Comparing things using percentages

Two numbers, by themselves, may not be easy to compare meaningfully. "$3$ marks out of $5$" vs "$80$ marks out of $100$" , which is better? Translating to percentages reveals $60\%$ vs $80\%$ instantly. This is the silent superpower of percent: it strips out the "out of how many" and lets us compare like with like.

Concept

Comparing test scores. Marks out of different totals must always be converted to a common base (typically $100$).

Example: Roll-A scored $42/50$ in maths, $56/80$ in science. Which subject did better?

• Maths: $\dfrac{42}{50} \times 100 = 84\%$.
• Science: $\dfrac{56}{80} \times 100 = 70\%$.
• Maths is the stronger subject.

Comparing population growth. Two cities of very different sizes can be compared by percent growth.

• City A: $50{,}000 \to 55{,}000$. Growth $= 5000/50000 = 10\%$.
• City B: $200{,}000 \to 210{,}000$. Growth $= 10{,}000/200{,}000 = 5\%$.

City A grew faster (in percent) even though City B added twice as many people.

Comparing prices. Same goods at different shops, with different markups.

• Shop X: cost $\mathbb{?}\,200$, SP $\mathbb{?}\,250$. Markup $= 25\%$.
• Shop Y: cost $\mathbb{?}\,500$, SP $\mathbb{?}\,575$. Markup $= 15\%$.

Shop Y has a smaller markup percentage even though the absolute markup is larger.

Performance over time. A bowler's strike rate, a batsman's average, a player's win percent are all percent-based summaries that allow fair comparison across different match counts.

Spotting "misleading" percentages. Some percentages are easy to mis-read.

• "$50\%$ of doctors recommend brand X" , out of how many doctors? If only $4$ doctors were asked, "$50\%$" is just $2$ people, not very compelling.
• "Sales grew $200\%$" sounds impressive but starting from $1$ unit means new sales of $3$ , modest.
• "$0\%$ side effects" , until you read the fine print: "in our $10$-person trial".

Always ask: percent of what? And how many people / items / rupees does that percent represent in absolute terms?

Worked examples

Example 1. Two students scored: Aisha $35/40$, Bina $48/60$. Who did better?

• Aisha: $87.5\%$. Bina: $80\%$. Aisha did better.

Example 2. Town A grew from $25{,}000$ to $28{,}000$. Town B from $80{,}000$ to $84{,}000$. Which grew faster in percent?

• A: $3000/25000 = 12\%$.
• B: $4000/80000 = 5\%$.
• Town A grew faster.

Example 3. A cricketer scored $510$ runs in $15$ innings of which $5$ were not out. Find her batting average (runs per completed innings).

• Completed innings $= 15 - 5 = 10$.
• Average $= 510 / 10 = 51$.

Example 4. Of $500$ students in a school, $42\%$ are girls. How many boys?

• Girls: $0.42 \times 500 = 210$. Boys: $500 - 210 = 290$.

Try it yourself

1. Riya scored $72/90$ in Hindi and $88/100$ in English. Which is her stronger subject?
2. India's population grew from $1.21$ to $1.39$ billion. Find the percent increase.
3. Two shops: X sells a bag for $\mathbb{?}\,540$ marked $\mathbb{?}\,600$; Y sells for $\mathbb{?}\,720$ marked $\mathbb{?}\,800$. Compare discount %.
4. A team won $14$ of $20$ matches. Their win percent is?
5. $35\%$ of the $400$ people in a hall are women. How many men?
6. Compare: A scored $25$ out of $30$. B scored $42$ out of $50$.
7. A bowler took $22$ wickets in $32$ overs vs $16$ in $24$. Compare wickets per over.
8. A class of $48$ has $30\%$ boys. How many boys?

Activity / Insight

Read the headline. Find a recent news headline that quotes a percent ("$+30\%$ growth", "$50\%$ off", "$80\%$ vaccinated"). Try to find the absolute numbers behind it. Compute the percent yourself. Often the headline either rounded heavily or omitted a key denominator , and discovering this is one of the most important real-world lessons of this chapter.