain
(a + 1) (b + 1) = (a + 1) b + (a + 1) 1 (23 + 1) (27 + 1) = (23 + 1) 27 + (23 + 1) 1
= ab + (b + a + 1) = 23 × 27 + (27 + 23 + 1)
Increase Increase
Thus, the product ab increases by a + b + 1 when each of a and b are
increased by 1.
What would we get if we had expanded (a + 1) (b + 1) by first taking (b + 1)
as a single term? Try it?
What happens when one of the numbers in a product is increased by 1
and the other is decreased by 1? Will there be any change in the product?
Let us again take the product ab of two numbers a and b. If a is
increased by 1 and b is decreased by 1, then their product will be (a + 1)
(b – 1). Expanding this, we get
If a = 23, and b = 27, we get
(a + 1) (b – 1) = (a + 1) b – (a + 1) 1 (23 + 1) (27 – 1) = (23 + 1) 27 – (23 + 1) 1
= ab + b – (a + 1) = 23 × 27 + 27 – (23 + 1)
= ab + b – a – 1 = 23 × 27 + 27 – 23 – 1
Increase Increase
Will the product always increase? Find 3 examples where the product
decreases.
What happens when a and b are negative integers?
Check by substituting different values for a and b in each of the above
cases. For example, a = –5, b = 8; a = –4, b = –5; etc.
We have seen that integers also satisfy the distributive property, that
is, if x, y and z are any three integers, then x (y + z) = xy + xz.
Thus, the expressions we have for increase of products hold when the
letter-numbers take on negative integer values as well.
Recall that two algebraic expressions are equal if they take on the
same values when their letter-numbers are replaced by numbers. These
138
Reprint 2026-27
, We Distribute, Yet Things Multiply
numbers could be any integers. Mathematical statements that express
the equality of two algebraic expressions, such as
a (b + 8) = ab + 8a,
(a + 1) (b – 1) = ab + b – a – 1, etc.,
are called identities.
By how much will the product of two numbers change if one of the
numbers is increased by m and the other by n?
If a and b are the initial numbers being multiplied, they become
a + m and b + n.
(a + m) (b + n) = (a + m)b + (a + m)n
= ab + mb + an + mn
The increase is an + bm + mn.
Notice that the product is the sum of the product of each term of
(a + m) with each term of (b + n).
Identity 1 (a + m) (b + n) = ab + mb + an + mn
This identity can be visualised as follows —
b columns n columns
..... ......
..... ......
..... ......
a rows . . . . . . . .
. . . . . . . .
. . . ab
. . . . an .
. . . . . . . .
..... ......
..... ......
....... .......
....... .......
. . . . . . .
m rows . . mb . . . . mn .
. . . . . . .
...... .......
(a + m) (b + n)
139
Reprint 2026-27
6 YET THINGS
MULTIPLY
We have seen how algebra makes use of letter symbols to write general
statements about patterns and relations in a compact manner. Algebra
can also be used to justify or prove claims and conjectures (like the many
properties you saw in the previous chapter) and to solve problems of
various kinds.
Distributivity is a property relating multiplication and addition that
is captured concisely using algebra. In this chapter, we explore different
types of multiplication patterns and show how they can be described in
the language of algebra by making use of distributivity.
6.1 Some Properties of Multiplication
Increments in Products
Consider the multiplication of two numbers, say, 23 × 27.
1. By how much does the product increase if the first number (23) is
increased by 1?
2. What if the second number (27) is increased by 1?
3. How about when both numbers are increased by 1?
Do you see a pattern that could help generalise our observations to the
product of any two numbers?
Let us first consider a simpler problem — find the increase in the
product when 27 is increased by 1. From the definition of multiplication
(and the commutative property), it is clear that the product increases by
23. This can be seen from the distributive property of multiplication as
well. If a, b and c are three numbers, then —
Reprint 2026-27
, We Distribute, Yet Things Multiply
a (b + c) = ab + ac
This property can be visualised nicely using a diagram:
b columns c columns
..... ......
..... ......
..... ......
. . . . . . . .
a rows . . . . . . . .
. . . ab
. . . . ac .
. . . . . . . .
..... ......
..... ......
a (b + c)
This is called the distributive property of multiplication over
addition. Using the identity a (b + c) = ab + ac with a = 23, b = 27, and
c = 1, we have
23 ( 27 + 1) = 23 × 27 + 23
Increase
Remember that here, a (b + c) and 23 (27 + 1) mean a × (b + c), and
23 × (27 + 1), respectively. We usually skip writing the ‘×’ symbol before
or after brackets, just as in the case of expressions like 5a, xy, etc.
We can also similarly expand (a + b) c using the distributive property
as follows —
(a + b) c = c (a + b) (commutativity of multiplication)
= ca + cb (distributivity)
= ac + bc (commutativity of multiplication)
We can use the distributive property to find, in general, how much a
product increases if one or both the numbers in the product are increased
by 1. Suppose the initial two numbers are a and b. If one of the numbers,
say b, is increased by 1, then we have —
a ( b + 1) = ab × a
Increase
Now let us see what happens if both numbers in a product are
increased by 1. If in a product ab, both a and b are increased by 1, then
we obtain (a + 1) (b + 1).
137
Reprint 2026-27
, Ganita Prakash | Grade 8
How do we expand this?
Let us consider (a + 1) as a single term. Then, by the distributive
property, we have
(a + 1) (b + 1) = (a + 1) b + (a + 1) 1
If a = 23, and, b = 27, we get
Again applying the distributive
property, we obtain
(a + 1) (b + 1) = (a + 1) b + (a + 1) 1 (23 + 1) (27 + 1) = (23 + 1) 27 + (23 + 1) 1
= ab + (b + a + 1) = 23 × 27 + (27 + 23 + 1)
Increase Increase
Thus, the product ab increases by a + b + 1 when each of a and b are
increased by 1.
What would we get if we had expanded (a + 1) (b + 1) by first taking (b + 1)
as a single term? Try it?
What happens when one of the numbers in a product is increased by 1
and the other is decreased by 1? Will there be any change in the product?
Let us again take the product ab of two numbers a and b. If a is
increased by 1 and b is decreased by 1, then their product will be (a + 1)
(b – 1). Expanding this, we get
If a = 23, and b = 27, we get
(a + 1) (b – 1) = (a + 1) b – (a + 1) 1 (23 + 1) (27 – 1) = (23 + 1) 27 – (23 + 1) 1
= ab + b – (a + 1) = 23 × 27 + 27 – (23 + 1)
= ab + b – a – 1 = 23 × 27 + 27 – 23 – 1
Increase Increase
Will the product always increase? Find 3 examples where the product
decreases.
What happens when a and b are negative integers?
Check by substituting different values for a and b in each of the above
cases. For example, a = –5, b = 8; a = –4, b = –5; etc.
We have seen that integers also satisfy the distributive property, that
is, if x, y and z are any three integers, then x (y + z) = xy + xz.
Thus, the expressions we have for increase of products hold when the
letter-numbers take on negative integer values as well.
Recall that two algebraic expressions are equal if they take on the
same values when their letter-numbers are replaced by numbers. These
138
Reprint 2026-27
, We Distribute, Yet Things Multiply
numbers could be any integers. Mathematical statements that express
the equality of two algebraic expressions, such as
a (b + 8) = ab + 8a,
(a + 1) (b – 1) = ab + b – a – 1, etc.,
are called identities.
By how much will the product of two numbers change if one of the
numbers is increased by m and the other by n?
If a and b are the initial numbers being multiplied, they become
a + m and b + n.
(a + m) (b + n) = (a + m)b + (a + m)n
= ab + mb + an + mn
The increase is an + bm + mn.
Notice that the product is the sum of the product of each term of
(a + m) with each term of (b + n).
Identity 1 (a + m) (b + n) = ab + mb + an + mn
This identity can be visualised as follows —
b columns n columns
..... ......
..... ......
..... ......
a rows . . . . . . . .
. . . . . . . .
. . . ab
. . . . an .
. . . . . . . .
..... ......
..... ......
....... .......
....... .......
. . . . . . .
m rows . . mb . . . . mn .
. . . . . . .
...... .......
(a + m) (b + n)
139
Reprint 2026-27
