Chapter 7: Proportional Reasoning-1

How much chutney for $20$ guests when the recipe is for $4$? If a shirt is $30\%$ off, how much do you pay? If $6$ pencils cost $\mathbb{?}\,42$, how much do $11$ pencils cost? Every one of these questions sits in the same family , proportional reasoning , and once you see the family resemblance you can solve them all the same way.

A ratio compares two quantities. A proportion says two ratios are equal. Percentage is just a ratio with a fixed second part: out of $100$. These three ideas, used together, run through almost everything practical you do with numbers in daily life , shopping, cooking, mixing colours, planning trips, reading news.

In this chapter you will learn to write and simplify ratios, scale them up or down, set up proportions, and convert smoothly between ratios, fractions, decimals, and percentages. You will also see that not all comparisons are proportional , some grow together, some grow opposite , and we will carefully separate the two cases (the second case, inverse proportion, will be tackled fully in Proportional Reasoning-2).

By the end you will be able to spot a "proportion problem" in disguise , a word problem, a recipe, a price tag , and crack it in two or three lines.

What's inside

1. Ratios , what they are, how to simplify and compare.
2. Proportion and the unitary method , solving for an unknown.
3. Percentages , fractions out of $100$, conversions, applications.
4. Profit, loss, discount, and tax , practical percentage problems.
5. Direct variation in the real world , when quantities scale together.

Key results

Direct proportion. When $y$ varies directly with $x$, we write $y \propto x$, meaning $y = kx$ for a constant $k$. Doubling $x$ doubles $y$.

How to read this chapter

Always pause to set up the unit value before scaling. Cost of $11$ pencils given cost of $6$? First find cost of $1$ pencil ($\mathbb{?}\,7$), then multiply by $11$ ($\mathbb{?}\,77$). This unitary method is the safest tool for any proportion problem.