Made2Master Digital School — General Mathematics

Part 1C — Fractions, Ratios & The Bridge to Algebra

Track: Foundations → University Level · Tone: Mentor · Edition: 2026–2036 future-proof

Why Fractions Are Secretly Advanced Mathematics

Fractions have traumatised more students than calculus ever did. The irony: fractions are where you quietly meet real algebra for the first time.

Think about it:

• A fraction like 3/5 is not just “3 out of 5 slices of pizza.” It is a ratio that can live on any quantity: money, time, distance.
• The line “/” is a disguised division sign. 3/5 literally means “3 ÷ 5.”
• As soon as you write x/5, you are in algebra: an unknown being split into five equal parts.

So this chapter isn’t “baby stuff.” It’s the bridge between counting and abstract reasoning. Master fractions and ratios, and algebra, percentages, and even calculus will come to you more naturally than they did to most people in school.

Three Faces of a Fraction

A single fraction can wear three different “faces.” Most confusion comes from mixing them up.

1. Part–whole: “3/4 of a cake.” The whole is known; we’re taking a portion.
2. Division: “6 ÷ 4 = 6/4.” We’re sharing or splitting something.
3. Ratio: “The mix is 3:4.” For every 3 parts of one thing, 4 parts of another.

Same symbol, different role. Part-1C goal: train you to spot which role a fraction is playing in any situation.

Mini-Drill: Identify the Face

For each statement, ask: part–whole, division, or ratio?

• “I drank 2/3 of the bottle.”
• “We’re sharing £50 between 3 people.”
• “The recipe is 2 parts sugar to 5 parts flour.”
• “The car used 3/4 of a tank on that journey.”
• “The map scale is 1:50,000.”

You don’t need to calculate anything — just name the role. You’re already thinking more like a mathematician than most adults.

Equivalent Fractions — Same Number, Different Clothes

1/2, 2/4, 3/6, 50/100 — these are all the same point on the number line. The world doesn’t care how you write it; the value is identical. This is why simplifying fractions is not “making them smaller”; it’s changing the outfit.

At the heart of equivalence is one idea:

If you multiply (or divide) the top and bottom of a fraction by the same non-zero number, you haven’t changed its value.

This principle is quietly the engine of algebraic manipulation later. When you “multiply both sides of an equation” by something, you’re doing the same balancing act.

Visual: Fractions on the Number Line

Picture 0 to 1 on a number line. Mark 1/2 in the middle. Now mark 2/4, 3/6, 50/100 — they all land exactly where 1/2 is.

That’s equivalence: different names, same location. The number line will be your best friend for the rest of this course.

Improper Fractions & Mixed Numbers — There Is No “Wrong” Form

School sometimes treats improper fractions (like 7/3) as ugly and mixed numbers (like 2⅓) as “nice.” In real maths, both are valid. They’re just useful at different times:

• Improper form (7/3): easier for computation and algebra.
• Mixed form (2⅓): easier for intuition and everyday interpretation.

Being able to switch between them is like being bilingual.

Decimals, Fractions & Percentages — Three Languages for the Same Idea

These are not three different topics. They’re three different outfits for the same quantity:

• Fraction: 3/4
• Decimal: 0.75
• Percentage: 75%

Conversions:

• Fraction → decimal: perform the division (3 ÷ 4).
• Decimal → percentage: multiply by 100 and add “%”.
• Percentage → fraction: write over 100 and simplify.

The trick isn’t remembering rules — it’s remembering that they’re just relabels of the same point on the number line.

Rare Insight: Why Some Fractions Have “Ugly” Decimals

1/2 → 0.5 (neat). 1/4 → 0.25 (neat). 1/3 → 0.333… (repeating). Why?

It’s not the fraction’s fault — it’s our base-10 system. Fractions whose denominators only have 2s and 5s as prime factors (like 1/2, 1/4, 1/5, 1/8, 1/10, 1/20) terminate neatly in base 10. Others (like 1/3, 1/6, 1/7) repeat forever. In a different number system (like base 3), 1/3 would look neat. This connects fractions to number bases and, later, to computer science.

Ratios — Fractions Turned Sideways

A fraction compares a part to a whole: “3 out of 5.” A ratio compares parts to each other: “3 to 5.” Both tell you how things scale, but the ratio is more “engineering-flavoured.”

Example: A drink mix says 1:4 (syrup:water). If you double both sides to 2:8, the taste is identical — same proportion, just a bigger batch. That’s a ratio version of equivalent fractions.

Key leap: as soon as you start saying “if I double this, then that doubles too,” you are doing linear thinking — the heart of algebra and graphs.

Scaling Ratios — Proportional Reasoning

Suppose the ratio of cats to dogs in a shelter is 3:5 and there are 24 animals in total. How many are cats? How many are dogs?

Total “parts” = 3 + 5 = 8. Each “part” = 24 ÷ 8 = 3 animals. Cats = 3 × 3 = 9. Dogs = 5 × 3 = 15.

You didn’t guess; you scaled a ratio. This process becomes formal algebra soon.

Proportion Statements — Where Algebra Sneaks In

A proportion is an equation saying “these two ratios are equal”:

a/b = c/d

Set up like that, you can solve for any unknown using cross-multiplication:

a × d = b × c

This isn’t a random trick; it’s the equivalence principle of fractions in action. You’re multiplying both sides by b and d to clear denominators — a move you’ll see repeatedly in algebra and calculus.

Example

If 3/5 = x/20, what is x?

Cross-multiply: 3 × 20 = 5 × x → 60 = 5x → x = 12.

You just performed algebra using fraction thinking.

Common Fraction Misconceptions (And How We’ll Fix Them)

• “Bigger denominator means bigger fraction.” Actually, with the same numerator, a bigger denominator means a smaller piece. 1/4 < 1/3 even though 4 > 3.
• Comparing unlike fractions by gut feel. Strategy: put them on the same denominator or visualise them on a number line.
• Adding tops and bottoms. 1/3 + 1/4 is not 2/7. You don’t add denominators — you find a common “unit of measure.”

Part of becoming “math mature” is catching your brain when it tries to misuse whole-number intuition on rational numbers.

Fractions as a Bridge to Algebra

Here is the quiet turning point of the whole track: When you write

x/3 = 5

you are:

• Thinking of x as a quantity.
• Understanding that “divide by 3” has been applied to it.
• Wanting to undo that division (by multiplying both sides by 3).

That is algebra in one line: “What operation has been applied, and how do I reverse it?”

Example — Solving a Fraction Equation

Solve 3x/4 = 9.

1. Clear the denominator: multiply both sides by 4 → 3x = 36.
2. Undo multiplication by 3: divide both sides by 3 → x = 12.

Every move you just made rests on fraction logic from this chapter.

Week-1 Fraction & Ratio Drills (Confidence Builders)

These drills are designed to feel achievable but powerful. Do them slowly and neatly — the goal is understanding, not speed.

Drill 1 — Name the Face (20 Questions)

Write 20 real-world sentences involving “out of,” “for every,” “shared between,” “percent,” etc. For each, label:

• Part–whole
• Division
• Ratio

This conditions your brain to parse context — a skill most textbooks skip.

Drill 2 — Fraction Families

Choose 5 base fractions (e.g., 1/3, 2/5, 3/7, 4/9, 5/8). For each, generate at least 4 equivalent fractions and place them between 0 and 1 on a hand-drawn number line. Label them clearly. You’re building the ability to see equivalence, not just compute it.

Drill 3 — Ratio Recipes

Invent 3 “recipes” with whole-number ratios (e.g., 2:5, 1:3, 4:7). For each:

• Write the ratio as a fraction.
• Scale it to a batch of at least 5 times the original.
• Explain in words why the taste/result doesn’t change even as the total grows.

Transformational AI Prompts for Fractions & Ratios

These prompts are designed to remain useful for at least 10 years, regardless of which AI system you use. They turn your model into a patient fraction tutor that respects understanding over shortcuts.

Prompt 1 — “Fraction MRI: Scan My Understanding”

Prompt:
“Act as a diagnostic mathematics mentor. I want you to scan my understanding of fractions and ratios. Ask me 10 questions: some conceptual, some visual, some numeric, some word problems. For each answer I give, classify my response as correct, partially correct, or based on a common misconception (and name the misconception). Then design a 7-day repair plan with daily micro-exercises to fix my specific weaknesses, using number lines, part–whole models, and ratio tables.”

Prompt 2 — “Ratios to Graphs Bridge”

Prompt:
“Help me see how ratios turn into straight-line graphs. Start with a simple ratio situation (for example, kilometres vs hours at constant speed). 1) Show the ratio as a sentence. 2) Show it as a table. 3) Show it as a fraction and a proportion. 4) Plot 5 points I could draw on a graph, and explain why they form a straight line. Finish by giving me 3 new ratio scenarios and asking me to build the table and verbal description myself.”

Prompt 3 — “Algebra from Fractions”

Prompt:
“Act as my bridge from fractions to algebra. Give me 5 fraction equations (like 3x/4 = 9 or (x + 2)/5 = 4) and walk me through solving them step by step. For each, explain: - what operation is being ‘undone’ at each step, - why we can multiply or divide both sides safely, - and how this is the same logic as simplifying or scaling a fraction. End with 5 similar practice problems for me to do alone, plus answers.”

Where We Go Next

With fractions, ratios, and proportions in place, you’re standing on the bridge into full algebra. Part 2 of the General Mathematics track will start introducing variables, equations, and functions formally — but you’ve already met them in disguise.

For now, the victory is this: fractions and ratios are no longer random rules. They are the grammar of relationships. Once you see that, the rest of mathematics stops feeling like a foreign language and starts feeling like an accent you can master.

Next in this track → Part 2A: Stepping into Algebra — Variables, Equations, and the Language of General Patterns.

Apply It Now (5 minutes)

1. One action: What will you do in 5 minutes that reflects this essay? (write 1 sentence)
2. When & where: If it’s [time] at [place], I will [action].
3. Proof: Who will you show or tell? (name 1 person)

You are my Micro-Action Coach. Based on this essay’s theme, ask me:
1) My 5-minute action,
2) Exact time/place,
3) A friction check (what could stop me? give a tiny fix),
4) A 3-question nightly reflection.
Then generate a 3-day plan and a one-line identity cue I can repeat.

🧠 AI Processing Reality… Commit now, then come back tomorrow and log what changed.