🎮 Maths Castle: The Wizard's Ratio Secret

Master sharing amounts in ratios step by step to solve wizard puzzles and ace your 11+ exam!

Welcome to Maths Castle, brave adventurer! 🧙 Maths Wizard here to guide you through one of the most powerful spells in all of mathematics: sharing in a ratio. Have you ever wondered how master bakers craft perfect cakes, how video game developers balance reward chests between players, or how architects design epic castles? They all use ratios! Ratio is simply a way of comparing two or more quantities to show how much of one thing there is compared to another. Imagine you and your best friend find a treasure chest filled with shiny gold coins after defeating a dragon in Maths Castle. If you worked twice as hard, it would only be fair for you to get two coins for every single coin your friend receives. That is a ratio of 2:1! Understanding ratios is not just an essential skill for your 11+ exam; it is a superpower that helps you split bill payments fairly, adjust delicious recipes for a giant feast, and calculate map scales when exploring new worlds. Today, we are going to master how to share any total amount into any ratio step by step!

Let us break down what a ratio actually represents. A **ratio** compares relative sizes or quantities using a colon symbol (:). For instance, if a magic potion recipe calls for 3 blue berries and 2 red berries, we write the ratio of blue to red berries as **3:2** (read aloud as 'three to two'). It is crucial to understand that a ratio does not tell you the total number of items on its own; instead, it tells you the proportions of the ingredients or shares. Think of a ratio like a set of building blocks or groups. In the ratio 3:2, one complete group contains 3 blue blocks and 2 red blocks, making a total of 5 blocks in that single group. If you wanted to make a double batch of potion, you would keep the exact same ratio balance by using 6 blue berries and 4 red berries. Notice how the proportions stay identical! Ratios allow us to scale amounts up or down without changing the fundamental recipe or relationship between the parts.

Now, let us explore how **sharing in a ratio** works when you are given a grand total to divide. Imagine 🧙 Maths Wizard gives you £50 to share between two apprentices, Oliver and Freya, in the ratio **2:3**. This ratio means that for every £2 Oliver gets, Freya gets £3. To solve this problem without guessing, we use the principle of **equal parts**. We view the ratio numbers as individual slices of a giant magical pie. Oliver receives 2 slices, and Freya receives 3 slices. Together, they are sharing a total of 5 equal slices (2 + 3 = 5). Since the whole pie represents £50, each individual slice (or **1 part**) must be worth £50 divided by 5, which equals £10! Once you know the value of 1 single part (£10), calculating everyone's final share is super easy. Oliver gets 2 parts, so 2 × £10 = £20. Freya gets 3 parts, so 3 × £10 = £30. To double-check, we add their amounts together: £20 + £30 = £50. It matches our original total perfectly!

Whenever you face a ratio sharing problem in your 11+ exam, always follow the Wizard's foolproof **4-Step Ratio Method**: 1. **ADD THE PARTS**: Add all the numbers in the ratio together to find the **total number of parts**. (For example, for 4:1, 4 + 1 = 5 parts). 2. **FIND ONE PART**: Divide the **total amount** given in the question by the **total number of parts**. This gives you the exact value of **1 part**. 3. **MULTIPLY UP**: Multiply the value of 1 part by each individual number in the original ratio to find each person's or item's share. 4. **CHECK YOUR TOTAL**: Add your calculated shares together. They MUST equal the original total amount! If they do not, re-check your division in Step 2. Memorise this rule: **Add, Divide, Multiply, Check!** If you chant this mantra during your revision, ratio questions will become some of the easiest marks you score in the entire exam paper.

Let us put our 4-Step Ratio Method into practice with a warm-up puzzle! Suppose two wizarding students, Leo and Mia, gather 36 magical glow-beans. They decide to share them in the ratio **1:3**. How many glow-beans does each student receive? Let us follow our steps carefully: Step 1 — **Add the parts**: The ratio is 1:3, so total parts = 1 + 3 = **4 parts**. Step 2 — **Find 1 part**: The total number of glow-beans is 36. Divide 36 by the total parts: 36 ÷ 4 = **9 glow-beans per part**. Step 3 — **Multiply up**: Leo gets 1 part, so 1 × 9 = **9 glow-beans**. Mia gets 3 parts, so 3 × 9 = **27 glow-beans**. Step 4 — **Check your total**: 9 + 27 = **36 glow-beans**. Brilliant! The answer matches our starting total of 36. Notice how Mia receives three times as many glow-beans as Leo, exactly as the ratio 1:3 required. You have just completed your first ratio calculation with total confidence!

Now let us tackle a two-step problem with three parts in the ratio! Three knights — Sir Gareth, Sir Lancelot, and Sir Tristan — share 60 silver coins in the ratio **2:3:5**. How many coins does Sir Tristan get, and how many more coins does Sir Tristan get than Sir Gareth? Let us apply our method step by step: Step 1 — **Add the parts**: 2 + 3 + 5 = **10 total parts**. Step 2 — **Find 1 part**: Divide total coins by total parts: 60 ÷ 10 = **6 coins per part**. Step 3 — **Multiply up**: - Sir Gareth (2 parts) = 2 × 6 = **12 coins**. - Sir Lancelot (3 parts) = 3 × 6 = **18 coins**. - Sir Tristan (5 parts) = 5 × 6 = **30 coins**. Step 4 — **Check**: 12 + 18 + 30 = 60 coins. Perfect! Now answer the second part of the question: 'How many more coins does Sir Tristan get than Sir Gareth?' Sir Tristan has 30 coins and Sir Gareth has 12 coins. 30 - 12 = **18 more coins**. You can also find this by taking the difference in parts: 5 - 2 = 3 parts, and 3 × 6 = 18 coins!

Here is a classic GL and CEM exam question that tricks many students because it does NOT give you the total amount! *Exam Question*: 'Amara and Noah share some wizard tokens in the ratio 3:7. Noah receives 20 more tokens than Amara. How many tokens did they share in total?' A) 35 tokens B) 50 tokens C) 20 tokens D) 70 tokens Let us solve this step by step: Notice that 20 is NOT the total number of tokens; it is the **difference** between Noah's and Amara's shares! Step 1 — Find the difference in ratio parts: Noah has 7 parts and Amara has 3 parts. 7 - 3 = **4 parts difference**. Step 2 — Find the value of 1 part: If 4 parts = 20 tokens, then 1 part = 20 ÷ 4 = **5 tokens**. Step 3 — Find total tokens: Total parts = 3 + 7 = **10 parts**. Total tokens = 10 × 5 = **50 tokens**. The correct answer is **B) 50 tokens**! Option A (35) is Noah's share alone. Option C (20) is just the given difference. Option D (70) happens if you mistakenly multiply 7 × 10. Always check whether you are given a total or a difference!

To reach top-grammar or scholarship standard, you must avoid the 3 most common traps students fall into: 1. **Dividing by the wrong number**: Many pupils divide the total amount by 2 just because there are two numbers in the ratio, instead of adding the parts together first! *Fix*: Always write 'Total parts = __' before doing any division. 2. **Misreading who gets what**: In ratios, order matters! In a 2:5 ratio between Ava and Ben, Ava gets 2 parts and Ben gets 5 parts. *Fix*: Write initials above ratio numbers (A:B = 2:5). 3. **Confusing difference with total**: As shown in our exam question, if someone gets '12 more', that represents the *difference in parts*, not the overall sum! 🧙 **Maths Wizard's #1 Exam Power Tip**: After finding each share, always perform the **Re-Addition Check**! Add all your individual final answers together. If they do not add up to the total in the question, stop and check your step 2 division immediately!

Common mistakes

• Wrong: Sharing £30 in ratio 1:2 gives £15 and £15 — Right: £10 and £20. Add parts first: 1 + 2 = 3 parts. £30 ÷ 3 = £10 per part. Shares are £10 and £20.
• Wrong: Sharing 40 marbles in ratio 3:5 gives 15 and 20 marbles — Right: 15 and 25 marbles. Total parts = 3 + 5 = 8. Value of 1 part = 40 ÷ 8 = 5 marbles. 3 × 5 = 15 and 5 × 5 = 25.
• Wrong: Sharing 72 coins in ratio 2:3:4 gives 24, 24, 24 coins — Right: 16, 24, and 32 coins. Total parts = 2 + 3 + 4 = 9. Value of 1 part = 72 ÷ 9 = 8. Shares: 2×8=16, 3×8=24, 4×8=32.
• Wrong: Ratio is 3:5. Person B gets 12 more sweets than Person A, so total sweets = 30 — Right: Total sweets = 48. Difference in parts = 5 - 3 = 2 parts. 2 parts = 12 sweets, so 1 part = 6. Total parts = 8, so 8 × 6 = 48.
• Wrong: Ratio is 2:5:8. Person C gets £30 more than Person A. Person B gets £30 — Right: Person B gets £25. Difference between C and A is 8 - 2 = 6 parts. If 6 parts = £30, 1 part = £5. B has 5 parts: 5 × £5 = £25!

Frequently asked questions

Why do we add the numbers in a ratio together first?

Adding the ratio numbers tells us how many equal slices or parts make up the whole amount. Knowing the total parts allows us to find the value of one part!

What if a ratio has three numbers like 1:2:3?

The exact same rule applies! Add all three numbers together (1 + 2 + 3 = 6 parts), divide the total by 6, then multiply by each ratio number. Easy!

How can I check if my answers are correct during the exam?

Add your individual calculated amounts back together! If their sum equals the total given in the question, you know your answer is 100% correct!

What if the question gives a difference instead of a total?

Subtract the ratio numbers to find the difference in parts! Divide the given difference amount by that part difference to find 1 part.

Does the order of numbers in a ratio matter?

Yes! The order of numbers matches the order of names or items in the sentence. For Ava and Ben in ratio 2:3, Ava gets 2 parts and Ben gets 3.

Can ratios be simplified like fractions?

Yes! You can divide all numbers in a ratio by their common factor. For instance, 4:6 simplifies to 2:3 by dividing both sides by 2.