Question

Find A) \[\left( -49 \right)\times 18\]; B) \[\left( -25 \right)\times \left( -31 \right)\]; C) \[70\times \left( -19 \right)+\left( -1 \right)\times 70\] using distributive property.

Answer 1

Hint: To apply distributive property to an expression containing one term convert the bigger number as a sum of two numbers or a difference of two numbers and then apply distributive property. And for the expression with two terms take the common term out and add or subtract the rest of the number.

Complete step by step answer:
A) For \[\left( -49 \right)\times 18\] express \[-49\] as a difference of 1 and 50.
Therefore,
\[\begin{align}
  & \left( -49 \right)\times 18=\left( 1-50 \right)\times 18 \\
 & \left( -49 \right)\times 18=1\times 18-50\times 18\text{ (By distributive property)} \\
 & \left( -49 \right)\times 18=18-900 \\
 & \left( -49 \right)\times 18=-882 \\
\end{align}\]

B) For \[\left( -25 \right)\times \left( -31 \right)\] express \[-31\] as a sum of \[-30\] and \[-1\].
Therefore,
\[\begin{align}
  & \left( -25 \right)\times \left( -31 \right)=\left( -25 \right)\times \left\{ \left( -30 \right)+\left( -1 \right) \right\} \\
 & \left( -25 \right)\times \left( -31 \right)=\left( -25 \right)\times \left( -30 \right)+\left( -25 \right)\times \left( -1 \right)\text{ (By distributive property)} \\
 & \left( -25 \right)\times \left( -31 \right)=750+25 \\
 & \left( -25 \right)\times \left( -31 \right)=775 \\
\end{align}\]

C) Now for \[70\times \left( -19 \right)+\left( -1 \right)\times 70\] take 70 common from both terms then add the remaining terms and multiply with the common term.
Therefore,
\[\begin{align}
  & 70\times \left( -19 \right)+\left( -1 \right)\times 70=70\times \left\{ \left( -19 \right)+\left( -1 \right) \right\}\text{ (by distributive property)} \\
 & 70\times \left( -19 \right)+\left( -1 \right)\times 70=70\times \left( -20 \right) \\
 & 70\times \left( -19 \right)+\left( -1 \right)\times 70=-1400 \\
\end{align}\]

Note: While expressing the number as a sum or difference of two numbers always make sure to express it in a simpler number like \[-49\] as a difference of 1 and 50.