The words per cent come from Latin: per centum – “out of a hundred”. You always divide the whole into 100 equal parts. So 100% is always the complete whole, no matter how big it really is.
💯 What does per cent mean?
45 out of 100 squares are yellow = 45%
Liri: “Whether it is 100 stars or 100,000 – I simply divide them in my mind into 100 equal heaps?”
Erklärus: “That is exactly the trick! 1% is always 1 of those 100 equal parts – no matter how big the whole really is.”
Three quantities, one tool
“40 per cent of the stars have been mapped,” Liri read. “But sometimes a different number is missing – sometimes the percentage, sometimes the whole.”
Erklärus: “There are three quantities: the total (T), the percentage (p%) and the value (V). If you know two, you can find the third – without learning three formulas by heart.”
✖️ The percentage cross – thinking instead of learning by heart
Instead of learning three different formulas, you only need a table and a cross. The principle: what lies diagonally opposite is multiplied – divided by the fourth number.
| Part (%) | Value | |
| 100 | T | |
| p% | V |
You always know two of the four boxes – multiply diagonally, divide by the fourth box.
🟡 yellow = multiplied · 🔵 blue = divided · 🟢 green = the result
Erklärus: “Whether T, p% or V is missing – the table stays the same. You simply fill in what you know and cross-multiply.”
🧠 Why does cross-multiplying work?
Liri: “But why am I allowed to simply multiply diagonally and then divide? That feels like magic.”
Erklärus: “Not magic, just logic. T stands for 100%. So: one per cent is T divided by 100. If you want to know how much p per cent is, you take that one per cent and multiply it by p.”
1% of T = T ÷ 100
p% of T = (T ÷ 100) × p = T × p ÷ 100
Erklärus: “Cross-multiplying is just a quick shortcut for exactly this thought: first divide by 100 (because T is 100%), then multiply by p.”
🧥 Example: finding the percentage
A jacket costs €180 (the total). In the sale you pay €126 (the value). What percentage is that?
| Part (%) | Value | |
| 100 | 180 | |
| ? | 126 |
T = €180 (the whole = 100 squares)
V = €126 (the filled squares)
p% = 70 (number of filled squares)
Cross-multiplying: 100 × 126 ÷ 180 = 70
🟡 multiply diagonally (100 × 126) → 🔵 divide by the remaining box (180) → 🟢 result
So it is 70% of the original price.
Total 50, value 10. What percentage?
🚀 Example: finding the value
A spaceship ticket costs €200. You get a 15% discount. How many euros is that?
| Part (%) | Value | |
| 100 | 200 | |
| 15 | ? |
T = €200 (the whole = 100 squares)
p% = 15 (the filled squares)
V = €30 (value of the filled squares)
Cross-multiplying: 200 × 15 ÷ 100 = 30
🟡 multiply diagonally (200 × 15) → 🔵 divide by the remaining box (100) → 🟢 result
The discount is €30.
Total 300, percentage 20%. Value?
💰 Interest – the same cross
Interest is nothing other than a percentage value: with €100 and 5% interest per year you fill in the same cross as before.
€200 at 4% interest – how much interest per year?
🌌 Per cent in the universe
Only about 5% of the universe is made of ordinary matter – the rest is dark matter and dark energy.
A solar sail with 20% more area also collects about 20% more energy.
❌ Common mistake
Many people put the numbers in the wrong place in the cross – then a wrong answer comes out.
Tip: 100 is ALWAYS next to the total (the whole). Mark the number you are looking for with a question mark before you cross-multiply.
✖️ Exercise 1 – Value
Total 160, percentage 25%. What is the value?
✖️ Exercise 2 – Percentage
Total 80, value 20. What percentage?
💰 Exercise 3 – Interest
€500 at 6% interest – how much interest per year?
One cross for everything
Liri no longer had to remember three formulas – whichever quantity was missing, the cross always led her to the answer.
Liri: “Now I understand why you said: learn the principle, not the formula.”
“Percentage cross: table with 100|T on top, p%|V underneath. Multiply across, divide by the fourth number – no matter which quantity you are looking for.”
What you learned today
100 | T · p% | V
Multiply across, divide by the fourth number
🌟
Chapter 41 mastered!
Liri can now solve every percentage and interest problem with the same cross.
Erklärus’s notebook – page 41:
“Percentage cross: table [100|T] above [p%|V]. Two boxes known, multiply diagonally, divide by the fourth box. It works equally for the value, the total and the percentage – also for interest.”
“Percentage cross: table [100|T] above [p%|V]. Two boxes known, multiply diagonally, divide by the fourth box. It works equally for the value, the total and the percentage – also for interest.”