Which is larger?
3/5, 0.62 or 61%?
The forms look different.
The comparison becomes easy once they are written on a common scale.
3/5 = 0.6 = 60%.
0.62 = 62%.
61% = 0.61.
So:
0.62 > 61% > 3/5.
Fractions, decimals and percentages are not separate kinds of numbers. They are different representations of the same quantities.
Conversion is therefore not mainly a memory task.
It is a representation task.
The quick answer
- fraction → decimal: divide numerator by denominator;
- decimal → percentage: multiply by 100%;
- percentage → decimal: divide by 100;
- percentage → fraction: write over 100 and simplify;
- decimal → fraction: write using place value and simplify.
But the most efficient route depends on the number.
Fraction to decimal: division reveals the decimal form
Convert 3/8.
3 ÷ 8 = 0.375.
Therefore:
3/8 = 0.375.
The fraction bar itself means division.
Fraction to percentage: sometimes direct scaling is faster
Convert 7/20 to a percentage.
Scale denominator to 100:
20×5=100.
7×5=35.
So:
7/20 = 35/100 = 35%.
No decimal step is needed.
Decimal to fraction: place value gives the denominator
Convert 0.375 to a fraction.
0.375 = 375/1000.
Simplify:
375/1000 = 3/8.
The denominator comes from decimal place value.
Percentage to fraction: percentages already have denominator 100
Convert 45%.
45% = 45/100.
Simplify:
45/100 = 9/20.
Benchmark conversions should become fluent
- 1/2 = 0.5 = 50%;
- 1/4 = 0.25 = 25%;
- 3/4 = 0.75 = 75%;
- 1/5 = 0.2 = 20%;
- 2/5 = 0.4 = 40%;
- 1/10 = 0.1 = 10%;
- 1/8 = 0.125 = 12.5%.
These relationships reduce later calculation load.
Conversion fluency is useful because it lets the learner choose the representation that makes the next step easiest.
Not every fraction has a terminating decimal
Consider 1/3.
1÷3 = 0.333…
The 3 repeats forever.
This decimal is recurring.
Writing 0.33 is only an approximation.
The exact value remains:
1/3.
Exactness matters
Suppose a calculation uses 1/3 repeatedly.
If 1/3 is replaced too early by 0.33, error enters immediately.
Example:
3×1/3 = exactly 1.
But:
3×0.33 = 0.99.
The approximation has changed the result.
Keep exact fractions where they are convenient, and round only when the problem requires an approximation.
When does a fraction terminate as a decimal?
After a fraction is simplified, its decimal terminates exactly when the denominator contains no prime factors other than 2 and 5.
Why?
Decimals are fractions with denominator powers of 10.
10 = 2×5.
So denominators built only from 2s and 5s can be scaled to a power of 10.
Examples:
- 3/8, denominator 2³ → terminates;
- 7/20, denominator 2²×5 → terminates;
- 1/6, denominator 2×3 → recurring because of factor 3;
- 2/7 → recurring.
Worked example: compare unlike forms
Order from smallest to largest:
5/8, 0.61, 63%.
Convert to decimals:
- 5/8 = 0.625;
- 0.61 = 0.61;
- 63% = 0.63.
Order:
0.61 < 5/8 < 63%.
Choose a common form that minimises work
To compare 1/4 and 30%, converting 1/4 to 25% is easy.
To compare 7/8 and 0.86, converting 7/8 to 0.875 is natural.
To add 1/3 and 0.25 exactly, converting 0.25 to 1/4 is better than approximating 1/3 as a decimal.
Representation choice should follow the next mathematical job.
Percentage change is not the same as percentage points
A rate rises from 40% to 50%.
The increase is:
10 percentage points.
Relative percentage increase:
(50−40)/40 = 10/40 = 25%.
These are different statements.
Fractions often preserve ratios exactly
If 2 out of every 3 items satisfy a condition:
fraction = 2/3.
percentage ≈ 66.666…%.
The fraction is exact and compact.
The percentage may be more communicative to a general audience but requires rounding.
Neither representation is universally superior.
Decimals are strong for place-value calculation
Money, measurements and calculators often make decimal representation convenient.
Example:
$4.75 + $2.30 = $7.05.
The decimal places align naturally with cents.
Percentages are strong for relative comparison
Class A: 18 out of 24 students succeed.
Class B: 28 out of 40 succeed.
Raw successes cannot be compared fairly because class sizes differ.
Class A:
18/24 = 75%.
Class B:
28/40 = 70%.
The percentage creates a common “out of 100” scale.
Reverse conversions test understanding
If 0.48 of a quantity is known:
0.48 = 48% = 12/25.
If 12/25 of a whole is 84:
1/25 = 84÷12 = 7.
Whole = 25×7 = 175.
Moving among forms can reveal a shorter solving route.
Calculator displays can hide exact values
A calculator may display:
0.666666667.
This does not mean the exact number is a terminating decimal with nine places.
It may be a rounded display of 2/3.
Students should distinguish display precision from mathematical exactness.
Worked example: exact arithmetic across forms
Evaluate exactly:
1/3 + 0.25.
Convert 0.25 to 1/4.
1/3 + 1/4 = 4/12 + 3/12 = 7/12.
If a decimal approximation is later required:
7/12 ≈ 0.5833.
Exact first; approximate later.
Common misconception 1: multiply by 100 changes the number
When converting 0.35 to 35%, multiplying the decimal number by 100 is paired with the percent unit, which means “per hundred”.
0.35 and 35% represent the same quantity.
Common misconception 2: every decimal is exact
A written decimal may be exact or rounded.
Context matters.
Common misconception 3: percentage can exceed 100% only by mistake
Percentages above 100% are valid when a quantity exceeds the reference whole.
1.5 = 150%.
Common misconception 4: a larger denominator means a larger fraction
For unit fractions, a larger denominator means smaller equal parts.
1/8 < 1/4.
Common misconception 5: convert everything to decimals
Recurring decimals can destroy exactness or make algebra harder.
Choose the form that best supports the next step.
A diagnostic ladder
- Can the learner connect common benchmark fractions, decimals and percentages?
- Can the learner convert a terminating fraction to a decimal?
- Can the learner convert decimals to simplified fractions?
- Can the learner convert percentages to fractions?
- Can the learner compare mixed forms efficiently?
- Can the learner preserve exact fractions when decimals recur?
- Can the learner identify terminating versus recurring decimal structure?
- Can the learner distinguish exact values from rounded displays?
- Can the learner choose the most useful representation for the next operation?
- Can the learner distinguish percentage change from percentage points?
How this fits Secondary Mathematics
Secondary Mathematics expects increasing fluency in moving among numerical representations. These forms support proportional reasoning, finance, probability, statistics, algebra and measurement.
Exact subject-level scope should be checked against the current MOE/SEAB Mathematics syllabus for the learner’s cohort.
The deeper lesson: representation should serve the mathematics
Fractions preserve exact ratios elegantly.
Decimals align with place value and measurement.
Percentages normalise comparisons to a shared hundred-scale.
Fluency is not converting everything into one favourite form. It is recognising that the number stays the same while the representation changes—and choosing the form that reveals the next relationship most clearly.