The linear model in money matters · Year 10
In The linear model in money matters (Year 10) you practise 3 skills: The rate and the starting value · Matching a story, a rule and a graph · Two plans: parallel, crossing or the same. Many charges are made of two parts: a fixed amount paid once, and an amount for each unit, such as a minute, a kilometre or a month. A rule like that is a straight line.
The number multiplying the variable is the rate: how much the price goes up for each unit. The number added on its own is the starting value: the price before any unit is used. On a graph, the starting value is where the line meets the vertical axis. Two plans with the same rate and different starting values are parallel lines: they never cost the same. Two plans with different rates are lines that cross: they cost the same once. To find when the price is the same, set the two rules equal and solve the equation. After that point, the plan with the smaller rate is cheaper.
Tip: Before you choose a plan, check how many units you will really need: the cheaper plan depends on it.
What you learn
The rate and the starting value
For example: A monthly phone bill is made up of a fixed charge and a charge for each minute of calls. The rule is y = 0.5x + 50 , where x is the number of minutes and y is the bill in pounds. What is the fixed charge?
Answer and solution
The answer: £50
The number added on its own is 50, so the answer is £50.
Matching a story, a rule and a graph
For example: A monthly phone bill is made up of a fixed charge and a charge for each minute of calls. The fixed charge is £50, and each minute of calls costs £0.50. Which rule gives y , the bill, from x , the number of minutes?
- y = 50x + 0.5
- y = x + 50
- y = 0.5x
- y = 0.5x + 50
Answer and solution
The answer: y = 0.5x + 50
The rate is 0.5 and the starting value is 50, so y = 0.5x + 50 .
Two plans: parallel, crossing or the same
For example: There are two plans for hiring a bike. In both plans x is the number of hours and y is the price in pounds. Plan A: y = 3x + 35 . Plan B: y = 35 + 3x . What is true of the graphs of the two plans?
- They are parallel: the price is never the same
- They cross: the price is the same once
- They are the same line: it is the same plan
Answer and solution
The answer: They are the same line: it is the same plan
Both have the rate 3 and the start 35. It is the same plan, written in a different order.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).