Question

Mary’s backyard vegetable garden measures 20 ft by 30 ft, so its area is $20\times 30=600{\mathrm{ft}}^2$. She decides to make it longer, as shown in the figure so that the area increases to $A=20(30+x)$. Which property of real numbers tells us that the new area can also be written $A=600+20x$?

Short answer

Distributive Property.

Step-by-step solution

Step 1. Apply the concept of the distributive property.

When we multiply a number by a sum of two numbers, we get the same result as we get if we multiply the number by each of the terms and then add the results.

$\begin{array}{l}a(b+c)=ab+ac\\ (b+c)a=ab+ac\end{array}$

Step 2. Example of the distributive property.

$\begin{array}{l}2\cdot (3+5)=2\cdot 3+2\cdot 5\\ (3+5)\cdot 2=2\cdot 3+2\cdot 5\end{array}$

Step 3. Analyze the given situation.

Given

Old length and width are

$\begin{array}{l}w=20\mathrm{ft}.\\ l=30\mathrm{ft}.\end{array}$

New length is

$l\text{'}=\left(30+x\right)\mathrm{ft}.$

So, the new area is given by

$\begin{array}{l}A=w\times l\\ \Rightarrow A=20\times \left(30+x\right)\end{array}$

Step 4. Identifying the property.

Now,

$\begin{array}{l}A=20\times (30+x)\\ \Rightarrow A=20\cdot 30+20\cdot x\mathrm{...............}\left[\mathrm{distributive}\mathrm{property}\right]\\ \Rightarrow A=600+20x\end{array}$

So, the distributive property of real numbers tells us that the new area can be written as

$A=600+20x$