Percentages and conversions

Percentage as a ratio out of 100. Whenever a problem compares a "part" to a "whole," the percentage form of that comparison is:

where: part is the quantity being measured, whole is the total it is measured against, and the result is expressed with a $\%$ sign meaning "divided by 100."

Fraction and decimal are the same idea in different clothes. Since $\%$ always means $\div 100$, converting between the three forms is nothing but multiplying or dividing by 100 in a different direction:

Fraction to percentage: multiply by 100 and simplify. Percentage to fraction: write over 100 and simplify. Decimal to percentage: shift the decimal point two places right. Percentage to decimal: shift the decimal point two places left. Because these are all the same value, you can compute in whichever form is most convenient and convert back only at the end.

Benchmarks worth memorising, so you recognise them instantly instead of recalculating every time: $\frac{1}{2}=50\%$, $\frac{1}{4}=25\%$, $\frac{1}{5}=20\%$, $\frac{1}{10}=10\%$, $\frac{1}{3}=33\frac{1}{3}\%$, and $\frac{3}{4}=75\%$.

Why bother rescaling to 100 at all? Because a common denominator turns comparison into a single glance. Two fractions with different denominators — say $\frac{9}{12}$ and $\frac{15}{20}$ — cannot be compared by eye; you would normally need a common denominator of 60 to line them up. Percentages fix the denominator at 100 once and for all, so every ratio, no matter what its original whole was, lands on the same 0-to-100 ruler. That is precisely why exam scores from a 50-mark test and a 80-mark test can be compared directly once both are written as percentages, and why a shop's "30% off" sign means the same discount whether the item costs ₹200 or ₹2,000.

In a class of 40 students, 26 walk to school. Express this as a percentage, a fraction in simplest form, and a decimal, and verify all three agree.

Step 1. Identify part and whole: part $= 26$ (students who walk), whole $= 40$ (total students).

Step 2. Apply the percentage formula:

Step 3. Simplify before multiplying: $\dfrac{26}{40} = \dfrac{13}{20}$, so $\dfrac{13}{20} \times 100 = 13 \times 5 = 65$.

Step 4. So $65\%$ of the class walks to school, and as a fraction this is $\dfrac{13}{20}$ — already simplest form, since 13 is prime and does not divide 20.

Step 5. Convert to a decimal: $65\% = 0.65$. As a check, multiply back: $0.65 \times 40 = 26$, matching the original count exactly, so the percentage, fraction, and decimal all agree.

Competitive stretch

Convert $\dfrac{7}{8}$ into a percentage and a decimal in one pass. Multiply by 100: $\dfrac{7}{8} \times 100 = \dfrac{700}{8} = 87.5$, so $\dfrac{7}{8} = 87.5\%$. Dividing 87.5 by 100 gives the decimal directly: $87.5\% = 0.875$. Answer: $\dfrac{7}{8} = 87.5\% = 0.875$, a reminder that percentages do not have to come out as whole numbers.

• Forgetting to simplify the fraction before or after converting — always reduce to lowest terms as a final check.
• Shifting the decimal point the wrong way, turning $0.6$ into $0.6\%$ instead of $60\%$.
• Assuming the "whole" is always literally 100 objects — the whole is whatever total the problem gives, and the percentage formula rescales it to 100, it does not require it to already be 100.
• Treating $\%$ as a free-standing number: $25\%$ is not the number 25, it is $\frac{25}{100}$, so it cannot be used in a calculation without first converting.
• Rounding too early in a multi-step conversion (e.g., rounding $87.5\%$ to $88\%$ before finding the decimal), which drags a small error into the rest of the problem.
• Mixing up "percentage of a number" with "percentage change" — the formula $\frac{\text{part}}{\text{whole}} \times 100$ only applies when you already have a fixed part and a fixed whole, not when a quantity is increasing or decreasing (that is a later topic in the same chapter).

Ask the tutor: if you are still unsure which way to shift the decimal point, or want more practice turning tricky fractions into percentages, ask your AI tutor to walk through a few extra conversions with you.