Converting between Fractions and Percentages

\( \frac{3}{5} \) means 3 ÷ 5 = 0.6, and 0.6 × 100 = 60. So \( \frac{3}{5} \) = 60%. Alternatively, find an equivalent fraction with denominator 100: multiply numerator and denominator by 20 to get \( \frac{60}{100} \), which is 60% by definition (“per cent” means “per hundred”).

You can also think of it as: \( \frac{1}{5} \) = 20%, so \( \frac{3}{5} \) = 3 × 20% = 60%. All three methods give the same answer, reinforcing that the conversion is about expressing the same proportion out of 100.

If \( \frac{1}{3} \) were exactly 33%, then three thirds would equal 3 × 33% = 99%, not 100%. But we know that \( \frac{1}{3} + \frac{1}{3} + \frac{1}{3} = 1 \) = 100%. So 33% must be too small. The exact percentage is 33.333…% (or 33⅓%), a recurring decimal that never terminates.

This addresses the “rounding equals exact” misconception. 33% = \( \frac{33}{100} \), which is slightly less than \( \frac{1}{3} \). Students who use 33% as an exact equivalent will get inaccurate answers — for example, “find \( \frac{1}{3} \) of 300” gives 100, but “find 33% of 300” gives 99.

\( \frac{7}{4} \) = 7 ÷ 4 = 1.75, and 1.75 × 100 = 175. Equivalently, \( \frac{7}{4} = \frac{175}{100} \) (multiply numerator and denominator by 25). Since \( \frac{7}{4} \) is greater than 1 whole, the percentage must be greater than 100%.

You can break it down: \( \frac{4}{4} \) = 100% and \( \frac{3}{4} \) = 75%, so \( \frac{7}{4} \) = 100% + 75% = 175%. Percentages over 100% are perfectly valid — they simply mean “more than the whole.” A test score of 175% would mean scoring 1.75 times the base amount.

45% means 45 out of 100, so 45% = \( \frac{45}{100} \). The highest common factor of 45 and 100 is 5. Dividing both by 5: 45 ÷ 5 = 9 and 100 ÷ 5 = 20, giving \( \frac{9}{20} \).

To verify, convert back: 9 ÷ 20 = 0.45, and 0.45 × 100 = 45%. ✓ The key insight is that “per cent” literally means “per hundred,” so any percentage immediately becomes a fraction with denominator 100 — the only remaining task is simplifying by the HCF.

The whole number 2 represents 2 wholes, which is 200%. The fractional part \( \frac{1}{4} \) is 25%. Adding them together gives 200% + 25% = 225%.

Alternatively, convert the mixed number to an improper fraction: \( 2\frac{1}{4} = \frac{9}{4} \). Since \( \frac{1}{4} \) = 25%, \( \frac{9}{4} \) = 9 × 25% = 225%.

Example: \( \frac{2}{5} \) (since 2 ÷ 5 = 0.4 = 40%)

Another: \( \frac{4}{10} \) (since 4 ÷ 10 = 0.4 = 40%)

Creative: \( \frac{6}{15} \) (since 6 ÷ 15 = 0.4) or \( \frac{40}{100} \) — less obvious equivalent fractions that students rarely consider. Also \( \frac{0.8}{2} \) using a decimal numerator.

Trap: \( \frac{1}{40} \) — a student sees the “40” and places it in the denominator, thinking “40 goes on the bottom.” But \( \frac{1}{40} \) = 0.025 = 2.5%, not 40%. The 40 in “40%” refers to 40 out of 100, not 1 out of 40.

Example: \( \frac{1}{2} \) = 50%

Another: \( \frac{1}{4} \) = 25%

Creative: \( \frac{1}{50} \) = 2% — students rarely think of larger denominators like 50 as giving neat whole number percentages. Or \( \frac{1}{25} \) = 4%.

Trap: \( \frac{1}{3} \) — many students believe \( \frac{1}{3} \) = 33%, but the true value is 33.333…% (recurring), which is not a whole number percentage. The denominator 3 is not a factor of 100, which is why it doesn’t convert cleanly.

Example: 12.5% = \( \frac{1}{8} \)

Another: 37.5% = \( \frac{3}{8} \)

Creative: 87.5% = \( \frac{7}{8} \) — using the complement (100% − 12.5% = 87.5%). Or 62.5% = \( \frac{5}{8} \).

Trap: 8% — a student sees “8” and assumes the denominator must be 8. But 8% = \( \frac{8}{100} = \frac{2}{25} \). The denominator in simplest form is 25, not 8. Having “8” as the percentage does not mean 8 appears in the denominator.

Example: \( \frac{3}{2} \) = 150%

Another: \( \frac{5}{4} \) = 125%

Creative: \( \frac{11}{10} \) = 110% — just barely exceeding 100%, showing that improper fractions can be very close to 1 while still exceeding it. Or \( \frac{200}{100} \) = 200%.

Trap: \( \frac{99}{100} \) = 99% — a student might think “99 is nearly 100, so this must be over 100%.” But \( \frac{99}{100} \) is a proper fraction (numerator < denominator), so its percentage is less than 100%. For the percentage to exceed 100%, the numerator must be strictly greater than the denominator.

It depends on the value of n. When n > 2, the unit fraction is less than \( \frac{1}{2} \) and so its percentage is below 50%: for example, \( \frac{1}{4} \) = 25% and \( \frac{1}{5} \) = 20%.

But when n = 2, we get \( \frac{1}{2} \) = 50% (not less than 50%), and when n = 1, we get \( \frac{1}{1} \) = 100%. Students often assume unit fractions are always “small” because the numerator is 1 — but the denominator determines the size.

If two fractions produce the same percentage, they must have the same value (since the percentage is just the fraction’s value × 100). Fractions with the same value are, by definition, equivalent. For example, \( \frac{1}{2} \) and \( \frac{3}{6} \) both give 50% — and they are equivalent since \( \frac{3}{6} \) simplifies to \( \frac{1}{2} \).

Students may think two “different-looking” fractions could coincidentally give the same percentage without being related. But the conversion is deterministic: same value in means same value out.

It depends on whether the denominator (in simplest form) is a factor of 100. If it is, the percentage will be a whole number: \( \frac{1}{4} \) = 25%, \( \frac{3}{5} \) = 60%, \( \frac{7}{20} \) = 35%. If it isn’t, the percentage will be a decimal or recurring: \( \frac{1}{3} \) = 33.333…%, \( \frac{1}{7} \) = 14.2857…%, \( \frac{3}{8} \) = 37.5%.

The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. Fractions with these denominators (in simplest form) give whole number percentages; all others do not. Fractions like \( \frac{1}{8} \) or \( \frac{1}{40} \) might look like they should yield whole numbers because 8 and 40 are “neat” even numbers, but because they do not divide perfectly into 100, they yield decimals (12.5% and 2.5%).

When the numerator exceeds the denominator, the fraction is greater than 1, so its percentage must exceed 100%. For example, \( \frac{5}{4} \) = 125% and \( \frac{3}{2} \) = 150%. A percentage of exactly 100% corresponds to numerator = denominator (e.g. \( \frac{4}{4} \) = 100%), and any percentage below 100% corresponds to numerator < denominator.

This addresses the common belief that percentages “can’t go above 100” — in fact, any improper fraction converts to a percentage greater than 100%.

The student has multiplied the numerator by the denominator instead of dividing. A fraction means “numerator ÷ denominator,” not “numerator × denominator.” The correct calculation is 3 ÷ 8 = 0.375, then 0.375 × 100 = 37.5%.

This error sometimes goes undetected because the resulting percentage (24%) looks plausible — it’s a reasonable-sounding number between 0 and 100. A quick sense check helps: \( \frac{3}{8} \) is a bit less than half (\( \frac{4}{8} \)), so the percentage should be a bit less than 50%. 24% is far too small.

The answer is correct, but the reasoning contains a flawed general rule. The student claims “you always divide by the percentage number” — this happens to work for 50% because the HCF of 50 and 100 is 50, which coincidentally equals the percentage.

But applying this rule to other percentages fails immediately. For 30%: the student would divide 30 and 100 both by 30, giving \( \frac{1}{3.33\ldots} \) — not a valid fraction. The correct method is to divide by the HCF of the numerator and 100. For 30%, HCF(30, 100) = 10, giving \( \frac{30}{100} = \frac{3}{10} \). The student’s rule only works when the percentage itself happens to equal the HCF, which is rare.

The student correctly calculated 3 ÷ 20 = 0.15 but then applied the wrong conversion. The error is moving the decimal point one place instead of two — confusing “multiply by 10” with “multiply by 100.” Converting a decimal to a percentage requires multiplying by 100 (moving two places right), not by 10.

The correct answer is 0.15 × 100 = 15%. A sense check confirms: \( \frac{3}{20} \) is a bit less than \( \frac{4}{20} = \frac{1}{5} \) = 20%, so the percentage should be close to but less than 20%. The student’s answer of 1.5% is far too small — that would mean \( \frac{3}{20} \) is roughly equal to \( \frac{1}{100} \), which is clearly wrong.

The student has divided the numerator and denominator by different numbers — the numerator by 5 and the denominator by 10. When simplifying a fraction, you must divide both the numerator and denominator by the same number to maintain equivalence.

The correct simplification: \( \frac{35}{100} \) — divide both by 5 (the HCF) to get \( \frac{7}{20} \). Checking: 7 ÷ 20 = 0.35 = 35% ✓. The student’s answer of \( \frac{7}{10} \) = 70%, which is double the original percentage. Dividing top and bottom by different numbers changes the value of the fraction — it’s no longer equivalent.

The student hasn’t simplified the fraction to integers. A fraction is strictly a ratio of whole numbers, so leaving a decimal like 12.5 in the numerator is incorrect.

To fix this, they need to find an equivalent fraction with whole numbers. Multiplying the numerator and denominator by 10 gives \( \frac{125}{1000} \). From there, dividing both by 125 gives the simplest form, \( \frac{1}{8} \). Alternatively, multiplying the top and bottom of \( \frac{12.5}{100} \) by 2 immediately gives \( \frac{25}{200} \), which simplifies to \( \frac{1}{8} \).