1. Expressions & Operations

Lesson

Many real-life situations involve problems with unknown amounts or values. By using algebra, we can express these kinds of problems as mathematical statements. Doing so allows us to more easily understand the problem, and we will later see a number of tools and tricks that can be used to solve such problems.

Say there is a certain amount of fuel left in the tank and at the gasoline station we pump 30 liters of fuel to fill the tank. Now because we don't know exactly how much fuel there was in the tank to begin with, we can use a variable to represent the amount. We could represent the total amount of fuel in the tank by the expression $x+30$`x`+30. Mathematical expressions that use variables are called algebraic expressions.

Remember!

When we multiply a number by a variable, we leave out the multiplication sign. For example, $2\times y$2×`y` can be written more simply as $2y$2`y`. When we do so, the number is often called the coefficient of the variable.

Similarly, we usually express division using fractions instead of the symbol $\div$÷. So we would write $\frac{x}{2}$`x`2 in place of $x\div2$`x`÷2.

Think of a store that sells laptops for $\$1000$$1000 each and smartphones for $\$500$$500 each. The amount they make in a day will depend on how many of each they sell.

If we let:

- $a$
`a`represent the number of laptops sold in a day, and - $b$
`b`represent the number of smartphones sold in a day,

then the total amount they make in a day is given by the expression $1000a+500b$1000`a`+500`b`.

If they sell $5$5 laptops and $6$6 smartphones in one day, then the total sales for the day will be:

$1000a+500b$1000a+500b |
$=$= | $1000\times5+500\times6$1000×5+500×6 |

$=$= | $5000+3000$5000+3000 | |

$=$= | $8000$8000 |

So the store will have made a total of $\$8000$$8000 in sales on this particular day.

If $x$`x` represents the number of elephants then write an expression for $4$4 times the number of elephants plus $10$10.

The other day Tina bought $v$`v` chart-topping albums from Entertainment R Us for $\$52$$52. Write an expression for the average cost of each album.

Algebraic expressions are applied almost everywhere in real life problems, so we need to learn how to form these expressions when situations are described to us in words.

Converting worded problems into mathematical expressions takes practice and it helps to be familiar with the language of math. Let's look at some examples below.

Let's start with something simple. We know that if I had an amount $x$`x`, 2 more than that would, of course, be $x+2$`x`+2. So then if I said that I had $x$`x` amount of apples and my friend has $2$2 more than me, then that means they have $x+2$`x`+2 apples. What if I had something like $x+47$`x`+47 apples? Treat $x+47$`x`+47 as one amount at first, and add $2$2 to it. So my friend will have $x+47+2=x+49$`x`+47+2=`x`+49 apples.

Similar techniques are used in multiplication. Imagine that a big bucket holds $3$3 times as much water as a smaller bucket and the smaller one has a volume of $6m$6`m` liters. To find how much the bigger bucket holds, try to see the $6m$6`m` as one number and not as $6\times m$6×`m` at first. Then the volume must be $3\times6m=18m$3×6`m`=18`m` liters.

Write an expression for $3$3 consecutive odd numbers, the first of which is $7k+1$7`k`+1.

**Think: **We know that consecutive odd numbers only have one pattern among them, and that is a difference of two. For example $3$3, $5$5, $7$7...etc.

**Do:** Therefore the next odd number after $7k+1$7`k`+1 is $7k+1+2=7k+3$7`k`+1+2=7`k`+3.

Following the same logic the third number is $7k+1+2+2=7k+5$7`k`+1+2+2=7`k`+5.

So the answer is $7k+1$7`k`+1, $7k+3$7`k`+3, $7k+5$7`k`+5.

The first of two consecutive numbers is $\frac{7+p}{2}$7+`p`2. Write an expression for the other number.

If Kelly is currently $x-1$`x`−1 years old, what was her age $10$10 years ago?

Now comes the really exciting part where you can use all the rules and principles of algebra that you have learned to solve problems!

Algebra is all about describing patterns and relationships using symbols. As you have seen before, these symbols are usually letters such as $x$`x` or $a$`a` and they describe an unknown quantity of something.

In this chapter you will need to think about how you can write a worded problem as an algebraic expression, or how you can describe a visual pattern using an algebraic expression.

The sum of three consecutive numbers is $99$99.

a) If the first number is $x$`x`, write an expression for the sum of of the three numbers, expressing your answer in its simplest form.

**Think:** **Consecutive **numbers are integers that come one after the other. So if the first number is $x$`x`, one number more than this can be written as ($x+1$`x`+1) and the number after this would be ($x+1+1$`x`+1+1) which we can simplify to ($x+2$`x`+2).

**Do:** So the sum of these three numbers can be expressed as:

$x+x+1+x+2$`x`+`x`+1+`x`+2 = $99$99

This can be simplified to:

$3x+3$3`x`+3 = $99$99

b) Hence or otherwise, find the value of $x$`x`.

**Think:** We can solve this equation by using inverse operations.

**Do:**

$3x+3$3`x`+3 = $99$99 (Subtract 3)

$3x$3`x` = $96$96 (Divide by 3)

$x$`x` = $32$32

c) Hence, what are the three numbers?

**Think:** We've just worked out that the first number, $x$`x`, is 32.

**Do:** The three numbers are $32$32, $33$33 and $34$34.

**Check:** $32+33+34=99$32+33+34=99

A $23$23cm wire is cut into $2$2 pieces such that the length of the smaller piece is $4$4 cm. The longer piece is then bent to form a rectangle of width $3$3 cm.

What is the length of the longer piece?

Hence, what is the length of the rectangle?

Each week, Aaron earns $\$m$$`m` more than Paul.

Each week, Uther earns $\$u$$`u` more than Aaron.

If Paul earns $\$440$$440 per week, how much does Uther earn per week?

There are lots of different measurement formulas we use for lots of different things in math. We can write a formula for a shapes' perimeter, area or volume, just to name a few. We've already learned how to write and substitute numbers into common formula.

Now we are going to look at examples of shapes where the side lengths contain algebraic terms, not just numbers. As long as you know your measurement formulas, we can substitute these terms in just the same way as we would if they were numbers, then simplify the expression by grouping any like terms.

Write an expression for the perimeter of the rectangle.

Question 10

Write an expression for the area of the triangle below.

Question 11

Consider the rectangular prism below.

Write an expression for its volume.

Write an expression for its surface area.

Interpret the structure of expressions to represent a quantity in terms of its context.