Foundations — Converting - fractions ↔ decimals ↔ percentages

This page builds — from absolutely nothing — every symbol, word, and picture that the main topic leans on. If a line of the parent note ever made you pause on a symbol, this is where that symbol is born. We go slow, in the right order, so each idea stands on the one before it.

0. The number line — the stage everything stands on

Before fractions, before decimals, before percents, there is one picture that holds them all: a straight road with evenly spaced marks.

WHY the topic needs this: the whole promise of converting is that $\frac{1}{2}$, $0.5$ and $50\%$ all land on the exact same point. You cannot believe that promise until you can see the single road they all point to. Look at the amber dot in the figure — it sits halfway between $0$ and $1$, and it has three name-tags. Same spot, three names.

1. The whole, and "how much of one whole?"

Every one of our three forms answers the same question: how much of one whole?

2. Equal parts — the idea hiding inside every symbol

You cannot say "3 out of 4" fairly unless the 4 pieces are the same size. This word "equal" is doing silent heavy lifting everywhere in the topic.

WHY the topic needs this: the parent's definition "$\frac{a}{b}$ = $a$ parts out of $b$ equal parts" collapses without this word. Related idea lives in Fractions - simplifying and equivalent fractions.

3. The fraction bar $\frac{a}{b}$ and the symbols $a$, $b$

Now we can read the first real notation.

Letters like $a$ and $b$ are just name-slots: they stand for "whatever number goes here", so we can state a rule once instead of for every fraction separately. Building fair pieces of equal size connects to Ratio and proportion.

4. The division sign $÷$ and what "divide" pictures

WHY the topic needs this: because $\frac{a}{b}=a÷b$ is the bridge from a fraction to a decimal. When that division doesn't come out neatly in your head, you turn the handle of Long division — that is literally what "$5÷8=0.625$" in the parent's Example 1 is doing.

5. Place value and the decimal point — building a decimal

A decimal is a number written using a special addressing system for pieces smaller than one.

Look at the figure: each step right makes a piece ten times smaller. So $0.75$ literally reads as "$7$ tenths and $5$ hundredths": $0.75=\frac{7}{10}+\frac{5}{100}=\frac{70}{100}+\frac{5}{100}=\frac{75}{100}.$

6. The multiply sign $\times$ and "scaling"

WHY the topic needs this: every hop to-and-from percent is a $\times 100$ or $÷100$. Understanding these as scaling the same number (not changing it — just re-measuring it in a different cup) is what keeps the direction straight.

7. The percent sign $\%$ — the master key, unlocked

Now we can meet the star of the topic.

8. The overline $\overline{}$ — writing "forever"

Some divisions never stop, like $1÷3=0.3333\dots$

WHY the topic needs this: without it we'd have to round $\frac{1}{3}$ to $0.33$ and lose exactness. The bar lets us stay exact. Turning such endless decimals back into neat fractions is its own skill — Recurring decimals to fractions.

9. The tools $\gcd$ and "lowest terms"

When a decimal becomes a fraction like $\frac{75}{100}$, we tidy it.

WHY the topic needs this: it doesn't change the number (same point on the line!) — it just gives the simplest name. Deeper practice: Fractions - simplifying and equivalent fractions.

How these foundations feed the topic

Equipment checklist

Connections

• Converting - fractions ↔ decimals ↔ percentages (parent topic)
• Fractions - simplifying and equivalent fractions
• Decimal place value
• Ratio and proportion
• Percentage increase and decrease
• Recurring decimals to fractions
• Long division