The distributive property is one of the most frequently used properties in mathematics. This property defines the relation between multiplication and addition, and is extensively used in arithmetic as well as algebraic simplifications.
The property states that when an integer or an expression is multiplied by a sum, the factor is multiplied to each term separately, and then the products are added.
For any three real numbers \(a\), \(b\) and \(c\)
- \(a(b +c) = ab + ac\)
This property is visually representated using the following diagram:
We can also find the product of \((a + b)c\) using the distributive property as follows:
\((a + b)c\) \(=\) \(c(a + b)\) [By commutative property of multiplication]
\(=\) \(ca + cb\) [By distributive property]
\(=\) \(ac + bc\) [By commutative property of multiplication]
- \((a + b)c\) \(=\) \(ac + bc\)
Increment in product:
The product of any two real numbers is represented by \(a \times b\).
In this chapter we will analyse the increase in product if either \(a\) or \(b\) or both \(a\) and \(b\) are increased by \(1\).
Case (i): In the product \(a \times b\), the second number \(b\) is increased by \(1\).
Let us find the increase in product for \(a (b +1)\).
Expand the product using distributive property \(a(b + c) = ab + ac\).
\(a (b + 1) = ab + \fbox{a}\)
Observe that the product \(ab\) is increased by \(a\).
Example:
Consider the product of \(a = 21\) and \(b = 19\).
Let us find the change in product if \(19\) is increased by \(1\).
\(21 (19 + 1) = 21 \times 19 + 21 \times 1\)
\(=\) \(21 \times 19\) \(+\) \(\fbox{21}\)
The product is increased by \(21\).
Let us find the increase in product for \((a + 1)b\).
Expand the product using distributive property \((a + b)c = ac + bc\).
\((a + 1)b = ab + \fbox{b}\)
Observe that the product \(ab\) is increased by \(b\).
Example:
Consider the product of \(a = 21\) and \(b = 19\)
Let us find the change in product if \(21\) is increased by \(1\).
\((21 + 1)19 = 21 \times 19 + 1 \times 19\)
\(=\) \(21 \times 19\) \(+\) \(\fbox{19}\)
The product is increased by \(19\).
Let us find the increase in product for \((a + 1)(b +1)\).
Expand the product using the distributive property of multiplication.
\((a +1)(b + 1) = (a + 1)b + (a + 1)1\)
\(=\) \(ab + \fbox{b + a + 1}\)
Observe that the product \(ab\) is increased by \(b + a + 1\).
Example:
Consider the product of \(a = 21\) and \(b = 19\).
Let us find the change in product if both \(21\) and \(19\) are increased by \(1\).
\((21 +1) (19 + 1) = (21 +1) 19 + (21 +1)1\)
\(=\) \(21 \times 19 + 19 + 21 + 1\)
\(=\) \(21 \times 19\) \(+\) \(\fbox{19 + 21 + 1}\)
The product is increased by \(19 + 21 + 1\).
Let us find the increase in product for \((a + 1)(b -1)\).
Expand the product using the distributive property of multiplication.
\((a + 1)(b - 1) = (a + 1)b - (a + 1)1\)
\(=\) \(ab + \fbox{b - a - 1}\)
Observe that the product \(ab\) is increased by \(b - a - 1\).
Example:
Consider the product of \(a = 19\) and \(b = 23\).
Let us find the change in product if \(19\) is increased by \(1\) and \(23\) is decreased by \(1\).
\((19 +1) (23 - 1) = (19 +1) 23 - (19 +1)1\)
\(=\) \(19 \times 23 + 23 - 19 - 1\)
\(=\) \(19 \times 23\) \(+\) \(\fbox{23 - 19 - 1}\)
The product is increased by \(23 - 19 - 1\).