Question

Solve the following using distributive property: \[136 \times 15 + \left( { - 10} \right) \times 136 + 136 \times 10\].
(a) \[2040\]
(b) \[1020\]
(c) \[510\]
(d) None of these

Answer 1

Hint: The given problem revolves around the concepts of algebraic solution, involving the use of properties such as commutative, distributive, associative, etc. here, we will use the distributive law (or, property) i.e. \[a \times \left( {b + c + d + ....} \right) = a \times b + a \times c + a \times d + a \times .....\] which includes less terms in the given expression (as a result, taking/dividing the common term outside the bracket), the desire value is obtained.

Complete step-by-step solution:
Since, we have given the expression that,
\[136 \times 15 + \left( { - 10} \right) \times 136 + 136 \times 10\]
Mathematically the expression becomes,
\[136 \times 15 - 10 \times 136 + 136 \times 10\]
Since, we have to solve the given expression so as to get the desired value(s)
Hence, from the given expression it seems that there is one term that is ‘\[136\]’ which is common in the entire expression,
So, there exists the Distributive Property to solve the required expressions which defines that to divide (i.e. to take common term/s) from the respective expression or any equation which leads to the solution easily.
As a result, the distributive property implies to the multiplication of a number involving the sum or difference with the other two i.e. distributive property equates that \[a \times \left( {b + c} \right) = a \times b + a \times c\] respectively.
Hence, using this property, we get
\[136 \times 15 + \left( { - 10} \right) \times 136 + 136 \times 10 = 136 \times \left( {15 - 10 + 10} \right)\]
Solving the equation algebraically, we get
\[136 \times 15 + \left( { - 10} \right) \times 136 + 136 \times 10 = 136 \times 15\]
\[136 \times 15 + \left( { - 10} \right) \times 136 + 136 \times 10 = 2040\]
\[\therefore \Rightarrow \] The option (a) is absolutely correct.

Note: One must able to know all the properties or, laws which includes (in terms of) in mathematics such as Associative law \[a + \left( {b + c} \right) = \left( {a + b} \right) + c\], Distributive law \[a \times \left( {b + c} \right) = a \times b + a \times c\], commutative law \[a + b = b + a\], respectively. which makes the solution more easy. Remember that, while solving in such cases there must be a common difference between the whole expressions which should not change the desired value of the respective expression, so as to be sure of our final answer.