Foundations — Ratio and proportion — equivalent ratios, dividing in a ratio

This page assumes you know nothing. We will unlock every symbol the parent note uses — the equals sign, the division symbol, the fraction bar, the colon, the letters $a,b,x$, the word "gcd" — one at a time, each with a picture, before it is ever used to prove anything. We introduce them in strict order, so no symbol is ever used before its own section.

1. The number line and "how big is a number?"

Before comparing two amounts, we must be able to see one amount. Picture a straight line with evenly spaced marks: $0,1,2,3,\dots$. A number is just how far along that line you walk from zero.

Why start here? Because a ratio compares two distances on this line, and we need the picture of "distance from zero" to make sense of "how many times bigger".

2. The equals sign, and two ways to compare

Before we compare anything, we need the equals sign, because we are about to write our very first true statement.

Now, suppose one rope is $6$ m and another is $2$ m. There are two honest questions you could ask:

• "How much longer?" → take the difference, and you find it is $4$ m longer. This is comparison by subtraction.
• "How many times as long?" → the long rope covers the short one exactly $3$ times. This is comparison by division (a ratio).

3. The division symbol and the fraction bar

The "how many times" question above is answered by division. Its first symbol is $÷$:

$6÷2=3.$

The fraction bar is the exact same instruction written vertically:

$6÷2=\frac{6}{2}=3.$

Why do we need fractions? Because the parent note's core sentence is "a ratio is the fraction $\frac{a}{b}$". If the fraction bar is fuzzy, the whole topic is fuzzy. Everything about ratios inherits from Fractions and simplification.

4. Letters that stand for numbers: $a$, $b$, $x$, $T$, $k$

The note writes formulas full of letters, such as $x=\frac{T}{a+b}$. A letter is just a box that holds a number we haven't chosen yet — a placeholder.

Why bother with letters instead of numbers? Because "multiply both terms by the same number" must be provable once, forever — not re-checked for $2$-to-$3$, then $4$-to-$5$, then $7$-to-$9$... A letter lets one line of algebra cover infinitely many cases.

5. The colon : — the ratio symbol

Now we have earned every piece — the equals sign, division, the fraction bar, and letters — so we can finally introduce the star of the show.

6. Multiplication and "scaling both terms"

The parent's key move is written $a:b=ka:kb$. One new symbol hides inside it.

• $ka$ means $k\times a$ (a number written right next to a letter means multiply).
• The $=$ sign is the balanced see-saw from section 2: both sides are the same value.

This picture is the whole engine of equivalent ratios, and later of Similar figures (all sides scaled by one $k$). Note we scale by a non-zero $k$ — multiplying both terms by $0$ would give $0:0$, which (bottom is zero!) is meaningless.

7. The plus sign in "total parts": $a+b$

When we split a total, the note writes $a+b$ ("sum of the terms"). The $+$ is ordinary addition. Here it counts how many equal parts the whole is cut into.

8. "gcd" — the greatest common divisor

The note says "divide both terms by $\gcd (a,b)$" to simplify. This word needs unpacking.

Why do we need it? To reduce a ratio to its simplest form, we divide both terms by their biggest shared factor — one clean step, guaranteed nothing further cancels. Full detail lives in Highest Common Factor (GCD).

9. The link to $\%$: a ratio out of 100

The note connects ratios to percentages. A percent is nothing new:

We flag it now so that when you meet $\%$ later you recognise it as an old friend in disguise.

How these foundations feed the topic

The map below is a flow of dependence: read an arrow "$X\to Y$" as "you need $X$ before you can understand $Y$". Start at the top ("a quantity is a distance") and every path eventually pours into the two big skills at the bottom — equivalent ratios and dividing a total in a ratio. In words: comparing by division gives the fraction bar; the fraction bar dressed with a colon gives the ratio; scaling a ratio gives equivalent ratios (and, with the gcd, simplest form); adding the terms gives the number of parts, which lets you split a total.

Equipment checklist

Test yourself — say the answer aloud before revealing.