Question

How do you use the distributive property to simplify $\left( 20+9 \right)5$? \[\]

Answer 1

Hint: We recall the distributive property of multiplication over addition which eliminates parenthesis by the working rule $a\left( b+c \right)=a\times b+a\times c$. We take $a=5,b=20,c=9$ and use the distributive property to get $a\times b+a\times c$.\[\]

Complete step by step answer:
We know there are four properties for the arithmetic operation called multiplication called closure, commutative, associative and distributive. The distributive property of multiplication requires one more operation either addition or subtraction.
The distributive property of multiplication over addition states that the product of a number say $a$ with sum of two numbers say $b,c$ is equal to the sum of products of the number $a$ multiplied separately with $b$ and $c$. It means
\[\begin{align}
  & a\times \left( b+c \right)=a\times b+a\times c \\
 & \Rightarrow a\left( b+c \right)=ab+ac \\
\end{align}\]
Here we give a parenthesis around $b+c$ to prioritize addition over multiplication so that we can find the sum first. It is called distributive property because the number $a$ outside the bracket is being distributed by each number inside the bracket which are$b,c$. The distributive property is used to multiply large numbers close to multiples of 10, 100 and so on. We are asked in the question to simplify
\[\left( 20+9 \right)5\]
We use the commutative property of multiplication which means $a\times b=b\times a$ in the above step for $a=\left( 20+9 \right),b=5$ to have
\[\left( 20+9 \right)5=5\left( 20+9 \right)\]
We use the distributive property of multiplication over addition $a\times \left( b+c \right)=a\times b+a\times c$ for $a=5,b=20,c=9$in the above step and open the bracket. We have;
\[\left( 20+9 \right)5=5\left( 20+9 \right)=5\times 20+5\times 9=100+45=145\]

Note: We can alternatively solve using distributive property of multiplication over subtraction which states that the product of a number say $a$ with difference of two numbers say $b,c$ is equal to the difference of products of the number $a$ multiplied separately with $b$ and $c$. It means
\[\begin{align}
  & a\times \left( b-c \right)=a\times b-a\times c \\
 & \Rightarrow a\left( b-c \right)=ab-ac \\
\end{align}\]
 We write the given expression as
\[\left( 20+9 \right)5=\left( 29 \right)\times 5=5\times \left( 29 \right)\]
We can write $29=30-1$ in the above step to have
\[\left( 20+9 \right)5=5\times \left( 30-1 \right)\]
We use the distributive property of multiplication over subtraction $a\times \left( b-c \right)=a\times b-a\times c$ for $a=5,b=30,c=1$in the above step and open the bracket. We
\[\left( 20+9 \right)5=5\times \left( 30-1 \right)=5\times 30-5\times 1=150-5=145\]
We should always try that a number within the bracket should be a multiple of 10.