Chapter 10: Proportional Reasoning-2

In Chapter 7 we worked with direct proportion: more pencils, more cost. In this companion chapter we meet a different , but equally common , kind: inverse proportion, where more of one means less of the other. More workers on a project means less time to finish. A car at higher speed takes less time. As pressure rises on a gas, its volume shrinks.

We also learn to visualise proportions using pie charts, the slice-of-pie picture where the size of each slice corresponds to the share of a total.

By the end you will be able to read, draw, and interpret pie charts; set up inverse-proportion equations; solve a variety of work-rate and speed-time problems; and decide quickly whether a real-life situation calls for direct or inverse reasoning.

What's inside

1. Pie charts , how to read and draw them.
2. Inverse proportion , definition, formula, examples.
3. Time and work , many workers, one job.
4. Speed, distance, and time , the most-used inverse pair.
5. Joint and combined variations , when both kinds happen at once.

Key results

Joint variation. If $z$ varies directly with $x$ and inversely with $y$, then $z = k \dfrac{x}{y}$.

How to read this chapter

For pie charts, always check that all the slice angles add to $360^\circ$. For inverse-proportion problems, the safest mental check is: if one quantity doubles, does the other halve? If yes, the inverse rule applies.