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🧙 The Wizard's Treasure-Sharing Quest

Master how to share amounts fairly using ratios — from splitting gold coins to scaling up magical recipes!

🧙 Welcome, brave apprentice! I am the Maths Wizard, and today we tackle one of the most useful spells in the whole castle: **sharing ratios**. Picture this — you and two friends find a treasure chest with £60 inside. But wait! You did most of the digging, so it wouldn't be fair to split it equally. Ratios solve this exact problem. They let us share things out fairly when 'fairly' doesn't mean 'equally'. Outside of school, ratios are everywhere. When a baker doubles a recipe to make more cakes, that's ratios. When you mix orange squash with water, that's a ratio. When football pundits say a team wins games in the ratio 3:1, that's ratios too. Even mixing paint colours to get the perfect shade uses ratios! Grown-ups use ratios to split bills, share profits in a business, and mix concrete on building sites. The reason the 11+ examiners love this topic is that it tests whether you can think in a smart, organised way. Get this spell right, and you'll unlock a skill you'll use for the rest of your life. Ready to become a Ratio Wizard? Grab your wand — let's begin the quest! ⭐

So what exactly is a **ratio**? A ratio is a way of comparing two or more amounts, showing how much of one thing there is compared to another. We write it with a colon, like this: 2:3. This means 'for every 2 of the first thing, there are 3 of the second thing'. Think of it like a recipe for lemonade: 2 cups of lemon juice to 3 cups of water — written 2:3. The magic number in every ratio is the **total number of parts**. In the ratio 2:3, you add the numbers together: 2 + 3 = 5 parts altogether. Imagine you have a chocolate bar and you slice it into these 5 equal chunks. One person gets 2 chunks, the other gets 3 chunks. That's the heart of sharing ratios! Here's a memorable way to picture it: a ratio is like a set of instructions for building a fair tower. Each block in the tower is one 'part', and the ratio tells you how many blocks each person should get. The whole tower is your total. Once you know how big one single block is, you can build anyone's share simply by stacking the right number of blocks. Simple, powerful magic!

Now, **how does sharing a ratio actually work?** The secret lies in finding the value of **one part**. Let's say we share £20 between two children in the ratio 2:3. First, we find the **total number of parts** by adding the ratio numbers: 2 + 3 = 5 parts. Next comes the clever bit — we work out what **one part** is worth. We do this by dividing the total amount by the total parts: £20 ÷ 5 = £4. So one single part equals £4. This is the golden key that unlocks everything! Now we simply give each person the right number of parts. The first child gets 2 parts: 2 × £4 = **£8**. The second child gets 3 parts: 3 × £4 = **£12**. Let's *check our magic worked*: £8 + £12 = £20. ✅ It matches the original total, so we know we're correct! Always do this check — it's your spell of protection against mistakes. The key terms to remember are **total parts** (add the ratio numbers), **value of one part** (divide the amount by total parts), and **each share** (multiply parts by the value of one part). Master these three ideas and no sharing question can defeat you!

Here is the exact **method** you must follow every single time — I call it the Wizard's Three-Step Spell. Follow it in order and you'll never go wrong. **Step 1 — ADD:** Add up all the numbers in the ratio to find the total number of parts. For example, in 3:5 the total is 3 + 5 = 8 parts. **Step 2 — DIVIDE:** Divide the total amount you're sharing by the total number of parts. This gives you the value of one single part. For example, sharing £40 in ratio 3:5 means £40 ÷ 8 = £5 per part. **Step 3 — MULTIPLY:** Multiply the value of one part by each number in the ratio to find each person's share. So the shares are 3 × £5 = £15 and 5 × £5 = £25. Finally, and this is your bonus safety step: **CHECK** by adding all the shares back together. They must equal the amount you started with (£15 + £25 = £40 ✅). I remember this as **ADD, DIVIDE, MULTIPLY, CHECK**. Some pupils use the phrase 'All Dragons Munch Cheese' to remember the order. Whatever helps it stick! Write your working clearly at every step — examiners award marks for showing your method, even if your final answer slips.

Let's cast the spell together on an easy question. **Share 24 sweets between Amy and Ben in the ratio 1:2.** Take a deep breath — we follow our three steps. **Step 1 — ADD:** the total parts are 1 + 2 = 3 parts. **Step 2 — DIVIDE:** the value of one part is 24 ÷ 3 = 8 sweets. So each single 'part' is worth 8 sweets. **Step 3 — MULTIPLY:** Amy gets 1 part, so 1 × 8 = **8 sweets**. Ben gets 2 parts, so 2 × 8 = **16 sweets**. Now the all-important check: 8 + 16 = 24. ✅ Brilliant — it matches, so our magic is correct! Notice how Ben gets twice as many sweets as Amy, which makes sense because his ratio number (2) is double Amy's (1). This is a great way to sanity-check your answer: does the bigger ratio number get the bigger share? If Amy had ended up with more sweets than Ben, we'd know something had gone wrong. Always glance at your final answer and ask, 'Does this feel sensible?' A ratio share should never give the smaller number a larger amount. You've just cast your first sharing spell perfectly — well done, apprentice! ⭐

Now let's try a trickier version with a hidden wrinkle. **A necklace is made using red and blue beads in the ratio 4:3. There are 21 blue beads. How many beads are there altogether?** Here's the twist — we're NOT told the total. We're only told one part of it! Many pupils panic here, but the spell still works, just in reverse. We know the blue beads are the '3' part of the ratio, and there are 21 of them. So **Step 1 — find one part:** if 3 parts = 21 beads, then one part = 21 ÷ 3 = 7 beads. **Step 2 — find the red beads:** red is the '4' part, so 4 × 7 = 28 red beads. **Step 3 — find the total:** add them together: 28 red + 21 blue = **49 beads altogether**. Let's check with parts: total parts = 4 + 3 = 7 parts, and 7 parts × 7 beads = 49. ✅ Both methods agree! The key lesson: when you're given just ONE share instead of the total, use that share to find the value of one part first, then build everything else from it. This 'work backwards' trick appears often in exams, so practise spotting it. Slow down and identify which part you've been given!

Let's see how examiners test this in a real GL or CEM paper. **Exam question: Money is shared between Tom, Uma and Vic in the ratio 2:3:5. Vic receives £45 more than Tom. How much money is shared in total?** Options: A) £90 B) £120 C) £150 D) £135. This is a HARD scholarship-style question because it uses the *difference* between two shares. Let's solve it. Vic has 5 parts and Tom has 2 parts, so the difference is 5 − 2 = 3 parts. We're told this difference is worth £45. So one part = £45 ÷ 3 = £15. The total parts are 2 + 3 + 5 = 10 parts, so the total money = 10 × £15 = **£150**. The answer is **C**. Now, why are the wrong options tempting? Option A (£90) comes from wrongly thinking £45 equals the total difference and doubling it. Option B (£120) happens if you use 8 parts instead of 10 (forgetting one number). Option D (£135) comes from dividing £45 by the wrong number of parts (using 3 parts as the total, giving 9 × £15). Each wrong answer is a genuine slip a rushing pupil makes. The lesson: read carefully — 'more than' means you use the *difference* in parts, not a single share!

Let's arm you against the three most common mistakes, apprentice! **Mistake 1: Forgetting to add ALL the ratio numbers.** Pupils sometimes divide the total by just one ratio number instead of the sum. Fix: always circle every number in the ratio and add them ALL before dividing. **Mistake 2: Confusing the total with one share.** When a question gives you one person's share (like 'Ben got 16 sweets') pupils treat it as the grand total. Fix: read carefully and ask 'Is this number the WHOLE amount or just PART of it?' Underline the exact words. **Mistake 3: Muddling 'difference' with 'total'.** In questions saying 'Vic gets £45 more than Tom', pupils use £45 as one share or the total. Fix: 'more than' always means subtract the parts first to find the difference in parts. Write it out: 'difference = big part − small part'. 🧙 My #1 power tip for exam day: **always find the value of ONE PART first, then check your final shares add back to the total.** That single check catches almost every error before the examiner ever sees it. Write 'ADD, DIVIDE, MULTIPLY, CHECK' at the top of your rough working, and you'll cast every ratio spell flawlessly. Now go forth and claim your treasure! 🏆

Common mistakes

Frequently asked questions

Why can't we just share things equally every time?

Because 'fair' doesn't always mean 'equal'! If one person worked twice as hard, a ratio like 2:1 shares things fairly based on effort. Ratios help us split things in exactly the right amounts. You're thinking like a true mathematician! 🌟

What if I forget whether to add, divide or multiply?

Just remember 'ADD, DIVIDE, MULTIPLY, CHECK' — try the phrase 'All Dragons Munch Cheese'! Write it at the top of your working before you start. Once it's written down, the whole spell falls into place. You've got this!

How do I know if a number is the total or just one share?

Read the words very carefully and underline them! Words like 'altogether' or 'in total' mean the whole amount. If it names one person's share, it's only part of it. Careful reading wins marks — keep it up!

What does 'more than' mean in ratio questions?

It means you use the DIFFERENCE between two shares. Subtract the smaller ratio number from the bigger one to get the difference in parts, then match it to the amount given. Spot this trap and you'll shine! ⭐

Do I always have to check my answer at the end?

Yes, please do — it's your magic shield! Add all the shares back together; they must equal the starting amount. This one quick step catches almost every mistake before the examiner sees it. It really is worth those few seconds!

What if the amount doesn't divide neatly by the parts?

In 11+ questions it almost always divides evenly, so if you get a messy decimal, gently check your parts total again. You've probably added the ratio slightly wrongly. Recheck calmly — you'll find it every time!