A linear model: a fixed cost and a cost per unit

Finance and economics, grade 10 (3 units) Lesson 4 of 15Learning

A linear model: a fixed cost and a cost per unit

A fixed cost and a cost per unit

In many services the price is made of two parts:

a fixed cost – an amount paid once, whatever the quantity;
a cost per unit – an amount paid for every unit: every kilometre, every minute, every hour.

A taxi charges 12 ₪ at the start of the ride, and 4 ₪ for every kilometre:

Kilometres	0	1	2	3	5	10
Price (₪)	12	16	20	24	32	52

Every kilometre adds the same amount to the price, 4 ₪.

The formula of a linear model

If $\displaystyle x$ is the number of kilometres and $\displaystyle y$ the price in shekels, the price of the ride is

y=4x+12

$\displaystyle 4x$ – the cost of $\displaystyle x$ kilometres, 4 ₪ each.
12 – the fixed cost.

A price described this way is a linear model: $\displaystyle m$ is the cost per unit, and $\displaystyle b$ the fixed cost.

\displaystyle y=mx+b

The graph of the model: a line

The graph of the price of the ride is a straight line. Every point on it is a ride: $\displaystyle x$ its kilometres and $\displaystyle y$ its price.

\displaystyle y=4x+12

\displaystyle A=(0,12)

\displaystyle B=(5,32)

The line meets the $\displaystyle y$ -axis at the point $\displaystyle (0,12)$ : a ride of 0 km costs 12 ₪ – the fixed cost, $\displaystyle b$ .
With every step of a kilometre to the right the line goes up 4 ₪ – the cost per unit, $\displaystyle m$ .
$\displaystyle (5,32)$ : a ride of 5 km costs $\displaystyle {4\cdot5+12}={32}$ ₪.

The slope: the price per unit

The slope of the line, $\displaystyle m$ , is how much $\displaystyle y$ grows when $\displaystyle x$ grows by 1. In a model of a price the slope is the price per unit.

Two points on the line give the slope: the change in the price, divided by the change in the quantity. For the ride, from the point $\displaystyle (2,20)$ to the point $\displaystyle (5,32)$ :

m=\frac{32-20}{5-2}=\frac{12}{3}=4

\displaystyle m=\frac{y_2-y_1}{x_2-x_1}

From the table to the formula

Check:

\displaystyle {90\cdot5+150}={600}

A plumber charges a fixed call-out fee and an amount for every hour of work.
A two-hour job cost 330 ₪.
A five-hour job cost 600 ₪.

The example below finds the price per hour and the call-out fee, and writes the formula:

The price per hour is the slope: $\displaystyle {m}={\frac{600-330}{5-2}}={\frac{270}{3}}={90}$ ₪.
The call-out fee: for two hours, $\displaystyle {330}={90\cdot2+b}$ , so $\displaystyle {b}={330-180}={150}$ ₪.
The formula: $\displaystyle {y}={90x+150}$ .

Exercise 1

A water bill charges a fixed 30 ₪ a month, and 8 ₪ for every cubic metre of water.
$\displaystyle x$ is the number of cubic metres in a month, and $\displaystyle y$ the bill in shekels.

Write the formula of the bill on the graph board.

What will the bill be in a month when 12 cubic metres were used?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The bill of another month was 190 ₪.
How many cubic metres of water were used that month?

Exercise 2

The graph is the price of a mobile plan in a month: $\displaystyle x$ the number of minutes, and $\displaystyle y$ the price in shekels.
The points $\displaystyle A$ and $\displaystyle B$ are on the line.

\displaystyle y=\,?

\displaystyle A=(0,20)

\displaystyle B=(40,40)

How much does the plan cost in a month with no calls at all?
How does the graph show it?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

What does a minute of calls cost in the plan?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Write the formula of the plan on the graph board above.

Exercise 3

A print shop charges a fixed amount for setting up a print job, and an amount for every copy.
Printing 100 copies cost 250 ₪.
Printing 300 copies cost 450 ₪.

What does each copy cost?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

What is the fixed amount?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

What will printing 500 copies cost?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

An electricity bill charges a fixed 40 ₪ a month, and 0.6 ₪ for every kilowatt-hour.

Write the formula of the bill, where $\displaystyle x$ is the number of kilowatt-hours in a month.
What is the slope, and what does it tell you?

What will the bill be in a month when 350 kilowatt-hours were used?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: finance and economics, grade 10 (3 units)

Open as a page

Buying and selling

The unit price – $\displaystyle P$ the price of the pack, $\displaystyle n$ the number of units in it (grams, litres, items)
$\displaystyle u=\frac{P}{n}$
A $\displaystyle p\%$ discount – $\displaystyle a$ the price before the discount, $\displaystyle b$ the price after it
$\displaystyle b$ $\displaystyle =$ $\displaystyle (100\%-p\%)\cdot a$
A price with 18% VAT – $\displaystyle a$ the price without VAT, $\displaystyle b$ the price with VAT
$\displaystyle b$ $\displaystyle =$ $\displaystyle 1.18\cdot a$

A linear model

The total cost – $\displaystyle x$ the number of units, $\displaystyle m$ the cost per unit (the slope), $\displaystyle b$ the fixed cost
$\displaystyle y=mx+b$
The cost per unit from two points $\displaystyle (x_1,y_1)$ , $\displaystyle (x_2,y_2)$ on the line
$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}$
Two plans cost the same – where their two lines meet
$\displaystyle m_1x+b_1$ $\displaystyle =$ $\displaystyle m_2x+b_2$

Profit and loss

The total cost – $\displaystyle F$ the fixed costs, $\displaystyle c$ the cost of one unit, $\displaystyle x$ the number of units
$\displaystyle C=F+c\cdot x$
The revenue – $\displaystyle s$ the selling price of one unit
$\displaystyle R=s\cdot x$
The profit – the revenue minus the cost (a negative profit is a loss)
$\displaystyle P=R-C$
The profit percent – out of the cost
$\displaystyle r\%=\frac{P}{C}$
The break-even point – the revenue equals the cost
$\displaystyle s\cdot x$ $\displaystyle =$ $\displaystyle F+c\cdot x$

Wages

Pay – $\displaystyle w$ the hourly wage, $\displaystyle h$ the number of hours
$\displaystyle S=w\cdot h$
The pay for an overtime hour, in the first two overtime hours of a day
$\displaystyle 1.25\cdot w$
The pay for an overtime hour, from the third overtime hour of a day
$\displaystyle 1.5\cdot w$
Net pay – $\displaystyle G$ the gross pay, $\displaystyle D$ the deductions
$\displaystyle N=G-D$

Measures of centre

The mean of $\displaystyle n$ values
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
The mean from a frequency table – $\displaystyle f_i$ the frequency of the value $\displaystyle x_i$
$\displaystyle \bar{x}=\frac{x_1f_1+x_2f_2+\dots+x_kf_k}{f_1+f_2+\dots+f_k}$
The place of the median in an ordered list of $\displaystyle n$ values
$\displaystyle \frac{n+1}{2}$