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🏰 Maths Castle: Squaring Numbers Ending in 5

Master the Vedic shortcut to instantly square any number that ends in 5, a favourite 11+ mental‑maths trick.

🧙 Welcome to the Maths Castle, brave learner! Imagine you’re at a medieval market and a merchant shouts, “Quick! What’s 75 × 75?” You could reach for a calculator, but the wizard knows a secret spell that gives the answer in a flash. Squaring numbers that end in 5 appears in GL, CEM, Kent, Bucks and ISEB papers, and it also helps in real life — like figuring the area of a square garden plot that’s 35 m on each side, or checking a discount that’s 15 % off a £250 coat. Knowing this trick saves precious seconds in the exam hall and builds confidence for tougher mental‑maths challenges. So grab your wand (or pencil) and let’s unlock the magic together! ⭐

🧠 What exactly is this shortcut? When a whole number finishes with the digit **5**, its square always ends in **25**. The digits before that final 25 come from a simple multiplication: take the **stem** (the number formed by all digits except the final 5), multiply it by the **next integer** (stem + 1), and then **append 25** to the product. Think of it like a two‑part recipe — first bake the “stem × (stem+1)” part, then frost it with “25”. For example, 45 → stem = 4, 4 × 5 = 20, add 25 → 2025. The rule works for any length — 115, 1005, even 9995 — because the algebra (10n+5)² = 100n(n+1)+25 guarantees the pattern. 🎯

🔎 Let’s see the algebra behind the magic so you understand *why* it works, not just *how*. Write a number ending in 5 as **10n + 5**, where **n** is the stem (for 35, n = 3). Squaring gives (10n + 5)² = 100n² + 100n + 25 = 100 × n(n + 1) + 25. The term **100 × n(n + 1)** shifts the product of n and n+1 two places to the left, leaving room for the final “25”. That’s why we only need to compute n × (n+1) and then tack on 25. No long multiplication, no carrying — just one small multiplication and a quick write‑down. 🧮

🪄 **The Wizard’s Step‑by‑Step Method** (memorise these three moves): 1️⃣ **Spot the stem** – drop the last digit 5. For 65 the stem is 6; for 115 the stem is 11. 2️⃣ **Multiply stem by stem + 1** – calculate stem × (stem + 1). For 6 → 6 × 7 = 42; for 11 → 11 × 12 = 132. 3️⃣ **Attach “25”** – write the product from step 2 and immediately put 25 after it. 42 → 4225; 132 → 13225. That’s it! Practise the three moves until they feel like a single smooth spell. ✅

📝 **Simple Worked Example – 25²** Step 1: Stem = 2 (drop the 5). Step 2: Multiply 2 × 3 = 6. Step 3: Attach 25 → 625. Check: 25 × 25 = 625 ✔️. Notice how the answer appears instantly — no column multiplication needed. This speed is exactly what the 11+ examiners love to see. 🎉

📈 **Medium Worked Example – 85²** (a two‑digit stem, still easy). Step 1: Stem = 8. Step 2: 8 × 9 = 72. Step 3: Attach 25 → 7225. Common pause: some pupils forget that 8 × 9 = 72, not 56. A quick mental‑times‑table check (8 × 9 = 72) keeps you on track. If you ever hesitate, recall the 9‑times‑table pattern: 9 × 8 = 72. The final answer 7225 is verified by 85 × 85 = 7225. 🏆

🧪 **Exam‑Level Example (GL/CEM style)** *Question:* “Which of the following equals 115²?” A) 11225 B) 13225 C) 1325 D) 11025 **Work‑through:** Stem = 11. 11 × 12 = 132. Attach 25 → **13225** (Option B). Why the others tempt you: A) 11225 comes from mistakenly using 10 × 11 = 110 then adding 25 → 11025, then a slip adds an extra 2. C) 1325 forgets the zero placeholder (100 × 132 = 13200). D) 11025 is the correct square of 105, not 115. Spotting the stem correctly (11, not 10) avoids the trap. 🎯

⚠️ **Common Mistakes & Power Tips** 1️⃣ *Mistake:* Using the wrong stem (e.g., for 115 using 1 instead of 11). *Fix:* Cover the final 5 with your finger; whatever remains is the stem. 2️⃣ *Mistake:* Adding 25 instead of appending it (e.g., 42 + 25 = 67). *Fix:* Say “write 25 after” not “plus 25”. 3️⃣ *Mistake:* Mis‑multiplying stem × (stem+1) under pressure. *Fix:* Practise the 1‑12 times tables daily; a 5‑second recall makes the whole shortcut instant. 🧙 **Wizard’s #1 Power Tip:** On exam day, write the three steps in the margin of your paper as a tiny checklist — stem, multiply, attach 25. It costs nothing and prevents panic errors. 🌟

Common mistakes

Frequently asked questions

Why does the answer always end in 25?

Because (10n+5)² expands to 100n(n+1)+25 — the +25 fixes the last two digits. You’ve got this! 🌟

What if the number has three digits, like 115?

Same rule: stem = 11, 11×12 = 132, attach 25 → 13225. Works for any length. 🎯

Do I have to memorise the times tables up to 12?

Knowing up to 12×12 makes the shortcut instant; a quick mental recall is all you need. Keep practising! 🏆

Can I use this trick for numbers ending in 0 or 1?

No — this shortcut is special to numbers ending in 5. Other endings have different patterns. 🌈

What if I accidentally add 25 instead of attaching it?

You’ll get a much smaller number. Remember: “write 25 after” not “plus 25”. Say it aloud while practising. ✅

Will this appear in the actual 11+ test?

Yes — GL, CEM, Kent, Bucks and ISEB often include a squaring‑ending‑in‑5 question. Master it and you’ll shine! ⭐