Worked examples
The question
Question: C = 12h + 30 is the cost, in dollars, to rent a bike for h hours.
What does 12 mean?
We write the equation big, so we can label it.
C = 12h + 30
1 What the letters mean
C is the cost, in dollars.
h is the number of hours.
2 Make a small table
To see what each number does, we try a few hours: 0, 1 and 2.
0 is where it starts.
h = 0: 12 times 0 is 0.
Plus 30 is 30.
h = 0: C = 12 × 0 + 30 = 0 + 30 = 30
30
h = 1: 12 times 1 is 12.
Plus 30 is 42.
h = 1: C = 12 × 1 + 30 = 12 + 30 = 42
42
h = 2: 12 times 2 is 24.
Plus 30 is 54.
h = 2: C = 12 × 2 + 30 = 24 + 30 = 54
54
Look at the values of C: 30, 42, 54.
Each time h goes up 1, C goes up by 12.
The same number every time.
3 Find the numbers in the table
Now we write the equation again, and label its numbers.
C = 12h + 30
And 30?
At h = 0, 12 times 0 is 0.
Only 30 is left: the first number in the table.
So 30 is the cost at 0 hours: the start.
Where does plus 12 each time come from?
From 12h: 12 times h.
Each extra hour adds one more 12.
Careful!
12 is multiplied by h, so each hour adds another 12.
The start is the number alone.
Careful: not $12 at the start ✗
4 Say it in words
So 12 means: the cost goes up 12 dollars for each extra hour.
5 Check
Let's check with h = 3.
In words: start at 30, then add 12 three times.
We get 66.
words: 30 + 12 + 12 + 12 = 66
The equation with h = 3: 12 times 3 is 36.
Plus 30 gives 66 too.
So the meaning is right!
equation: 12 × 3 + 30 = 36 + 30 = 66
Let's look back.
First: what the letters mean.
Then: make a small table.
Then: find the numbers in the table.
Then: say it in words.
Last: check.
▶ Watch it solved
The question
Question: W = 500 − 20t is the water left in a tank, in liters, after t minutes.
What does 20 mean?
We write the equation big, so we can label it.
W = 500 − 20t
1 What the letters mean
W is the water left, in liters.
t is the number of minutes.
2 Make a small table
To see what each number does, we try a few minutes: 0, 1 and 2.
0 is where it starts.
t = 0: 20 times 0 is 0.
500 minus 0 is 500.
t = 0: W = 500 − 20 × 0 = 500 − 0 = 500
500
t = 1: 20 times 1 is 20.
500 minus 20 is 480.
t = 1: W = 500 − 20 × 1 = 500 − 20 = 480
480
t = 2: 20 times 2 is 40.
500 minus 40 is 460.
t = 2: W = 500 − 20 × 2 = 500 − 40 = 460
460
Look at the values of W: 500, 480, 460.
Each time t goes up 1, W goes down by 20.
The same number every time.
3 Find the numbers in the table
Now we write the equation again, and label its numbers.
W = 500 − 20t
And 500?
At t = 0, 20 times 0 is 0.
Only 500 is left: the first number in the table.
So 500 is the water left at 0 minutes: the start.
Where does minus 20 each time come from?
From 20t: 20 times t, taken away from 500.
Each extra minute takes away one more 20.
Careful!
There is a minus before 20.
So the water left goes down, not up.
Careful: not + 20 liters each minute ✗
4 Say it in words
So 20 means: the water left goes down 20 liters for each extra minute.
5 Check
Let's check with t = 3.
In words: start at 500, then take away 20 three times.
We get 440.
words: 500 − 20 − 20 − 20 = 440
The equation with t = 3: 20 times 3 is 60.
500 minus 60 gives 440 too.
So the meaning is right!
equation: 500 − 20 × 3 = 500 − 60 = 440
Let's look back.
First: what the letters mean.
Then: make a small table.
Then: find the numbers in the table.
Then: say it in words.
Last: check.
▶ Watch it solved
The question
Question: C = 0.25m + 40 is the cost, in dollars, of a phone plan with m minutes of calls.
What does 0.25 mean?
We write the equation big, so we can label it.
C = 0.25m + 40
1 What the letters mean
C is the cost of a phone plan, in dollars.
m is the number of minutes.
2 Make a small table
To see what each number does, we try a few minutes: 0, 1 and 2.
0 is where it starts.
m = 0: 0.25 times 0 is 0.
Plus 40 is 40.
m = 0: C = 0.25 × 0 + 40 = 0 + 40 = 40
40
m = 1: 0.25 times 1 is 0.25.
Plus 40 is 40.25.
m = 1: C = 0.25 × 1 + 40 = 0.25 + 40 = 40.25
40.25
m = 2: 0.25 times 2 is 0.5.
Plus 40 is 40.5.
m = 2: C = 0.25 × 2 + 40 = 0.5 + 40 = 40.5
40.5
Look at the values of C: 40, 40.25, 40.5.
Each time m goes up 1, C goes up by 0.25.
The same number every time.
3 Find the numbers in the table
Now we write the equation again, and label its numbers.
C = 0.25m + 40
And 40?
At m = 0, 0.25 times 0 is 0.
Only 40 is left: the first number in the table.
So 40 is the cost of a phone plan at 0 minutes: the start.
Where does plus 0.25 each time come from?
From 0.25m: 0.25 times m.
Each extra minute adds one more 0.25.
Careful!
0.25 is multiplied by m, so each minute adds another 0.25.
The start is the number alone.
Careful: not $0.25 at the start ✗
4 Say it in words
So 0.25 means: the cost of a phone plan goes up 0.25 dollars for each extra minute.
5 Check
Let's check with m = 3.
In words: start at 40, then add 0.25 three times.
We get 40.75.
words: 40 + 0.25 + 0.25 + 0.25 = 40.75
The equation with m = 3: 0.25 times 3 is 0.75.
Plus 40 gives 40.75 too.
So the meaning is right!
equation: 0.25 × 3 + 40 = 0.75 + 40 = 40.75
Let's look back.
First: what the letters mean.
Then: make a small table.
Then: find the numbers in the table.
Then: say it in words.
Last: check.
▶ Watch it solved
