Directly proportional quantities

Ratio, proportion and percent, grade 8 Lesson 3 of 12Learning

Directly proportional quantities

A kilogram of tomatoes costs 6 ₪. This is how the price changes with the weight:

Weight (kg)	1	2	3	5
Price (₪)	6	12	18	30

When the weight doubles, the price doubles too; when the weight is 5 times as much, so is the price.
So the ratio of price to weight is the same in every column: $\displaystyle {\frac{6}{1}}={\frac{12}{2}}={\frac{18}{3}}={\frac{30}{5}}={6}$ .
Two quantities whose ratio is constant are called directly proportional.

A price with a fixed charge

A taxi ride costs 10 ₪ to get in, and another 5 ₪ for every kilometre:

Distance (km)	1	2	4
Price (₪)	15	20	30

The distance doubles, from 1 km to 2 km, but the price does not double: from 15 ₪ to 20 ₪.
$\displaystyle \frac{15}{1}$ and $\displaystyle \frac{20}{2}$ are not equal – the 10 ₪ to get in is paid on every ride, short or long.
So here the price and the distance are not directly proportional.

Exercise 1

In which table are the quantities directly proportional? Explain.

Number of tickets	2	4	6
Price (₪)	50	100	150

Age (years)	2	4	6
Height (cm)	85	100	115

Unit price

In 6 ₪ a kilogram of tomatoes, the 6 is the unit price: the constant ratio of price to amount.
That is how prices of packs of different sizes are compared.
A six-pack of water bottles costs 15 ₪, and four single bottles cost 11 ₪.

In the six-pack a bottle costs $\displaystyle {15:6}={2.5}$ ₪.
As single bottles a bottle costs $\displaystyle {11:4}={2.75}$ ₪.
Each bottle in the six-pack is cheaper.

Exercise 2

In one supermarket 4 yogurts cost 18 ₪.
In another, 6 of the same yogurts cost 24 ₪.
Where is one yogurt cheaper?

One yogurt, first supermarket

/ – 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

One yogurt, second supermarket

/ – 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 proportion for directly proportional quantities

The same order in each fraction: notebooks in the numerator, shekels in the denominator.

When quantities are directly proportional, any two columns of their table form a proportion.
5 notebooks cost 35 ₪.
How many notebooks can you buy for 56 ₪?

Let $\displaystyle x$ be the number of notebooks you can buy for 56 ₪:

\frac{x}{56}=\frac{5}{35}

By cross-multiplication $\displaystyle {35x}={280}$ , so $\displaystyle {x}={8}$ : you can buy 8 notebooks.

Exercise 3

A car travels 100 km on 7 litres of fuel.
How many litres does it need to travel 250 km?

Exercise 4

A printer prints 45 pages in 3 minutes.
How many minutes does it take to print 120 pages?

Formulas: ratio, proportion and percent, grade 8

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Ratio and proportion

A ratio is also a fraction
$\displaystyle a:b$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
Cross-multiplying a proportion
$\displaystyle a\cdot d$ $\displaystyle =$ $\displaystyle b\cdot c$ $\displaystyle \frac{a}{b}=\frac{c}{d}$
Quantities in the ratio $\displaystyle a:b$ – $\displaystyle k$ the size of one part
$\displaystyle ak:bk$ $\displaystyle =$ $\displaystyle a:b$
Directly proportional quantities – the ratio $\displaystyle k$ stays the same (the unit price, for one)
$\displaystyle \frac{y}{x}=k$

Scale

At the scale $\displaystyle 1:n$ – $\displaystyle a$ the length in the drawing, $\displaystyle b$ the real length, in the same unit
$\displaystyle b$ $\displaystyle =$ $\displaystyle a\cdot n$
An area at the scale $\displaystyle 1:n$ – it grows $\displaystyle n^2$ times, not $\displaystyle n$ times
$\displaystyle S_b$ $\displaystyle =$ $\displaystyle S_a\cdot n^2$

Percent and percent change

$\displaystyle p$ percent are $\displaystyle p$ hundredths
$\displaystyle p\%$ $\displaystyle =$ $\displaystyle \frac{p}{100}$
A rise of $\displaystyle p\%$ – $\displaystyle a$ the amount before the change, $\displaystyle b$ the amount after it
$\displaystyle b$ $\displaystyle =$ $\displaystyle (100\%+p\%)\cdot a$
A fall of $\displaystyle p\%$ – $\displaystyle a$ the amount before the change, $\displaystyle b$ the amount after it
$\displaystyle b$ $\displaystyle =$ $\displaystyle (100\%-p\%)\cdot a$
The percent change – a part of the amount before the change
$\displaystyle p\%=\frac{b-a}{a}$
Two changes one after the other – a rise of $\displaystyle p\%$ , then a rise of $\displaystyle q\%$ (for a fall: minus instead of plus)
$\displaystyle c$ $\displaystyle =$ $\displaystyle (100\%+p\%)\cdot(100\%+q\%)\cdot a$