Chapter 8: Fractions in Disguise

The fraction $\dfrac{1}{4}$, the decimal $0.25$, the percent $25\%$, and the ratio $1 : 4$ are all the same number. They are four "disguises" of the same idea. The catch is that one disguise is more convenient than another depending on what you need to do.

If you're slicing a cake into eight pieces, $\tfrac{1}{4}$ tells you to take two slices. If you're comparing a test result with a friend's, $87\%$ is easier than $\tfrac{87}{100}$. If you're mixing concrete in a $1:3:5$ mix, ratios beat percentages.

This chapter , the part-II companion to Proportional Reasoning-1 , takes a closer look at percentages disguised as fractions and vice versa, drilling into real situations where switching disguises makes the maths simpler. It also brings together earlier ideas: comparing quantities, simple discounts, taxes, and percentage change.

By the end you will be able to convert smoothly between the four forms; pick the most useful form for a given problem; and avoid the most common percentage traps (like "$20\%$ increase then $20\%$ decrease is not back to where you started").

What's inside

1. Four disguises of one number , fraction, decimal, percent, ratio.
2. Quick percentage tricks , finding 10%, then 1%, then anything.
3. Successive percentage change , why discounts and increases compound.
4. Comparing things using percentages , population, prices, performance.
5. Building intuition , mistakes to avoid and stories that stick.

Key results

Successive change. If a quantity changes by $p\%$ then by $q\%$:

$\text{final} = \text{initial} \times \left(1 + \frac{p}{100}\right)\left(1 + \frac{q}{100}\right).$

Negative $p$ for a decrease.

The classic trap. A $20\%$ increase followed by a $20\%$ decrease gives $0.8 \times 1.2 = 0.96$ , a $4\%$ decrease overall, not zero change.

How to read this chapter

Convert constantly between forms while reading. Whenever you see a percent, mentally express it as a fraction in lowest terms (e.g., $40\% = \tfrac{2}{5}$). Many problems become trivial in fraction form and confusing in percent form, or vice versa.