Building intuition

We end the chapter with a small collection of percentage misconceptions, traps, and clever solutions. The aim is not new theory , it is to make sure the theory sticks in everyday life.

Concept

Trap 1: Adding percents that come from different bases.

"In our class, $30\%$ play cricket and $40\%$ play football. So $70\%$ play one or the other?" Wrong , unless nobody plays both. If $10\%$ play both, then by Venn-diagram thinking, $30 + 40 - 10 = 60\%$ play at least one.

Trap 2: "$200\%$ more" vs "$200\%$ of".

"Sales of $200\%$ more" means new is $300\%$ of old (you add the original $100\%$ to the $200\%$ increase). "Sales of $200\%$ of" means new is $200\%$ of old, a $100\%$ rise. Sloppy headlines often confuse these.

Trap 3: Ratios as percents.

"$3$ out of every $5$" is not $35\%$. It is $\dfrac{3}{5} = 60\%$.

Trap 4: Averaging percents.

A class of $20$ students has average $80\%$ marks; another class of $50$ has $70\%$. The overall average is not $\tfrac{80 + 70}{2} = 75\%$. You have to weight by class size:

$\text{overall} = \frac{20 \cdot 80 + 50 \cdot 70}{20 + 50} = \frac{1600 + 3500}{70} = \frac{5100}{70} \approx 72.9\%.$

Trap 5: Reversing percentages without inversion.

"Price went up $25\%$ and then we want to bring it back to the original." A reduction of $25\%$ off the new price would give $0.75 \times 1.25 = 0.9375$, a $6.25\%$ shortfall. The correct decrease is $\dfrac{1}{1.25} = 0.8$, i.e., a $20\%$ decrease (not $25\%$) to undo a $25\%$ increase.

A story to remember the trap. Imagine a $\mathbb{?}\,100$ shirt. Rise by $25\%$: now $\mathbb{?}\,125$. To bring it back to $\mathbb{?}\,100$ we need to subtract $\mathbb{?}\,25$ from $\mathbb{?}\,125$, which is $20\%$ off the new price. A $25\%$ decrease would only remove $\mathbb{?}\,31.25$, leaving $\mathbb{?}\,93.75$.

Useful inversion formulas.

• To undo $+p\%$: apply $-\dfrac{p}{1+p/100}\%$ (less than $p$).
• To undo $-p\%$: apply $+\dfrac{p}{1-p/100}\%$ (more than $p$).

Final tip: estimate first. Before computing, ask whether the answer should be bigger or smaller, roughly half, roughly double, etc. This catches the majority of percentage mistakes you might otherwise make.

Worked examples

Example 1. A salary of $\mathbb{?}\,20{,}000$ is increased $20\%$. By what percent must the new salary be decreased to get back to $\mathbb{?}\,20{,}000$?

• New $= \mathbb{?}\,24{,}000$. To get back: decrease by $\dfrac{4000}{24000} \times 100 \approx 16.67\%$.

Example 2. A school of $300$ has class averages: $40\%$ scored $80$, $60\%$ scored $50$. Overall average?

• Number of students: $120 \cdot 80 + 180 \cdot 50 = 9600 + 9000 = 18600$.
• Average: $18600 / 300 = 62$.

Example 3. "Newspaper readership rose $300\%$" , final readership is what fraction of original?

• A $300\%$ rise means new $= 4 \times$ original (the original $100\%$ plus the $300\%$ increase). So $400\%$ of original.

Example 4. Of $50$ students $30$ like maths, $25$ like science, $15$ like both. How many like neither?

• Like at least one $= 30 + 25 - 15 = 40$.
• Like neither $= 50 - 40 = 10$.

Try it yourself

1. A price rises by $50\%$. What percent decrease undoes it?
2. Class A ($40$ students) averaged $80\%$, Class B ($60$ students) averaged $70\%$. Overall average?
3. A discount of $40\%$ is offered. By what percent must the SP be increased to get back to MP?
4. "$2$ out of every $5$ people own a smartphone" , what percent is that?
5. A shop's sales doubled. What is the percent change?
6. Town: $40\%$ read newspaper A, $35\%$ read B, $15\%$ read both. What % read at least one?
7. A salary fell $25\%$ then rose $25\%$. Net change?
8. Marks rose from $80$ to $96$. Percent change?

Activity / Insight

Spot-the-trap weekend. Over a weekend, look for percent claims in three places: a shop's discount sign, a news article, and a product label. For each, ask: percent of what? With what comparison? Is the headline correct? You will become surprisingly good at spotting clever (and not-so-clever) misuse of percentages , a real-world maths superpower.