Visual walkthrough — Ratio and proportion — equivalent ratios, dividing in a ratio

We will build a strip of chocolate, cut it into equal pieces, and watch the formula appear.

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Step 1 — What does "$a:b$" ask us to do?

WHAT: we picture the quantity as a long strip. WHY: a strip lets us see "equal pieces" as equal-width blocks — the colon becomes a picture instead of a rule. PICTURE: the strip below is one whole chocolate bar. It has not been cut yet; the labels $a$ and $b$ just remind us how many pieces each person is owed.

Notice: we have not invented any numbers yet. $a$ and $b$ are just counts of pieces. Everything that follows is bookkeeping on those counts.

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Step 2 — Count the pieces: introduce $a+b$

WHAT: we glue A's block and B's block end-to-end and count the total number of equal pieces. WHY: to split a total fairly we first need to know how many equal pieces the total breaks into. That count is $a+b$, and it is the hinge of the entire derivation. PICTURE: the two coloured blocks sit side by side; the number line under them counts pieces $1,2,\dots ,a+b$.

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Step 3 — What is one piece worth? Introduce $x$

WHAT: we name the size of one equal piece $x$. WHY: if we know what one piece is worth, we can rebuild any share by multiplying — that is far easier than working with fractions directly. We use multiplication, not addition, because equal pieces stack multiplicatively ($5$ pieces of $\text{₹}20$ is $5\times 20$, never $5+20$). PICTURE: every block on the strip is now stamped with the same value $x$.

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Step 4 — The pieces must rebuild the whole: the key equation

Here is the one equation everything rests on. Read each term with the colour of its block:

$$\underset{\begin{array}{c}\text{A’s share:}\\ a\text{ pieces},\text{each }x\end{array}}{\underbrace{ax}}+\underset{\begin{array}{c}\text{B’s share:}\\ b\text{ pieces},\text{each }x\end{array}}{\underbrace{bx}}=\underset{\begin{array}{c}\text{the whole}\\ \text{quantity}\end{array}}{\underbrace{T}}$$

WHAT: we say "A's chocolate plus B's chocolate equals the whole bar." WHY: the pieces came from the total, so they must add back to the total — no chocolate created, none destroyed. This conservation is what pins $x$ to a definite value. PICTURE: the coloured blocks slide back together to reform the full length labelled $T$.

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Step 5 — Factor out $x$ and solve for one piece

Now we do algebra on the equation from Step 4. Watch $x$ come loose:

$$ax+bx=T\stackrel{\text{take out }x}{\to }\underset{\text{(one piece)}\times \text{(number of pieces)}}{\underbrace{x(a+b)}}=T\stackrel{÷(a+b)}{\to }\boxed{x=\frac{T}{a+b}}$$

WHAT: we pulled the common $x$ out of $ax+bx$, then divided both sides by $a+b$. WHY factor? Because $ax+bx$ is literally "$x$ counted $(a+b)$ times". Factoring turns the picture "total = pieces × piece-size" into a sentence we can solve. Dividing by $(a+b)$ undoes the multiplication and isolates the one thing we wanted: the size of a single piece. PICTURE: the whole strip $T$ is sliced by vertical cuts into $a+b$ equal pieces, and one lone piece is pulled out and measured as $T/(a+b)$.

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Step 6 — Multiply back to get each share

We know one piece is worth $x=\frac{T}{a+b}$. A owns $a$ of them, B owns $b$ of them:

$$\text{A}=ax=a\cdot \frac{T}{a+b}=\underset{\text{A’s slice of the whole}}{\underbrace{\frac{a}{a+b}}}T,\text{B}=bx=\frac{b}{a+b}T$$

WHAT: we scaled the single-piece value back up by each person's piece-count. WHY: this is the "multiply back" half of Unitary method — one piece → many pieces. The fraction $\frac{a}{a+b}$ is exactly A's portion of the whole, which is why it also equals A's share written as a percentage: $\frac{a}{a+b}\times 100\%$. PICTURE: A's stack of $a$ pieces (height $=ax$) beside B's stack of $b$ pieces (height $=bx$); together they refill $T$.

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Step 7 — Edge cases: what if a term is zero, or there are three people?

We must never leave a scenario unshown.

WHAT / WHY: we tested the boundaries so the reader is never ambushed — zero terms, an impossible zero-sum, and the many-person generalisation all fall straight out of the same strip picture. PICTURE: left panel — a strip with A's block shrunk to nothing ($a=0$); right panel — a strip cut into three coloured blocks $a:b:c$.

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The one-picture summary

Everything on this page is a single sentence made of pictures:

cut the whole into $(a+b)$ equal pieces → one piece is $\frac{T}{a+b}$ → hand $a$ pieces to A and $b$ pieces to B.

Recall Feynman retelling — say it to a 12-year-old

Imagine a chocolate bar you have to share as "2 for me, 3 for you". First you decide every piece will be the same size — that is what the ratio secretly demands. Since you want 2 and I want 3, you must cut the bar into $2+3=5$ equal pieces. Now ask: how much is one piece? Just the whole bar split five ways, $T÷5$. Finally, hand out pieces by the recipe — you take 2 of them, I take 3. That is the entire formula: add the parts, find one piece, multiply back. If a friend's number were $0$, they'd simply get no pieces; if you tried to make both numbers $0$, you'd be asking to cut the bar into zero pieces, which is nonsense — that's the only case the machine refuses.

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Connections

• Ratio and proportion — equivalent ratios, dividing in a ratio — the parent result we derived here.
• Unitary method — Steps 3–6 are "find one, then scale".
• Fractions and simplification — each share is the fraction $\frac{a}{a+b}$ of the whole.
• Highest Common Factor (GCD) — used when checking $210:350=3:5$.
• Percentages — a share written as $\frac{a}{a+b}\times 100\%$.
• Direct and inverse proportion — why both shares move together as $x$ changes.
• Similar figures — same "equal ratio" idea in geometry.