NCERT Class 8 Mathematics Ganita Prakash Part 1 Chapter 06 We Distribute Yet Things Multiply PDF Download

Chapter 6: We Distribute, 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, \( 23 \times 27 \).

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?
4. 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 -

\[ a (b + c) = ab + ac \]

This property can be visualised nicely using a diagram:

[Figure: A rectangular array divided into two sections with \( b \) columns and \( c \) columns, containing \( a \) rows, showing \( ab \) and \( ac \) regions, See in your textbook]

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

\[ 23 (27 + 1) = 23 \times 27 + 23 \]

Remember that here, \( a (b + c) \) and \( 23 (27 + 1) \) mean \( a \times (b + c) \), and \( 23 \times (27 + 1) \), respectively. We usually skip writing the '\( \times \)' 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)

Teacher's Note

When you apply the distributive property, check that you multiply the single number by every term inside the brackets. A common mistake is to multiply only the first term. For example, in \( 5(x + 3) \), you need both \( 5x + 15 \), not just \( 5x + 3 \). Try it with numbers: \( 5(2 + 3) = 5 \times 5 = 25 \), and \( 5 \times 2 + 5 \times 3 = 10 + 15 = 25 \).

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 -

\[ a (b + 1) = ab + a \]

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 \( (a + 1) (b + 1) \).

How do we expand this?

Let us consider \( (a + 1) \) as a single term. Then, by the distributive property, we have

\[ (a + 1) (b + 1) = (a + 1) b + (a + 1) 1 \]

Again applying the distributive property, we obtain

\[ (a + 1) (b + 1) = (a + 1) b + (a + 1) 1 = ab + (b + a + 1) \]

If \( a = 23 \), and \( b = 27 \), we get

\[ (23 + 1) (27 + 1) = (23 + 1) 27 + (23 + 1) 1 = 23 \times 27 + (27 + 23 + 1) \]

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

\[ (a + 1) (b - 1) = (a + 1) b - (a + 1) 1 = ab + b - (a + 1) = ab + b - a - 1 \]

If \( a = 23 \), and \( b = 27 \), we get

\[ (23 + 1) (27 - 1) = (23 + 1) 27 - (23 + 1) 1 = 23 \times 27 + 27 - (23 + 1) = 23 \times 27 + 27 - 23 - 1 \]

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.

Teacher's Note

The distributive property works the same way whether your numbers are positive, negative, or zero. When you expand \( (a + 1)(b + 1) \), you always get \( ab + a + b + 1 \), no matter what values \( a \) and \( b \) have. Try \( a = -2 \) and \( b = 3 \): you get \( (-2 + 1)(3 + 1) = (-1)(4) = -4 \), and also \( (-2)(3) + (-2) + 3 + 1 = -6 - 2 + 3 + 1 = -4 \). Both give the same answer.

Recall that two algebraic expressions are equal if they take on the same values when their letter-numbers are replaced by numbers. These

Key Points

• The distributive property of multiplication over addition states that \( a(b + c) = ab + ac \), which lets you break down complex multiplications into simpler parts.
• When you increase one number in a product by 1, the product increases by the other number; when you increase both by 1, the product increases by \( a + b + 1 \).
• The distributive property holds for all integers, including negative numbers, so algebraic identities work no matter what values you substitute.
• You can visualise the distributive property using rectangular arrays, where multiplying out a sum corresponds to splitting a rectangle into sections.

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