Comparing two plans

Finance and economics, grade 10 (3 units) Lesson 5 of 15Learning

Comparing two plans

Two mobile plans

A mobile company offers two monthly plans:
In plan A you pay 30 ₪ and 0.4 ₪ for every minute of calls.
In plan B you pay 50 ₪ and 0.15 ₪ for every minute of calls.

Minutes a month	0	40	80	120	160
Plan A (₪)	30	46	62	78	94
Plan B (₪)	50	56	62	68	74

In plan A the fixed cost is low, but every minute is dear.
In plan B the fixed cost is high, but every minute is cheap.
At 80 minutes the two plans cost the same, 62 ₪.

Where the lines meet

Each plan is a line: plan A is $\displaystyle {y}={0.4x+30}$ , and plan B is $\displaystyle {y}={0.15x+50}$ (the dashed one).

\displaystyle y=0.4x+30

\displaystyle y=0.15x+50

\displaystyle A=(80,62)

At the meeting point, $\displaystyle (80,62)$ , the two plans cost the same.
To its left plan A’s line is lower: up to 80 minutes plan A is cheaper.
To its right plan B’s line is lower: above 80 minutes plan B is cheaper.
Plan A’s line is steeper, because its slope – the price per minute – is greater.

The meeting point by an equation

An equation finds the meeting point exactly: the price of plan A equals the price of plan B.

\displaystyle m_1x+b_1

\displaystyle =

\displaystyle m_2x+b_2

The example below solves the equation of the two plans:

0.4x+30=0.15x+50

We subtract $\displaystyle 0.15x$ and 30 from both sides: $\displaystyle {0.25x}={20}$ .
We divide by 0.25: $\displaystyle {x}={80}$ minutes.
The price: $\displaystyle {0.4\cdot80+30}={62}$ ₪.

A decision by use

The better plan depends on the number of minutes you talk:

If you talk less than 80 minutes a month – plan A is for you.
If you talk more than 80 minutes – plan B is for you.
If you talk about two hours, 120 minutes, plan B costs 68 ₪ instead of 78 ₪ – a saving of 10 ₪ a month.

Exercise 1

Two taxi companies in town:
“Or” taxis charge 10 ₪ at the start of the ride and 5 ₪ for every kilometre.
“Derech” taxis charge 25 ₪ at the start of the ride and 3.5 ₪ for every kilometre.
On the graph, $\displaystyle x$ is the number of kilometres and $\displaystyle y$ the price in shekels; “Derech” is the dashed line.

Mark where the lines meet on the graph.

\displaystyle y=5x+10

\displaystyle y=3.5x+25

For a ride of how many kilometres do the two companies cost the same?
Solve the equation.

Which company is better for a ride of 6 km?
And which for a ride of 15 km?

Exercise 2

Two companies hire out a car by the day:
Company A charges 150 ₪ a day and 0.5 ₪ for every kilometre.
Company B charges 100 ₪ a day and 0.9 ₪ for every kilometre.

Write an equation and solve it: for how many kilometres a day do the two companies cost the same?

The Mizrahi family is planning a day trip of 200 km.
What will it pay in each company?
Which company should it hire from?

Company A

/ – 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

Company B

/ – 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 3

There are two ways to pay for calls abroad:
Without a package you pay 0.5 ₪ for every minute.
With a package you pay 20 ₪ a month and 0.3 ₪ for every minute.
$\displaystyle x$ is the number of minutes a month, and $\displaystyle y$ the price in shekels.

Write the formulas of the two ways on the graph board.

Write an equation and solve it: for how many minutes a month do the two ways cost the same?

Itai’s grandmother talks with her granddaughter abroad for about an hour a month.
Which is better for her: a package or no package?

Exercise 4

In one plan you pay 60 ₪ a month, and talk as much as you like.
In another plan you pay 20 ₪ a month and 0.25 ₪ for every minute.
From how many minutes a month is the unlimited plan cheaper?

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