Question

Find \[69 \times 78 + 22 \times 69\] using distributive property.

Answer 1

Hint: In this particular question take the number common which is multiplied with two different numbers and then solve the equation or in other words apply distributive property which states that \[a \times b + a \times c = a \times \left( {b + c} \right)\] to find the solution of the problem.

Complete step-by-step answer:
According to the definition of distributive property, which states that the value of product of a number and sum of two numbers is calculated by multiplying each addend separately with the number and then add the products
In mathematical terms, if x, y and z are the two numbers then \[y \times \left( {x + z} \right) = y \times x + y \times z\]
Where x and z are the addends because the number y is multiplied by each addend (x and y) and then the sum of the products is calculated on the RHS of the above equation.
So, now let us solve the given equation by using distributive property.
\[ \Rightarrow 69 \times 78 + 22 \times 69\] (1)
Now as we can see in the above equation that 69 is multiplied with two numbers 78 and 22 separately and the sum of the product is calculated.
So, we can take 69 common and then multiply that with the sum of number 78 and 22.
So, \[69 \times 78 + 22 \times 69 = 69 \times \left( {78 + 22} \right)\]
Now solving the above equation.
\[69 \times 78 + 22 \times 69 = 69 \times \left( {100} \right) = 6900\]
Hence, \[69 \times 78 + 22 \times 69 = 6900\]

Note:Whenever we face such types of questions the key concept is to recall the formula for the distributive property. In general, the distributive property of multiplication of integers is divided into two categories: over addition and over subtraction. Like if a, b and c are three integers then from distributive property of multiplication of integers over addition \[a \times \left( {b + c} \right) = a \times b + a \times c\] and from distributive property of multiplication of integers over subtraction \[a \times \left( {b - c} \right) = a \times b - a \times c\]. So, from these properties we will get the required answer.