Practice: a linear model and comparing plans

Finance and economics, grade 10 (3 units) Lesson 6 of 15Practice

Practice: a linear model and comparing plans

A plumber charges 120 ₪ for coming, and 90 ₪ for every hour of work.
$\displaystyle x$ is the number of hours of work, and $\displaystyle y$ the price in shekels.
Write the formula of the price on the graph.

How much does a 3-hour repair cost?

/ – 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 for another repair was 525 ₪.
How many hours did the plumber work? Write an equation and solve it.

A family’s electricity bill is a fixed cost a month plus a price for every kWh (kilowatt-hour).

Use in the month (kWh)	100	300
The bill (₪)	85	205

What is the price of one kWh?

/ – 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 cost a month?

/ – 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 the bill be in a month when the family uses 450 kWh?

/ – 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 graph shows the price of renting a car for a day: $\displaystyle x$ is the number of kilometres driven, and $\displaystyle y$ the price in shekels.
The formula is hidden, and the points $\displaystyle A$ and $\displaystyle B$ are on the line.
Write the formula of the line on the graph.
What do the slope and the fixed cost tell you about the price of renting the car?

\displaystyle y=\,?

\displaystyle A=(0,150)

\displaystyle B=(200,310)

A gym has two plans:
In plan A you pay 150 ₪ a month, and come as often as you like.
In plan B you pay 30 ₪ a month, plus 20 ₪ for every visit.
$\displaystyle x$ is the number of visits in a month; plan B is the dashed line.
Mark where the lines meet on the graph.
Write an equation and solve it: for how many visits do the two plans cost the same?
Which plan is cheaper for someone who comes 4 times a month? And for someone who comes 10 times?

\displaystyle y=150

\displaystyle y=20x+30

Two taxi companies:
In company A you pay 10 ₪ at the start of the ride, plus 4 ₪ for every km.
In company B you pay 16 ₪ at the start of the ride, plus 3.5 ₪ for every km.
Solve the equation: for what distance does a ride cost the same in both companies?
Which company is cheaper for a 20 km ride? By how much?

/ – 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

A film website has two plans:
In plan A you pay 12 ₪ for every film, with no monthly payment.
In plan B you pay 49 ₪ a month, and watch as many films as you like.
Write an equation and solve it: for what number of films would the two plans cost the same?
From how many films a month is plan B cheaper?

/ – 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

A mobile company offers three plans a month:
Plan A: 25 ₪, plus 0.2 ₪ for every minute of calls.
Plan B: 45 ₪, plus 0.1 ₪ for every minute of calls (the dashed line).
Plan C: 75 ₪, and you talk as much as you like.
Mark the three points where the lines meet on the graph.
Write which plan is the cheapest for every number of minutes a month.

\displaystyle y=0.2x+25

\displaystyle y=0.1x+45

\displaystyle y=75

Noa says: “A plan with a higher fixed cost is always more expensive.”
Is she right?
Explain with one of the plans of question 4 or question 7.

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
$\displaystyle \frac{n+1}{2}$