Percentage bar

What is it about?

Percent means hundredths: $1\,\% = \frac{1}{100}$, so $25\,\% = \frac{25}{100} = \frac14 = 0.25$. Every percentage problem has three quantities:

• the base value $G$ – the whole, it corresponds to $100\,\%$ (purple),
• the rate $p\,\%$ (orange),
• the percentage value $W$ – the part as a quantity, e.g. in euros (blue).

On the percentage bar, percentages are on top and values underneath. Rule of three on the bar (25 % of €80): split the bar into 100 equal parts, so $1\,\% \;\hat{=}\; €0.80$, and $25\,\% \;\hat{=}\; 25 \cdot €0.80 = €20$. To find the base value, go backwards: from the percentage value to $1\,\%$ and then to $100\,\%$.

With 20 % off you pay $80\,\%$ of the old price ($G \cdot 0.8$); after a 20 % rise it costs $120\,\%$ ($G \cdot 1.2$). German VAT is usually $19\,\%$, so the gross price is $119\,\%$ of the net price. Pitfall: if a bike costs €84 after a 20 % rise, then $€84 \;\hat{=}\; 120\,\%$, so it cost $€84 : 1.2 = €70$ before – not $€84 - 20\,\% = €67.20$.

Try it

1. Start: press “Rule of three on the bar”. The bar splits into $100$ parts, then $25$ parts are counted. Watch the hundred square at the same time.
2. Drag the orange handle, tap the bar or tap squares in the hundred square: set $10\,\%$, $25\,\%$, $50\,\%$ and $75\,\%$. Which fractions belong to them? Three quarters as a circle
3. 1 % of 250: why is $1\,\%$ of $250$ equal to $2.5$? What is $10\,\%$ then?
4. Find the base value: start the animation – now it counts backwards, from $15\,\%$ via $1\,\%$ to $100\,\%$.
5. Above 100 %: the percentage value is larger than the base value. Where is the handle now?
6. Everyday life: discount, price increase, receipt, savings, survey. Which part of the bar is the new price each time?
7. Pitfall: compare the two bars. Why does the red bar end before the purple line?

Tasks

Task 1: Percentage value

What is $25\,\%$ of €80? Load task

Task 2: In the sale

A jumper costs €60. There is $20\,\%$ off. What does it cost now? Load task

Task 3: The receipt

a) Headphones cost €50 before tax. What do they cost with $19\,\%$ VAT? Load task

b) A receipt shows €119 including $19\,\%$ VAT. What is the net price? Load task

Task 4: The savings account

There is €1200 in a savings account at an interest rate of $2.5\,\%$ per year. How much interest is paid after one year, and what is the balance then? Load task

Task 5: The pitfall

a) After a price rise of $20\,\%$ a bike costs €84. What did it cost before? Load task

b) A price is first raised by $20\,\%$ and then lowered by $20\,\%$. Is it back to where it started?

Ideas for the classroom

• With “Show the solution” switched off (show=0), the unknown appears as “?” – estimate on the bar first, then calculate and check with the animation. “New task” creates practice problems with friendly numbers.
• Bar, hundred square and circle show the same share: a bridge from $\frac34$ via $0.75$ to $75\,\%$.
• Model: amounts of money are rounded to the cent; values that are not exact appear with “≈”. Interest for one year only, without compound interest.
• Common stumbling blocks: mixing up base value and percentage value; confusing the discount or tax with the new price; subtracting the rate from the new value when working backwards ($€84 - 20\,\% \ne €70$); thinking that +20 % and −20 % cancel out; believing rates above 100 % are impossible; results such as $9.24$ children – a good reason to talk about sensible answers.