Proportional Reasoning-1
This chapter explores one of the most far-reaching ideas in mathematics: comparing two quantities by how many times one is the other, rather than by how much they differ. We open with a set of posters of the same elephant, some that look right, some that look stretched or squat, and let them guide us to ratios and proportion. From there we assemble a toolkit: reducing ratios to simplest form, testing whether two ratios are proportional by cross multiplication, solving “find the missing fourth number” puzzles with the ancient Indian Rule of Three (Trairāśika), sharing a quantity in a given ratio, and converting units so comparisons are fair. Along the way we meet Āryabhaṭa’s neat rule, a puzzle from the classical Līlāvatī, and the surprising fact that not every problem which looks like a proportion really is one.
Learning Outcomes
By the end of this chapter, you will be able to:
- write a comparison of two quantities as a ratio a : b, and reduce it to simplest form by dividing by the HCF;
- decide whether two ratios are proportional, both by comparing simplest forms and by cross multiplication (a:b :: c:d \iff ad = bc);
- use proportional reasoning and the Rule of Three to find a missing fourth quantity in real-life problems;
- divide a quantity in a given ratio m : n, and adjust mixtures to a new ratio;
- convert units of length, area, volume and temperature so that two quantities can be compared fairly;
- recognise when a situation is not a direct proportion (such as speed and time), and explain why.