The unit price and comparing offers

Finance and economics, grade 10 (3 units) Lesson 2 of 15Learning

The unit price and comparing offers

The unit price

The unit price is the price of one unit – one litre, one kilogram, one item. When packs come in different sizes, the unit price shows which of them is cheaper.

A 1.5-litre bottle of juice costs 9 ₪, and a 1-litre bottle costs 7 ₪.

The large bottle: $\displaystyle {9\div1.5}={6}$ ₪ a litre.
The small bottle: 7 ₪ a litre.
The large bottle is cheaper – by a shekel for every litre.

When $\displaystyle P$ is the price of the pack and $\displaystyle n$ the number of units in it:

\displaystyle u=\frac{P}{n}

The price per 100 grams

On supermarket shelves the unit price of goods sold by weight is usually given per 100 grams.

Cheese in a 250-gram pack costs 15 ₪, and in a 400-gram pack 22 ₪.

250 grams are 2.5 times 100 grams: $\displaystyle {15\div2.5}={6}$ ₪ per 100 grams.
400 grams are 4 times 100 grams: $\displaystyle {22\div4}={5.5}$ ₪ per 100 grams.
The large pack is cheaper: 5.5 ₪ per 100 grams against 6 ₪.

Offers

In an offer the unit price is what you pay, divided by the number of items you get.

A shampoo costs 18 ₪.

“Buy one get one free”: you pay for one and get two – $\displaystyle {18\div2}={9}$ ₪ a bottle, half price: a 50% discount.
“3 for 2”: you pay for two and get three – $\displaystyle {36\div3}={12}$ ₪ a bottle. You save 6 ₪ out of 18, a third of the price: a discount of about 33%.
“Buy one get one free” is better value than “3 for 2” – but only if you need all the bottles.

A decision: more than the price

The unit price is where a decision starts, and three questions complete it:

How much do I need? A large pack is cheaper per unit, but what is thrown away is money in the bin.
How long does it keep? Food that goes off before it is finished makes what is eaten dearer.
Do I have the money now? A large pack costs more at the till, even when it is cheaper per unit.

Exercise 1

A 1-kg bag of rice costs 12 ₪.
A 2.5-kg bag of rice costs 27 ₪.

a

What does a kilogram of rice in the large bag 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

b

By how many shekels is a kilogram in the large bag cheaper than a kilogram in the small bag?

/ – 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

Exercise 2

Breakfast cereal is sold in two packs: 375 grams for 18 ₪, and 500 grams for 22 ₪.
Work out the price per 100 grams in each pack.
Which pack is cheaper?

a

375 grams for 18 ₪

/ – 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

b

500 grams for 22 ₪

/ – 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

Exercise 3

A chocolate bar costs 6 ₪, and Omer needs 6 bars for the class trip.
Three shops have an offer.
Work out what Omer pays for 6 bars in each shop.
In which shop should he buy them?

a

“Buy one get one free”

/ – 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

b

“3 for 2”

/ – 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

c

20% off every bar

/ – 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

Exercise 4

Yellow cheese is sold in a 200-gram pack for 16 ₪, and in a 1-kg pack for 60 ₪.
The Cohen family eats 200 grams of cheese a week.
Open cheese keeps for two weeks, and the family has no freezer.

a

The family buys the 1-kg pack, eats from it for two weeks and throws the rest away.
What did 100 grams of the cheese it ate 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

b

Which pack should the family buy?
Explain with the price per 100 grams of each pack.

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

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