Converting - fractions ↔ decimals ↔ percentages: Worked examples

The scenario matrix

Think of every conversion problem as a point on a grid. The two things that make a problem feel different are: (a) what form you start in, and (b) what kind of number it is (does the decimal stop? is it bigger than one whole? is it zero? is it a word problem?).

Here is every cell we must cover:

Cell	Case class	A number that lands here
A	Fraction → decimal, terminating	$\frac{5}{8}$
B	Fraction → decimal, recurring	$\frac{2}{3}$
C	Decimal → fraction, one/two places	$0.35$
D	Percentage → fraction, needs simplifying	$65%$
E	Percentage over 100 (improper)	$250%$
F	Zero / degenerate input	$0$ , and $\frac{0}{7}$
G	Percentage with a fractional/decimal part	$12.5%$
H	Whole number as a percentage (limiting: the whole)	$1 = 100%$
I	Word problem (real-world comparison)	"17 out of 25 marks"
J	Exam twist (which is biggest? mixed forms)	compare $\frac{3}{5}, 0.58, 62%$

The one fact behind all of it, from the parent note: $p % = \frac{p}{100}, \frac{a}{b} = a \div b, decimal \times 100 percent .$

The number line below reminds you these are all the same points wearing different costumes — a conversion never moves a number, it only relabels it.

Figure — Converting - fractions ↔ decimals ↔ percentages

Cell A — Fraction → terminating decimal

Worked example Cell A · Convert

$\frac{5}{8}$ to a decimal and a percentage. Forecast: Before reading on, guess: will the decimal stop, and roughly how big is it (between $0$ and $1$ ? closer to $\frac{1}{2}$ or to $1$ )?

Step 1. Rewrite the fraction as a division: $\frac{5}{8} = 5 \div 8$ . Why this step? A fraction $\frac{a}{b}$ is " $a$ shared into $b$ equal parts" — division is its definition, not a trick. See Long division.

Step 2. Do the long division: $5.000 \div 8 = 0.625$ . It reaches remainder $0$ , so it terminates. Why this step? $8 = 2^{3}$ has only the prime factor $2$ . A fraction terminates exactly when its denominator (in lowest terms) is built only from $2$ s and $5$ s — the primes hiding inside $10$ .

Step 3. To a percent, multiply by $100$ : $0.625 \times 100 = 62.5%$ . Why this step? "Percent" = number of hundredths; $\times 100$ counts them.

Verify: Go backwards — $62.5% = \frac{62.5}{100} = 0.625$ , and $0.625 = \frac{625}{1000} = \frac{5}{8}$ after dividing top and bottom by $125$ . Full loop closes. ✓ Answer: $\frac{5}{8} = 0.625 = 62.5%$ .

Cell B — Fraction → recurring decimal

Cell C — Decimal → fraction (place value)

Cell D — Percentage → fraction (simplify)

Cell E — Percentage over 100 (improper)

The picture below stacks the "one whole" bar so you can see $250%$ as two full bars plus a half.

Cell F — Zero and degenerate inputs

Cell G — Percentage with a fractional part

Cell H — The whole as a limiting percentage

Cell I — Word problem (real-world comparison)

Cell J — Exam twist (compare mixed forms)

The bar chart below shows all three heights side by side so the ordering is unmistakable.

Recall Which cell is this? (test yourself)

" $\frac{7}{9}$ to a decimal" — which cell? ::: Cell B (recurring: $9$ has no factors of $2$ or $5$ ). " $0.04$ to a fraction" — which cell? ::: Cell C (two places → $\frac{4}{100} = \frac{1}{25}$ ). " $0.5%$ to a decimal" — which cell? ::: Cell G (fractional/decimal percent → $0.005$ ). " $\frac{0}{12}$ to a percent" — which cell? ::: Cell F (degenerate → $0%$ ). " $300%$ to a fraction" — which cell? ::: Cell E (over 100 → $3 = \frac{3}{1}$ ).

Connections

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