Fractions — proper, improper, mixed numbers: Worked examples

Tools we lean on (built before we use them)

Before the matrix names any case, let's pin down the two symbols the whole page rests on.

The scenario matrix

Think of every fraction question as landing in one box of this table. If we work at least one example per box, you are covered.

Cell	Scenario class	What makes it tricky	Example
A	Improper → mixed, positive	plain quotient $q$ and remainder $r$	Ex 1
B	Mixed → improper, positive	(whole × denom) + num	Ex 2
C	Boundary: numerator = denominator	equals exactly one whole	Ex 3
D	Boundary: numerator is a multiple of denom	remainder $r = 0$ , no fraction part	Ex 4
E	Negative improper → mixed	where does the minus sit?	Ex 5
E′	Negative mixed → improper	carry the minus back through	Ex 6
F	Compare / order fractions	need a common denominator	Ex 7
G	Degenerate: numerator $= 0$ , denom $\to$ large	value $\to 0$ , limiting behaviour	Ex 8
H	Word problem (real world)	translate words → fraction	Ex 9
I	Exam twist: mixed-number trap	notation means add, not multiply	Ex 10
J	Simplify / reduce after conversion	strip common factors to lowest terms	Ex 11
K	Proper fraction — no conversion needed	already "mixed" with zero whole part	Ex 12
L	Negative denominator	the minus lives on the denominator	Ex 13

Figure — Fractions — proper, improper, mixed numbers

Read the map above before diving in. Follow the horizontal Number line: the magenta dot marks $\frac{23}{4}$ . Notice the violet double-arrow underneath spanning $0 \to 5$ — that length is the whole-number quotient $q = 5$ (five complete steps). The orange double-arrow from $5$ to the dot is the leftover $\frac{3}{4}$ . So "mixed" is nothing more than reading the same point as "which whole did I pass, and by how much extra?" Every example below is just this picture with different numbers.

Cell A — Improper → mixed (positive)

Cell B — Mixed → improper (positive)

Cell C — Boundary: numerator equals denominator

Cell D — Boundary: remainder is zero

Cell E — Negative improper → mixed

What to see in this figure. The magenta dot sits at $- \frac{23}{4}$ , wedged between $- 6$ and $- 5$ . The orange quarter-ticks show it lands three-quarters of the way down from $- 5$ toward $- 6$ . The violet caption underneath is the whole point of the example: the minus sign covers both the $- 5$ and the $- \frac{3}{4}$ — the point is more negative than $- 5$ , never between $- 5$ and its right neighbour. If you'd read $- 5 \frac{3}{4}$ as " $- 5 + \frac{3}{4}$ " you'd land at $- 4.25$ , the wrong side entirely.

Cell E′ — Negative mixed → improper

Worked example Ex 6. Convert

$- 3 \frac{2}{7}$ to an improper fraction. Forecast: This is the reverse of Ex 5. Will the numerator be $- 23$ exactly, or something else?

Strip the sign, work with the size $3 \frac{2}{7}$ first. Why this step? Same policy as always: convert the positive size, reattach the minus at the end so it wraps everything.
Mixed → improper on the size: $3 \frac{2}{7} = \frac{3 \times 7 + 2}{7} = \frac{23}{7}$ . Why this step? Turn $3$ wholes into $21$ sevenths, add the $2$ leftover sevenths → $23$ sevenths.
Reattach the minus around the whole thing: $- 3 \frac{2}{7} = - \frac{23}{7}$ . Why this step? $- (3 + \frac{2}{7}) = - \frac{23}{7}$ . The minus was never "just on the whole" or "just on the fraction" — it scales the total.

Verify: $- \frac{23}{7}$ back to mixed: $23 = 3 \cdot 7 + 2$ , so size $= 3 \frac{2}{7}$ , giving $- 3 \frac{2}{7}$ ✓. Sanity check: $- \frac{23}{7} \approx - 3.29$ , which lies between $- 4$ and $- 3$ , exactly where a "minus three and a bit" should sit.

Cell F — Compare / order fractions

Cell G — Degenerate: zero numerator, limiting behaviour

Worked example Ex 8. Evaluate

$\frac{0}{7}$ , then describe what happens to $\frac{3}{n}$ as $n$ grows huge. Forecast: Is $\frac{0}{7}$ allowed? (Contrast with $\frac{7}{0}$ .) And does $\frac{3}{n}$ head toward $0$ or blow up?

$\frac{0}{7}$ : take $0$ of the seventh-sized pieces. That's nothing, so $\frac{0}{7} = 0$ . Why this step? Zero numerator is perfectly legal — the denominator (piece size) still exists. Only a zero denominator is undefined, because there'd be no piece size at all.
Now $\frac{3}{n}$ : as $n$ gets larger the pieces $\frac{1}{n}$ get tinier, so $3$ of them shrink toward nothing. As $n \to \infty$ , $\frac{3}{n} \to 0$ . Why this step? Slice a cake into more and more pieces and each piece — even three of them — becomes vanishingly small.

Verify: $\frac{3}{1000} = 0.003$ , $\frac{3}{1 , 000 , 000} = 0.000003$ — clearly marching to $0$ ✓. Note the asymmetry: $\frac{0}{7} = 0$ but $\frac{7}{0}$ is undefined.

Cell H — Word problem (real world)

Cell I — Exam twist: the notation trap

Worked example Ex 10. A test asks: "Compute

$2 \frac{1}{3} + \frac{2}{3}$ ." A classmate answers $\frac{4}{9}$ . Find the real answer and spot their error. Forecast: Should the answer be near $3$ , or near $\frac{1}{2}$ ?

Read the notation correctly: $2 \frac{1}{3}$ means $2 + \frac{1}{3}$ , not $2 \times \frac{1}{3}$ . Why this step? Mixed-number notation is an addition shorthand. The classmate multiplied the whole by the fraction — the classic trap from the parent note.
Convert to improper to add safely: $2 \frac{1}{3} = \frac{2 \cdot 3 + 1}{3} = \frac{7}{3}$ . Why this step? All in thirds, so the two terms share a denominator.
Add: $\frac{7}{3} + \frac{2}{3} = \frac{9}{3} = 3$ . Why this step? Same-size pieces add their tops; $9$ thirds rebuild exactly $3$ wholes ( $r = 0$ , a Cell D echo).

Verify: $\frac{7}{3} + \frac{2}{3} = \frac{9}{3} = 3$ ✓. The classmate's $\frac{4}{9}$ came from $\frac{2 \times 1}{3 \times 3}$ — treating $2 \frac{1}{3}$ as multiplication. The answer is $3$ , near the forecast, nowhere near $\frac{1}{2}$ .

Cell J — Simplify / reduce after conversion

Worked example Ex 11. Convert

$\frac{18}{8}$ to a mixed number in lowest terms. Forecast: A raw conversion gives some $\frac{r}{8}$ . Will that fraction part already be simplest, or can it shrink?

Convert first: $18 = 2 \cdot 8 + 2$ , so $\frac{18}{8} = 2 \frac{2}{8}$ . Why this step? Standard improper→mixed: quotient $2$ , remainder $2$ riding over $8$ .
Reduce the fraction part. Find the biggest number dividing both $2$ and $8$ : that's $2$ . Divide top and bottom by it: $\frac{2}{8} = \frac{2 \div 2}{8 \div 2} = \frac{1}{4}$ . Why this step? $\frac{2}{8}$ and $\frac{1}{4}$ mark the same point — see Equivalent fractions. "Lowest terms" means no common factor is left to cancel, so the fraction is described with the fewest, biggest pieces possible.
Final answer: $2 \frac{1}{4}$ . Why this step? The whole part $2$ is untouched by reducing the fractional part; only the leftover simplified.

Verify: $2 \frac{1}{4} = \frac{2 \cdot 4 + 1}{4} = \frac{9}{4}$ , and $\frac{18}{8}$ also reduces to $\frac{9}{4}$ (divide by $2$ ) ✓. Both routes agree, so $2 \frac{1}{4}$ is genuinely $\frac{18}{8}$ in simplest form.

Cell K — Proper fraction: nothing to convert

Cell L — Negative denominator

Recall Which matrix cell is each example?

Ex1 → A (improper→mixed) ::: Ex2 → B (mixed→improper) ::: Ex3 → C (num = denom) ::: Ex4 → D (remainder 0) ::: Ex5 → E (negative improper→mixed) ::: Ex5b → E with r=0 (negative, even division) ::: Ex6 → E′ (negative mixed→improper) ::: Ex7 → F (compare) ::: Ex8 → G (zero/limit) ::: Ex9 → H (word problem) ::: Ex10 → I (notation trap) ::: Ex11 → J (simplify) ::: Ex12 → K (proper, no conversion) ::: Ex13 → L (negative denominator)

Connections

Division with remainder — powers Cells A, C, D, E, E′, J, K.
Equivalent fractions — the rescaling in Cell F and the reducing in Cell J.
Adding and subtracting fractions — Cells B, H, I need common denominators.
Decimals — the verify trick in Cell F.
Ratios and proportions — the "per batch" logic of Cell H.
Number line — where every worked point actually sits, including Cells E, L.
Fractions — proper, improper, mixed numbers — the parent this deep-dive expands.