Proportional Reasoning-2
Class 8 Mathematics · Ganita Prakash Part 2 · Chapter 3
2- Mathematics
- Mathematics
- Science
- + 3 more
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
Summary
Building on the first proportional reasoning chapter, this one extends ratios to more than two terms and introduces inverse proportion and pie charts. Ratios like \(8 : 4 : 2 : 1\) keep their proportion when every term scales by the same factor, and dividing a whole \(x\) among parts \(a : b : c : \ldots\) gives each part as \(x \times \dfrac{a}{a+b+c+\cdots}\). Map scales (Representative Fractions like \(1 : 60{,}00{,}000\)) connect map distance to real distance. A pie chart turns a ratio into slices by dividing \(360^\circ\) in that ratio. The major new idea is inverse proportion: two quantities \(x\) and \(y\) are inversely proportional when their product is constant, \(xy = k\); as one increases by a factor \(n\), the other decreases by \(\dfrac{1}{n}\). This models speed-and-time, workers-and-days and pumps-and-hours problems, where \(x_1 y_1 = x_2 y_2\). The chapter contrasts this with direct proportion, where the quotient \(\dfrac{x}{y}\) stays constant, and uses both to solve everyday work-rate problems.