The core idea
A fraction records division, a decimal uses place value, and a percentage expresses a quantity per hundred. They can represent the same number. Correct calculations preserve the value, identify the reference whole, and keep units attached to measured quantities.
1. Start with the whole
In 3/8, the slash means division: three divided by eight. The top number, 3, is the numerator; the bottom number, 8, is the denominator. If a strip is divided into eight equal lengths, three lengths represent 3/8 of that strip. Equal parts matter. Three pieces chosen from eight unequal pieces do not necessarily make three-eighths of the total length. A denominator cannot be zero because division by zero is undefined.
The whole is a reference, not always one physical object. If the reference collection contains 24 pencils, 3/8 of it is 9 pencils. A fraction may exceed one: 11/8 means eleven eighths, or 1 whole and 3/8. Comparing half a small sheet with half a large sheet requires their original sizes; equal fractions of different wholes need not be equal amounts.
Sources: NCERT Class VII Mathematics Exemplar: Fractions and Decimals ↗ · OpenStax: Prealgebra 2e: Visualize Fractions ↗
2. Change the representation without changing the amount
Multiplying the numerator and denominator by the same nonzero number creates an equivalent fraction. Splitting each eighth into two makes 3/8 become 6/16. Dividing both by a common factor reverses that process: 18/24 becomes 3/4 after division by 6. The fraction bar groups the whole numerator and denominator, so changing just one of them changes the value.
To compare 5/6 and 7/9, describe both using eighteenths: 5/6 = 15/18 and 7/9 = 14/18. Fifteen equal pieces exceed fourteen, so 5/6 is larger by 1/18. Comparing only numerators would have suggested the opposite. On a number line, equivalent fractions share one point; the line helps distinguish a new name from a new number.
3. Decimals are fractions organised by place value
The first three places after the decimal point represent tenths, hundredths and thousandths. Thus 0.375 means 375/1000, which simplifies to 3/8. To go the other way, divide 3 by 8. Adding a zero at the right end does not change a decimal: 0.6 = 0.60. Therefore 0.6 exceeds 0.58 because sixty hundredths exceed fifty-eight hundredths.
Some decimal forms continue indefinitely. One-third is 0.333…; the dots mean the digit continues. Writing 0.33 is an approximation, not an exact replacement. The symbol ≈ means approximately equal. Three amounts of 1/3 total exactly 1, while three amounts rounded first to 0.33 total 0.99. Keep exact fractions during calculation when practical, and round the final result to a precision appropriate for the task.
4. Solved example: combine lengths
An illustrative craft task uses 2/3 metre of ribbon for one decoration and 3/4 metre for another. A metre is the length unit; both measurements must use the same unit. Thirds and quarters have different sizes, so convert to twelfths before adding: 2/3 = 8/12 and 3/4 = 9/12. The total is 17/12 metres, or 1 and 5/12 metres. Adding the denominators would incorrectly give 5/7.
If the starting ribbon is 2 metres, write that as 24/12 metres. The remainder is 24/12 − 17/12 = 7/12 metre. Check by adding the used and remaining lengths: 17/12 + 7/12 = 24/12 = 2. An estimate also helps: both pieces exceed half a metre, so their total must exceed one metre.
Multiplication answers a different question: taking a fraction of a quantity. Two-thirds of three-fifths means divide three-fifths into three equal groups and take two, giving 2/3 × 3/5 = 6/15 = 2/5. Division can ask how many pieces fit. A ribbon of length 3/4 metre contains six pieces of length 1/8 metre because 3/4 = 6/8. Thus (3/4) ÷ (1/8) = 6. This explains why division by a positive fraction smaller than one can increase the numerical answer: you are counting smaller units, not making the ribbon longer.
Sources: OpenStax: Prealgebra 2e: Add and Subtract Fractions with Different Denominators ↗ · NCERT Class VII Mathematics Exemplar: Fractions and Decimals ↗
5. Percent means per hundred
The symbol % means per hundred: 37.5% = 37.5/100 = 0.375 = 3/8. To calculate a percentage of an amount, multiply that amount by the decimal form of the percentage. For 15% of 240 pages, calculate 0.15 × 240 = 36 pages. Here × means multiplication. A mental check separates 15% into 10% and 5%: 24 + 12 = 36.
To find what percentage one quantity is of another, divide the part by the reference whole, then multiply by 100%. Eighteen completed tasks out of 24 give 18/24 × 100% = 75%. This describes completion only if all 24 tasks belong in the count. A percentage can exceed 100% when an amount exceeds its reference, such as 30 books compared with a target of 20.
One amount, three ways to write it
3/8 = 0.375 = 37.5%
6. Solved example: identify the starting value
An illustrative club has 80 members and later 92. The increase is 12 members. Percentage increase uses the original value as its reference: 12/80 × 100% = 15%. Dividing by 92 answers a different question: what fraction of the final membership the increase represents. Always finish the phrase “percent of…” before choosing the denominator.
Now suppose a score rises from 40% to 50%. That is an increase of 10 percentage points, but the relative increase is 10/40 × 100% = 25%. The two statements use different comparisons. Similarly, reducing 100 by 20% gives 80; increasing 80 by 20% gives 96. The percentage is unchanged, but the reference amount has changed, so the operations do not cancel.
Sources: OpenStax: Prealgebra 2e: Solve General Applications of Percent ↗
7. Choose a form that makes the reasoning clear
Use fractions when equal parts or exact division matter, decimals when place-value calculation is convenient, and percentages when comparing proportions with clearly stated wholes. These are choices of representation, not competing kinds of truth. Before trusting an answer, check its unit, likely size and reference quantity. A claim that 25% of 60 is 240 fails because a quarter of a positive amount must be smaller than the whole.
Sources: OpenStax: Prealgebra 2e: Decimals and Fractions ↗ · OpenStax: Prealgebra 2e: Understand Percent ↗
PUT IT INTO PRACTICE
Plan an illustrative reading week
- Start with a 160-page book. Plan to read 3/8 of it on day one and 25% on day two. Convert both targets to page counts, showing the common reference whole.
- Calculate the total completed, the fraction still unread, and the percentage still unread. Check that completed and unread pages add to 160.
- Explain the result: 60 + 40 = 100 pages completed, leaving 60 pages, or 3/8 = 37.5%. Then calculate 25% of the remaining 60: it is 15 pages, illustrating why the reference matters.
Check your understanding
Which is larger, 0.45 or 4/9? Explain without rounding both to one decimal place.
0.45 = 45/100. Compare 45 × 9 = 405 with 4 × 100 = 400, so 0.45 is larger.
A 2-litre jug is 3/4 full. How much liquid does it contain?
It contains 2 × 3/4 = 1.5 litres. The fraction refers to the jug’s 2-litre capacity.
Attendance grows from 30 to 36. What is the percentage increase?
The increase is 6, so 6/30 × 100% = 20%, using the starting attendance.
Why is 1/4 + 1/4 not 2/8?
The pieces remain quarters. Two quarters give 2/4 = 1/2; 2/8 would equal only one quarter.
Two groups complete 80% of 10 tasks and 60% of 20 tasks. Which completes more tasks?
The first completes 8 and the second 12. The higher percentage does not imply the higher count when totals differ.