Fractions — proper, improper, mixed numbers: Question bank

For the machinery behind the conversions, see the parent note. If a question leans on splitting a number into "whole groups plus leftover", that's Division with remainder at work.

Figure — Fractions — proper, improper, mixed numbers

The picture above is worth holding onto: the number line we keep referencing is just this same strip of wholes, with each whole finely subdivided into $b$ -sized pieces. Proper fractions live inside a single gap between two whole marks; improper fractions reach past the first whole mark.

And "equivalent fractions" — same amount, different-looking name — is just the area picture of re-cutting one shaded region into more, smaller tiles without changing how much is shaded:

True or false — justify

Every improper fraction can be written as a mixed number.

False —

\frac{9}{9} = 1

and

\frac{8}{4} = 2

are improper but become whole numbers, not a whole-plus-proper-fraction. Mixed form needs a nonzero leftover (the remainder

r \neq = 0

from

a = q b + r

A proper fraction is always positive.

False — "proper" only means

∣ a ∣ < ∣ b ∣

, i.e. its value sits strictly between

- 1

and

1

. So

- \frac{3}{4}

is proper too; the label is about size, not sign.

\frac{5}{5}

is a proper fraction because the pieces "fit exactly".

False — numerator equals denominator, so it is improper (

∣ a ∣ \geq ∣ b ∣

) and equals exactly one whole. Proper requires the top to be strictly smaller.

Making the denominator bigger always makes the fraction smaller.

False — only if the numerator stays fixed.

\frac{1}{4} < \frac{1}{2}

(smaller pieces), but

\frac{6}{4} > \frac{1}{2}

because the numerator changed too. See Equivalent fractions (and the re-tiling picture above) for how top and bottom scale together.

1 \frac{3}{4}

and

\frac{7}{4}

are the same amount.

True — a mixed number is a shorthand sum:

1 \frac{3}{4} = 1 + \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4}

. They are one quantity written two ways.

An improper fraction is a mistake that must be corrected.

False — it is fully valid and often easier for multiplying and dividing. You only convert to mixed for human readability or when a question demands it.

Between

\frac{2}{3}

and

\frac{3}{4}

there are no other fractions.

False — between any two different fractions live infinitely many more (e.g. their average). On the Number line there is never a "next" fraction with no gaps.

\frac{0}{5}

is undefined.

False — it equals

0

(take zero of the five pieces). Only a zero denominator is undefined, because there's then no piece-size to count.

Spot the error

2 \frac{3}{4}

means

2 \times \frac{3}{4} = \frac{6}{4}

The error: mixed notation is addition, not multiplication.

2 \frac{3}{4} = 2 + \frac{3}{4} = \frac{11}{4}

. Two symbols side-by-side meaning "multiply" is an algebra habit that doesn't apply here.

4 \frac{3}{8} = \frac{4 + 3}{8} = \frac{7}{8}

The 4 wholes were never converted into eighths. First

4 = \frac{32}{8}

, then add:

\frac{32 + 3}{8} = \frac{35}{8}

. The rule is (whole × denominator) + numerator, all over the denominator.

\frac{2}{3} + \frac{1}{4} = \frac{3}{7}

— just add tops and bottoms."

You can only add counts when the pieces are the same size. Thirds and quarters differ, so you need a common denominator first:

\frac{8}{12} + \frac{3}{12} = \frac{11}{12}

. See Adding and subtracting fractions.

\frac{17}{5} = 3 \frac{2}{5}

, so I write it as

3 + \frac{2}{5} = \frac{5}{5} = 1

Nonsense arithmetic — you can't add a whole to a fraction by mashing the digits.

3 + \frac{2}{5}

stays

3 \frac{2}{5}

; converting back gives

\frac{17}{5}

, not

1

"To convert

\frac{7}{4}

to mixed, the quotient goes on the bottom:

\frac{1}{3}

The quotient is the whole part, and the remainder rides on top over the original denominator.

7 = 1 \cdot 4 + 3

, so

\frac{7}{4} = 1 \frac{3}{4}

, keeping the

4

\frac{6}{4}

is proper because

6

and

4

are both even and it simplifies."

Simplifying to

\frac{3}{2}

doesn't change its value, which is still greater than

1

— so it's improper either way. "Proper vs improper" is decided by value relative to one whole, not by whether it reduces.

3 \frac{2}{5}

converts to

\frac{3 \cdot 5 + 2}{2} = \frac{17}{2}

Wrong denominator underneath. The result must stay over the original denominator

5

, giving

\frac{17}{5}

. The bottom names the piece-size and mustn't change during conversion.

\frac{3}{- 4}

is a totally different fraction from

- \frac{3}{4}

Same value — a single minus sign can sit on top, on the bottom, or out front:

\frac{3}{- 4} = \frac{- 3}{4} = - \frac{3}{4}

. By convention we keep the denominator positive and put the sign in front for readability;

\frac{- 3}{- 4} = \frac{3}{4}

, because two negatives cancel.

Why questions

Why must the denominator be nonzero, but the numerator may be zero?

The denominator sets the piece-size by cutting a whole into

b

parts — cutting into

0

parts is meaningless. The numerator just counts, and counting zero pieces is perfectly fine (

= 0

Why does "improper" not mean "incorrect"?

It only describes shape: the numerator "overflows" past a full whole (

∣ a ∣ \geq ∣ b ∣

). The value is exact and legitimate; the word is historical, not a verdict on correctness.

Why does division-with-remainder give exactly the mixed number?

Writing

a = q b + r

(with

b > 0

) splits

a

into

q

full groups of

b

(the whole part) plus a leftover

r < b

(the numerator of the proper piece). Dividing through by

b

hands you

q + \frac{r}{b} = q \frac{r}{b}

Why can two very different-looking fractions be equal?

Scaling top and bottom by the same factor re-cuts the same amount into more, smaller pieces:

\frac{1}{2} = \frac{2}{4} = \frac{50}{100}

. These are Equivalent fractions — same point on the Number line.

Why do we need a common denominator to compare fractions but not to compare whole numbers?

Whole numbers already count in identical units. Fractions count in piece-sizes, and thirds vs fifths aren't directly comparable — you must first re-express both in one shared piece-size, then compare the counts.

Why is a fraction just a division waiting to happen?

\frac{a}{b}

literally means "

a

shared into

b

equal parts," which is

a \div b

. That's exactly why

\frac{a}{b}

can be rewritten as a decimal by carrying out the division.

Why does the sign of a fraction not care whether it sits on the top or the bottom?

A fraction is

a \div b

, and division follows the same sign rule as multiplication: one negative makes the result negative, two negatives cancel. So

\frac{- 3}{4}

\frac{3}{- 4}

and

- \frac{3}{4}

are all the same, while

\frac{- 3}{- 4} = \frac{3}{4}

Why does making a mixed number improper before multiplying make life easier?

A single

\frac{a}{b}

is one clean division; a mixed number is secretly a sum, and multiplying a sum (

2 + \frac{3}{4}

) forces you to distribute. Converting first avoids that hidden expansion.

Edge cases

Classify

\frac{9}{9}

: proper, improper, or mixed?

Improper (numerator

\geq

denominator) and it equals exactly

1

whole. It has no mixed form because the remainder

r

0

What is

\frac{0}{7}

as a mixed number?

Just

0

. The division

0 = 0 \cdot 7 + 0

gives quotient

0

and remainder

0

, so there's no whole part and no leftover.

\frac{4}{1}

a fraction, and is it proper or improper?

Yes — a denominator of

1

is allowed (one whole = one part). It equals

4

and is improper since

4 \geq 1

; every whole number is secretly an improper fraction over

1

Where does

- \frac{7}{4}

sit, and is it proper?

It's improper (

∣ - 7∣ \geq ∣4∣

) and lands between

- 2

and

- 1

on the Number line — its mixed form is

- 1 \frac{3}{4}

, i.e. one and three-quarters below zero.

\frac{- 3}{- 4}

positive or negative, and is it proper?

Positive — two negatives cancel, so it equals

\frac{3}{4}

, a proper fraction between

0

and

1

. The signs on top and bottom are not two separate minus signs on the value; they multiply.

Can a mixed number have a whole part of

0

Not usefully —

0 \frac{3}{4}

is just the proper fraction

\frac{3}{4}

. Mixed notation is only meaningful when there's at least one full whole to pull out.

What happens to

\frac{a}{b}

as the numerator

a

grows while

b

stays fixed?

The value climbs without bound: it passes

1

(becomes improper) at

a = b

, then

2

a = 2 b

, and so on — each whole reached exactly when

a

hits a new multiple of

b

Recall One-line self-test

Which single feature decides "proper vs improper"? ::: Whether the fraction's value is less than one whole ( $∣ a ∣ < ∣ b ∣$ ) or one whole and up ( $∣ a ∣ \geq ∣ b ∣$ ) — nothing about signs, simplifying, or looks.