3.03 Rewriting expressions to understand relationships
Simplifying an expression doesn't always make it easier to understand the relationship between two things. Different relationships can be represented by expressions in different but equivalent forms.
Exploration
Imagine that you have $\$10$$10 to spend on after school snacks for a week, and each snack from the vending machine costs $\$2$$2.
First, let's consider some different expressions that give the amount of snack money you have left after buying $s$s snacks.
| $10-2s$10−2s | $2\left(5-s\right)$2(5−s) | $10-s-s$10−s−s | $-2s-10$−2s−10 |
Although the expressions appear different, all four expressions represent the amount of money you have left after buying $s$s snacks. Let's interpret the terms of each expression in the context of our situation.
- In the expression $10-2s$10−2s, the number $10$10 represents the amount of money you have before buying any snacks. The number $-2$−2 represents the cost of buying one snack. The term $2s$2s represents the amount of money you have to subtract from your initial amount after buying $s$s snacks.
- In the expression $2\left(5-s\right)$2(5−s), the number $5$5 represents the total number of snacks you can buy. The expression $5-s$5−s represents the number of snacks you have left, and the $2$2 represents the amount of money for each snack you have left.
Practice questions
Question 1
In a particular year, Re-source Waste Recovery produces sewage treatment plants at a cost of $\$u$$u per plant. Re-source Waste Recovery also pays a fee of $\$b$$b per year for its use of the production facilities.
Using only addition, write an expression that represents the total production cost from $1$1 year if there are $3$3 sewage treatment plants produced in that year.
Which of the following expressions is also equivalent to the production cost in that year?
$3u+b$3u+b
A$3\left(u+b\right)$3(u+b)
B$b+b+b+u$b+b+b+u
C$bu$bu
D
Question 2
Xavier travels to a foreign country, where the value of currency is described in terms of the values ‘$x$x’ and ‘$y$y’. When Xavier arrives, he has plenty of coins worth $2y$2y and plenty of notes worth $x+y$x+y, but none of the several other types of coins or notes.
Xavier goes to a cafe to buy a hot breakfast and wants to work out whether he can buy one without getting any change back.
The display price of the meal is $4x+8y$4x+8y. Rewrite the display price in the form $\editable{}\left(x+y\right)+\editable{}\left(2y\right)$(x+y)+(2y).
Xavier looks at the rest of menu and thinks about which other items he could buy. Which two of these items could he buy without receiving any change back?
$2x+14y$2x+14y
A$x+3y$x+3y
B$16x+15y$16x+15y
C$2x+15y$2x+15y
D$3x+y$3x+y
E
Question 3
Consider the diagram below.
Hermione found the total blue shaded area by considering the length and width of the overall rectangle. Complete the values for the expression she found.
Area $=$= $\editable{}\left(6+\editable{}\right)$(6+)
Irene found the total blue shaded area by adding the areas of the two smaller rectangles. Complete the values for the expression she found.
Area $=$= $6\cdot\editable{}+\editable{}\cdot4$6·+·4 $=$= $\editable{}+4x$+4x