Ratios of three or more quantities at once

What to assume they know

• What a two-term ratio claims, and testing two of them against each other — What a ratio claims, and why it is not a difference, Direct proportion restated: the quotient that stays constant
• Equivalent ratios as scaled copies, and simplest form — Simplest form, and using it to test whether two ratios are proportional
• Multiplying and dividing by 2, 3 and by simple decimals such as 1.5 and 0.5
• Halving a quantity that is an odd number of units, and being comfortable with the half that results
• Adding decimals to a total

What they should be able to do

• Write a mixture of three or more ingredients as a single multi-term ratio, in a stated order
• Read a multi-term ratio as a set of pairwise claims, and pick out the claim between any two of its terms
• Given one quantity in a scaled mixture, find the common factor and use it to find every remaining quantity
• Explain why one common factor must apply to all terms, and give a concrete failure that shows what goes wrong otherwise
• Write two proportional multi-term ratios with the chapter's double-colon notation, and state the equal-quotients relation it stands for
• Work with a ratio whose terms are not whole numbers, and produce an equivalent whole-number form for it
• Compute the total of a scaled mixture and say why the total is not part of the ratio's own information
• State how many independent pieces of information a ratio of n terms carries

Where it usually goes wrong

• "a : b : c only compares neighbours." It compares every pair. Coriander to fenugreek is 8 : 1 in this mix, and a student who cannot produce that number has not read the ratio.
• "8 : 4 : 2 : 1 tells me how much powder I get." It tells you nothing about amounts. There is no total until one quantity is fixed, which is exactly why both worked examples begin by handing you one.
• "Ratio terms have to be whole numbers." 0.5 and 1.5 are terms in this chapter. A ratio is a statement about relative size, and relative sizes are not obliged to be integers.
• "Then 4 : 2 : 1 : 0.5 is a different, worse recipe than 8 : 4 : 2 : 1." They are the same recipe. Double every term and see.
• "Halving the chillies means halving the chillies." It means halving the batch. This is the single most common error in the section and section 4 exists for it.
• "You scale by adding or removing the same amount from each ingredient." Take one spoon off each of 8, 4, 2, 1 and the fenugreek disappears entirely while the coriander barely notices. Multiplying preserves a ratio; adding does not.
• "You can order the ingredients however you like." Once the ratio is written, the order is data. Example 1 names the colour order before any number appears, and reading 2 : 3 : 5 as white-red-blue gives a different paint.
• "4.5 bags and half a spoon are mistakes." They are what the arithmetic says. Whether you can buy half a bag of cement is a separate, real question, and worth naming as such rather than hiding.

Questions to check understanding

• Write a stated mixture of three or four ingredients as a ratio in a named order
• Given a multi-term ratio, state the ratio between two named terms that are not adjacent
• Given a multi-term ratio and one quantity of a scaled batch, find all the others
• Given two multi-term ratios, decide whether they are proportional and show the quotients
• Find the total of a scaled mixture, and say why the ratio alone could not have given it
• Produce an equivalent ratio with whole-number terms for a ratio containing a decimal
• Say what would go wrong if one ingredient were scaled by a different factor from the rest — the reasoning item this section supports

Examples worth working on the board

Values marked printed are worked out on the page. Values marked not in the book are worked out here on the chapter's inputs.

• The spice mix (Part II §3.3, p.57). Viswanath grinds four things together — coriander seeds, 8 spoons; red chillies, 4; toor dal, 2 spoons; fenugreek (methi) seeds, 1 spoon. Printed: the chapter fixes the ingredient order as coriander, then chillies, then toor dal, then fenugreek, writes the mixture 8 : 4 : 2 : 1, and points out that the ratio has four terms. The section is illustrated with a drawing of two children and a grinder on a kitchen counter with bowls of ingredients.
• The pairwise reading (not in the book). Inside 8 : 4 : 2 : 1 sit six pairwise claims, and a student should be able to name any of them: coriander to chillies is 2 : 1, chillies to toor dal is 2 : 1, toor dal to fenugreek is 2 : 1, and — the ones that are easy to miss because the terms are not adjacent — coriander to toor dal is 4 : 1, chillies to fenugreek is 4 : 1, coriander to fenugreek is 8 : 1.
• Puneet's batch. He has 2 red chillies and wants the same-tasting powder. Printed: two chillies is half of four, so every other quantity is halved too — coriander down to 4 spoons, chillies to 2, toor dal to 1 spoon, fenugreek to half a spoon — which is written 4 : 2 : 1 : 0.5.
• The failing mixture (not in the book — no non-example is printed in §3.3). Suppose Puneet halved everything except the fenugreek and used 4 : 2 : 1 : 1. The three pairs that do not involve fenugreek are untouched — coriander to chillies is still 2 : 1, chillies to toor dal still 2 : 1, coriander to toor dal still 4 : 1 — but all three pairs that do involve it have moved: toor dal to fenugreek from 2 : 1 to 1 : 1, chillies to fenugreek from 4 : 1 to 2 : 1, and coriander to fenugreek from 8 : 1 to 4 : 1. The powder now carries twice the fenugreek it should, relative to everything else. One term out of step is enough to break the mixture, and it breaks exactly the pairs that term belongs to — all three of them, since four terms make six pairs and fenugreek sits in half of them — and no others. This is the argument of section 4, and the chapter supplies nothing like it.
• The notation (Part II p.58, top). Printed: the chapter writes the two mixtures as 8 : 4 : 2 : 1 :: 4 : 2 : 1 : 0.5, then states the general case for a : b : c : d against p : q : r : s and gives the chain of equal quotients — a over p, b over q, c over r and d over s all equal. Not in the book: every one of those quotients here is 2, and that shared value is the factor. Say it out loud; the chapter prints the chain and never names its common value.
• The whole-number twin (not in the book). Doubling every term of 4 : 2 : 1 : 0.5 gives back 8 : 4 : 2 : 1, so the decimal in the ratio is an artefact of which batch you happened to name first, not a property of the recipe. Likewise the concrete ratio 1 : 1.5 : 3 is the same recipe as 2 : 3 : 6. The chapter never writes the concrete ratio in whole-number colon notation — it keeps 1 : 1.5 : 3 and 3 : 4.5 : 9 and carries the decimals through to 4.5 and 16.5 bags. But it does reach a whole-number instance without ever saying that it has: Example 3 on Part II p.59 multiplies the mixture out to 20, 30 and 60 units of the three ingredients taken in the printed order — cement, then sand, then gravel — and 20 : 30 : 60 is 2 : 3 : 6 scaled by ten. Section 7 should build on that rather than deny it — it is the stronger hook, because the book arrives at the whole-number form and never remarks on it. Deciding whether the explanation writes it out as a colon ratio is a real editorial choice; see Notes.
• Example 1, the purple paint (Part II §3.3, p.58). Inputs: the shade is fixed by Red : Blue : White in the ratio 2 : 3 : 5, and Yasmin has 10 litres of white paint. Printed: white corresponds to 5 parts, so 10 ÷ 5 = 2 litres is one part; red is 2 parts, which is 4 litres; blue is 3 parts, which is 6 litres. The chapter then asks for the total and answers 4 + 6 + 10 = 20 litres.
• Example 2, the concrete (Part II §3.3, p.58). Inputs: cement, sand and gravel mixed as 1 : 1.5 : 3, the mix the chapter says is used where a structure needs to be stronger — pillars, beams, roofs — and reinforced with steel rods; and 3 bags of cement in hand. Printed: multiplying through by 3 gives 3 : 4.5 : 9, and the total is 3 + 4.5 + 9 = 16.5 bags of mixture.
• Two directions, one operation (not in the book, and the hinge of the topic). In Example 1 the known quantity is the last term; in Example 2 it is the first. Neither position is special. In both cases you divide the known amount by its own term to get one part, then multiply every term by that. Yasmin: 10 ÷ 5 = 2. The builder: 3 ÷ 1 = 3.
• Counting the information (not in the book). A ratio of four terms carries three independent numbers, not four — fix the ratios of the second, third and fourth to the first and the ratio is completely determined, because scaling all four together changes nothing. This is why one measured quantity is exactly what is needed to finish the job, no more and no less.

Figures to have open

• A ratio strip for 8 : 4 : 2 : 1 built from unit blocks — eight, four, two and one — sitting above a second strip at half the size. The eye should see the shape survive the shrinking. Standard schematic, and it carries sections 3 to 7.
• An arc diagram over the four terms showing all six pairwise ratios. Standard schematic; this is the figure the chapter does not have and the topic needs.
• The chapter's kitchen illustration (Part II p.57) is decorative and can be replaced by simple ingredient icons carrying their quantities.
• Three paint tins for Example 1, with white pre-filled at 10 litres and red and blue empty until the factor is computed. Standard schematic.
• Bags of cement, sand and gravel for Example 2, with the half bag drawn honestly as a half. Standard schematic. Do not smooth 4.5 into 5 in the artwork.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 3, "Proportional Reasoning-2", §3.3 "Ratios with More than 2 Terms", Part II pp.57–58. §3.3 begins in the lower half of Part II p.57, below the boxed map-making activity, and runs to the start of §3.4 near the foot of Part II p.58. Examples 1 and 2 are both on Part II p.58.
• Part II p.69, the chapter SUMMARY, first bullet, for the reading of a multi-term ratio as "so many units of the first for so many of the second, and so on".
• Backward pointer: Part I printed Chapter 7, §7.2 and §7.3, for ratio and simplest form.
• Forward pointer inside this chapter: §3.4 opens at the foot of Part II p.58 and is covered by Dividing a quantity in a given ratio; its reuse of the concrete ratio 1 : 1.5 : 3 is Example 3 on Part II p.59, which is the page to cite for that item. (1 : 1.5 : 3 also appears on Part II p.58 itself, but in §3.3, above the §3.4 heading.) The two topics share that ratio deliberately — see Notes.