🧙 Maths Castle: Master Percentages

Learn to find any percentage of any amount with wizard‑level confidence for 11+ exams.

🧙 Welcome, brave apprentice, to the Maths Castle! Imagine you’re at the village market and a sparkling sign reads “30 % off all wizard hats.” You have 120 gold coins in your pouch. How many coins will you actually spend? Percentages are the secret code that lets you unlock discounts, calculate VAT on a new spell‑book, or split a dragon‑fruit pie fairly among friends. In the 11+ exams, percentage questions appear in shopping problems, journey‑time puzzles, and even recipe‑scaling challenges. Mastering them now means you’ll never be caught off guard when a question asks, “What is 37 % of 250?” – you’ll simply wave your wand and produce the answer. So grab your quill, steady your mind, and let’s turn percentages into your most reliable spell!

📖 A **percentage** is just a way of expressing a number as parts out of 100. The word comes from the Latin *per centum*, meaning “by the hundred.” Think of a giant chocolate bar split into 100 equal squares. If you eat 25 squares, you’ve eaten 25 % of the bar – the same as the fraction 25/100, which simplifies to 1/4. The % symbol is a shortcut that tells you “out of 100.” So 50 % = 50/100 = 1/2, 10 % = 10/100 = 1/10, and 5 % = 5/100 = 1/20. Once you see the link between percentages, fractions, and decimals, every problem becomes a simple translation exercise.

⚙️ The **rule** is beautifully simple: to find *P %* of a number *N*, first turn *P %* into a decimal by dividing by 100 (or moving the decimal point two places left). Then multiply that decimal by *N*. For example, to find 35 % of 240: 35 % → 0.35 (because 35 ÷ 100 = 0.35). Next, 0.35 × 240 = 84. You can also break the percentage into friendly chunks: 10 % of 240 = 24, so 30 % = 72, and 5 % = 12; 72 + 12 = 84. Both routes give the same answer – choose the one that feels fastest for you.

🪄 **The Wizard’s Three‑Step Method** 1️⃣ **Convert** – Write the percentage as a decimal (divide by 100). 2️⃣ **Multiply** – Multiply the decimal by the given amount. 3️⃣ **Check** – Does the answer make sense? (e.g., 50 % should be half, 10 % should be a tenth). Follow these steps every time and you’ll never lose your way in the castle corridors.

✨ **Simple Worked Example** – Find 25 % of £80. Step 1: 25 % → 0.25 (25 ÷ 100). Step 2: 0.25 × 80 = 20. Step 3: 25 % is a quarter, and a quarter of 80 is indeed 20. ✅ The answer is **£20**.

🌟 **Medium Worked Example** – A cloak costs £60. It’s on a 15 % discount, then VAT at 20 % is added. What is the final price? Step 1: Discount = 15 % of £60 → 0.15 × 60 = £9. Price after discount = £60 – £9 = £51. Step 2: VAT = 20 % of £51 → 0.20 × 51 = £10.20. Step 3: Final price = £51 + £10.20 = **£61.20**. Notice we applied the percentage to the *new* amount after discount, not the original – a common exam twist!

🎯 **Exam‑Level Example (GL/CEM style)** – *What is 27 % of 350?* A) 94.5 B) 9.45 C) 945 D) 9450 Work through: 27 % → 0.27. 0.27 × 350 = 94.5. So **A** is correct. Why the others tempt you: B moves the decimal one place too far (forgetting to multiply by 10). C forgets to divide by 100 (27 × 350 = 9450, then drops a zero). D is the raw product 27 × 350 without any conversion. Spotting these traps saves precious seconds.

🛡️ **Common Mistakes & Power Tips** 1️⃣ **Mistake:** Forgetting to convert % to decimal (e.g., 30 % × 200 = 6000). *Fix:* Always write “÷ 100” or move the decimal two places left first. 2️⃣ **Mistake:** Applying a second percentage to the original amount instead of the updated amount (as in the cloak example). *Fix:* Read the question carefully – “then” means a new base. 3️⃣ **Mistake:** Mixing up “increase by” and “decrease by” (multiply by 1 + decimal vs 1 – decimal). *Fix:* Use the phrase “add on” → 1 + decimal; “take off” → 1 – decimal. 🧙 **Wizard’s #1 Power Tip:** On exam day, jot a tiny “% → ÷100” reminder at the top of your working space. It’s a lightning‑fast safeguard that keeps your spells accurate! 🏆

Common mistakes

• Wrong: 30% of 200 = 6 — Right: 30% of 200 = 60. Move the decimal two places left when converting 30% to 0.30, then multiply.
• Wrong: 15% of 80 = 1.2 — Right: 15% of 80 = 12. 10% = 8, 5% = 4 → 8+4 = 12.
• Wrong: Increase 50 by 20% then decrease by 20% returns to 50 — Right: Increase 50 by 20% → 60; decrease 60 by 20% → 48. The base changes after the first step.
• Wrong: After a 25% discount the price is £75, so the original was £100 — Right: After a 25% discount the price is £75, original = £75 ÷ 0.75 = £100. Divide by (1‑discount) to reverse a percentage decrease.
• Wrong: A shop raises a price by 25% then offers 25% off – final price equals original — Right: Final price = 93.75% of original (lower). Percentages are applied to different bases; the second 25% is taken off a larger amount.

Frequently asked questions

Why do we have to turn a percent into a decimal?

Because multiplying by a decimal is the same as taking that many hundredths of the number. It makes the calculation simple and exact. You’ve got this! 🌟

What if the percentage is more than 100%?

It works the same way – just convert to a decimal (e.g., 150% = 1.5) and multiply. The answer will be larger than the original amount. Great thinking! 🚀

Can I use fractions instead of decimals?

Absolutely! 25% = 1/4, 20% = 1/5, etc. Use whichever feels faster, then multiply. Flexible minds win! 🧠

How do I reverse a percentage (find the original amount)?

Divide the final amount by (1 ± decimal). For a 20% increase, divide by 1.2; for a 20% decrease, divide by 0.8. You’re becoming a wizard! 🔮

What’s the quickest way to find 10%?

Just move the decimal point one place left (divide by 10). Then you can build any other percent from there. Speedy and smart! ⚡

What if I forget the steps during the exam?

Write a tiny reminder: “% → ÷100 → × amount”. A quick glance brings the method back instantly. Trust your preparation! 🏆