Equivalent fractions, simplifying fractions: Question bank

Before we start, one word we will lean on again and again. A factor of a number is a whole number that divides it exactly (no remainder) — for the machinery of factors and the GCD, keep that note open. A term is a chunk being added (the $2$ and $3$ in $2 + 3$ ); a factor is a chunk being multiplied (the $2$ and $3$ in $2 \times 3$ ). That single distinction is behind half the traps here.

True or false — justify

Every fraction has infinitely many equivalent forms.

True — multiply top and bottom by

k = 2, 3, 4, \dots

endlessly; each

\frac{k}{k} = 1

leaves the value untouched, so the list never ends.

A fraction in lowest terms has exactly one equivalent form that is also in lowest terms.

True (up to sign) — dividing out the whole GCD lands you at a unique simplest pair, e.g. every form of one-half collapses to

\frac{1}{2}

\frac{0}{5}

and

\frac{0}{9}

are equivalent.

True — both mean "take

0

of the pieces", which is

0

regardless of how the whole is cut; cross-multiply gives

0 \times 9 = 5 \times 0

\frac{5}{0}

can be simplified to

\frac{1}{0}

False — a

0

denominator means "cut into

0

equal parts", which is undefined, so

\frac{5}{0}

is not a number to simplify in the first place.

If two fractions have the same numerator, the one with the bigger denominator is smaller.

True — the same number of pieces cut from a whole into more parts makes each piece tinier, so

\frac{3}{8} < \frac{3}{5}

Simplifying a fraction makes it a smaller amount.

False — it changes only the labels (numerator and denominator), never the value;

\frac{18}{24}

and

\frac{3}{4}

are the identical amount.

Multiplying top and bottom by a negative number gives an equivalent fraction.

True —

\frac{k}{k} = 1

holds for any non-zero

k

, including negatives, so

\frac{3}{5} = \frac{- 3}{- 5}

; the two minus signs cancel.

A whole number like

7

can be written as a fraction.

True —

7 = \frac{7}{1}

(cut the whole into

1

part, take

7

wholes' worth); every integer is a rational number.

Cross-multiplication (

a d = b c

) only works when both fractions are already simplified.

False — it tests equality of value directly and works for any fractions;

\frac{6}{9}

and

\frac{10}{15}

pass (

90 = 90

) though neither is simplified.

Spot the error

Find the flaw: "

\frac{2}{4} = \frac{2 + 3}{4 + 3} = \frac{5}{7}

, since I did the same thing to both."

You may only multiply or divide both by the same number, because that hides a factor of

\frac{k}{k} = 1

; adding shifts the ratio —

\frac{2}{4} = 0.5

but

\frac{5}{7} \approx 0.71

Find the flaw: "

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

— I cancelled the

2

s."

The

2

is a term being added, not a factor multiplying the whole top and bottom, so it cannot be cancelled; the fraction is

\frac{5}{7} \neq = \frac{3}{5}

Find the flaw: "

\frac{18}{24} = \frac{9}{12}

, done — it's simpler."

Simpler is not simplest; lowest terms needs

g cd = 1

, but

g cd (9, 12) = 3

, so keep dividing to reach

\frac{3}{4}

Find the flaw: "To write

\frac{3}{7}

over

28

I multiply the bottom by

4

\frac{3}{28}

You must multiply the top too or the value changes; correct is

\frac{3 \times 4}{7 \times 4} = \frac{12}{28}

Find the flaw: "

\frac{6 x}{9 x} = \frac{6}{9}

because I cancelled the

x

, but

\frac{6}{9}

won't simplify further."

Cancelling

x

was valid (it is a common factor), but

g cd (6, 9) = 3

, so it does simplify to

\frac{2}{3}

Find the flaw: "

\frac{45}{60}

ends in

5

and

0

, so I divide top by

5

and bottom by

10

You must divide both by the same number; using

5

and

10

breaks the

\frac{k}{k} = 1

rule and changes the value.

Find the flaw: "

\frac{4}{8} = \frac{1}{2}

, and

\frac{4}{8}

also equals

\frac{2}{6}

since I halved the top and the difference."

Equivalence comes from dividing both by the same factor (

4 \div 4, 8 \div 4

), not from tinkering with differences;

\frac{2}{6} = \frac{1}{3} \neq = \frac{1}{2}

Why questions

Why does multiplying top and bottom by

k

leave the value unchanged?

Because

\frac{k}{k} = 1

and multiplying any number by

1

never changes it — you are re-cutting each piece into

k

smaller ones, gaining pieces and cuts in step.

Why does dividing by the GCD finish the job in one move, while dividing by a smaller common factor may not?

The GCD is the largest shared factor, so nothing common is left afterward; a smaller factor leaves some shared factor behind, forcing more steps.

Why can we not have a

0

denominator, but a

0

numerator is fine?

The denominator sets the size of each piece — "cut into

0

parts" is meaningless — while the numerator only counts pieces, and taking

0

pieces is a perfectly good amount (

0

Why does cross-multiplication decide equivalence?

Starting from

\frac{a}{b} = \frac{c}{d}

and multiplying both sides by

b

and

d

gives

a d = b c

, and every step reverses, so equality of the fractions and

a d = b c

stand or fall together.

Why is Prime factorisation a reliable route to lowest terms?

Writing top and bottom as products of primes exposes exactly the shared prime factors; cancel each shared prime once and what remains has

g cd = 1

automatically.

Why does making denominators equal (via equivalent fractions) come before adding fractions?

You can only count pieces of the same size; equivalent fractions re-cut both fractions to a common piece size so the numerators can simply be added.

Why is a simplified fraction the same idea as a simplified ratio?

Both strip out the common factor to name a relationship in the smallest whole numbers;

\frac{6}{9}

and the ratio

6 : 9

both reduce by

3

\frac{2}{3}

and

2 : 3

Edge cases

Simplest form of

\frac{0}{7}

It is

\frac{0}{1}

(just

0

); the numerator is

0

, and

g cd (0, 7) = 7

, so dividing both by

7

gives

\frac{0}{1}

Simplest form of

\frac{7}{7}

It is

\frac{1}{1} = 1

; the whole is taken completely, and

g cd (7, 7) = 7

divides both down to

1

over

1

\frac{- 3}{5}

the same as

\frac{3}{- 5}

Yes — both equal

- \frac{3}{5}

; a single minus sign can sit on top, on the bottom, or in front, since only the number of minus signs matters.

\frac{9}{6}

(an improper fraction, top bigger than bottom) allowed to be simplified?

Yes — the same rule applies regardless of size;

g cd (9, 6) = 3

gives

\frac{3}{2}

, which is one-and-a-half wholes.

What is the simplest form of a fraction whose numerator and denominator are already coprime, like

\frac{4}{9}

It is already in lowest terms —

g cd (4, 9) = 1

, so no common factor exists to divide out, and nothing changes.

Can

\frac{1}{1000000}

be simplified?

No —

g cd (1, 1000000) = 1

, and a numerator of

1

shares no factor above

1

with anything, so every such fraction is already simplest.

Recall One-line survival kit

Same to both, times or divide only. Denominator never $0$ . Cancel factors, never terms. Lowest terms means $g cd = 1$ — check by simplifying again and confirming nothing moves.

Connections

Equivalent fractions, simplifying fractions — the parent this bank drills.
Factors, multiples and GCD/HCF — the GCD decides "how far to simplify".
Prime factorisation — exposes shared factors cleanly.
Adding and subtracting fractions — needs equivalent fractions first.
Ratios and proportion — same cancel-the-common-factor idea.
Decimals and percentages — equivalent forms of one value.
Number Systems – rational numbers — every rational is a fraction in lowest terms.