What a ratio claims, and why it is not a difference

What to assume they know

• Why some resized images look right and others look stretched — two quantities changing by one shared factor is what the chapter calls proportional
• Multiplying and dividing whole numbers, including by a fraction
• Comparing two quotients to decide which is larger
• Reading a unit off a measurement, and knowing that 1 kg is 1000 g
• Elementary use of a letter for an unknown number

What they should be able to do

• Write the comparison of two quantities as a ratio with a colon, in the order the question asks for
• Say in words what a given ratio claims, using a "for every … there is …" form
• Name the terms of a ratio and identify which quantity each term measures
• Show that a ratio and its reverse are different claims, and give a case where reversing changes the answer
• Produce a second pair of numbers that makes the same claim as a given ratio, and explain why it does
• Demonstrate on given data that adding or subtracting the same number from both terms produces a different claim
• Decide, for a stated question, whether the useful comparison is a difference or a ratio, and defend the choice
• Compare two quantities of different kinds — length against bags, students against kilograms — and state the ratio with both units named

Where it usually goes wrong

• "A ratio is a fraction of the whole." Kesang's 6 glasses to 10 spoons of sugar is not "6 out of 10" of anything; the two terms count different kinds of thing and there is no whole they are parts of. Sharing a whole in a ratio is a separate move and gets its own topic.
• "Add the same to both and the ratio is unchanged." This is the chapter's main target. Give the class Neelima's numbers and let them watch one tenth become four thirteenths.
• "The ratio changed, so their age gap changed." The gap is the one thing that cannot change. Students routinely swap the two.
• "Ratios can only compare things of the same kind." Feet against bags of cement, students against kilograms of rice, teachers against students. The units of the two terms are usually different, and both must be said aloud.
• "7 : 12 and 12 : 7 are the same because they use the same numbers." They are opposite claims. Make the class say each one out loud in the "for every" form and the difference becomes audible.
• "Fewer bags means a weaker wall." The printed example exists to break this habit of comparing one column of a table instead of the pair.
• "A ratio tells you how much there is." It tells you only how the two quantities go together. Nothing in 20 : 1 says whether the wall is 20 feet or 200 feet long.

Questions to check understanding

• Write the ratio of two given quantities in the order asked, with units
• State in words what a given ratio claims
• Given a ratio, produce another pair of numbers making the same claim, and one making a different claim
• Given two pairs, decide whether they make the same claim and justify it
• Given a ratio, compute what it becomes when the same number is added to both terms, and say which way the claim has moved
• Age-ratio problems: given two ages now, find when their ratio takes a stated value, using the fixed difference
• Decide whether a stated real-world comparison should be reported as a difference or as a ratio, with a reason
• Spot the error in a worked solution that adds to both terms to keep a ratio

Examples worth working on the board

Values marked verified are worked out here; the chapter supplies the inputs and, for most of these, leaves the conclusion to the reader.

• The tinted definition box (Part I, §7.2, p.161). One boxed sentence gives the reading of a : b as a claim about how many units of the second quantity accompany a stated number of units of the first. This box is the topic's centre of gravity; everything else tests it.
• The image ratios (Part I, §7.2, p.161). Width to height: A is 60 : 40, C is 30 : 20, D is 90 : 60. The page also spells out what A's ratio claims in millimetres.
• Nitin and Hari's compound wall (Part I, §7.4, Example 3, p.163). Nitin builds the long side, 60 ft, using 3 bags of cement. Hari builds the short side, 40 ft, using 2 bags. Nitin thinks the shorter wall must be weaker because less cement went into it. Verified: 60 : 3 and 40 : 2 both say twenty feet of wall per bag, so the two walls carry cement at the same rate and Nitin's worry is unfounded. Hari is a woman in this example — the printed passage has Nitin comparing Hari's wall with the one he built and attributing the difference to the cement she used, which is internally consistent, with "he" the worrier and "she" the mason. The explanation should keep it: anyone who "regularises" the pronouns erases a female mason from the chapter.
• Teachers and students (Part I, §7.4, Example 4, p.163). A school with 5 teachers and 170 students, stated as 5 : 170; the student is then asked to count their own school and write the pair in two printed blanks, and to judge whether the two schools make the same claim. No answer is printed, and the second pair cannot be supplied by an explanation — it is a data-collection task.
• Neelima and her mother (Part I, §7.4, Example 6, p.164). At Neelima's age 3, her mother's age is ten times hers, so 3 : 30, which the page reduces to 1 : 10. Nine years later Neelima is 12 and her mother is 39, giving 12 : 39, which the page reduces to 4 : 13. The section closes by stating that adding or subtracting the same number from both terms need not leave a proportional ratio.
• The general fact behind section 9. Verified algebraically: for positive a < b and positive k, the claim (a + k) : (b + k) is closer to 1 : 1 than a : b, because b(a + k) − a(b + k) equals k(b − a), which is positive. Subtracting a common k moves the ratio the other way, further from 1 : 1 — **provided k is smaller than a, the smaller term, so that both remainders stay positive.** State that condition whenever the subtraction half of the rule is shown: at k = a the comparison has a zero term and says nothing, and past it the terms turn negative and the conclusion reverses. So on the Neelima data the ratio must rise from one tenth towards one, and it does: one tenth becomes four thirteenths. The chapter states the negative fact and does not give this reason; the reason is what makes the topic a topic.
• Harmain and her brother (Part I, p.177, exercise item 6). Harmain is 1 and her brother is 5. The question asks at what age of Harmain's the two ages stand as 1 : 2. This is section 9 run backwards — the gap of 4 years is permanent, so the question is really "when is your brother's age twice yours", and it has one answer. Verified: it happens once and only once; the point to make is that the ratio passes through 1 : 2 on its way towards 1 : 1 and never returns.
• Order matters (Part I, p.165, exercise item 1(iii)). One of six printed proportion statements sets 7 : 12 against 12 : 7. It is the item that tests whether the class has understood that a ratio is directed.
• Filter coffee, as a pair of claims (Part I, §7.4, pp.164–165). The regular cup takes 15 mL of coffee decoction and 35 mL of milk. The strong cup uses 20 mL and 30 mL, the light cup 10 mL and 40 mL. Read as claims, these are three different rates of decoction against milk, and section 6 needs only that much. The strength table and its comparison belong to Solving a proportion problem, and the Trairasika rule of three, which owns the "Filter Coffee!" material.
• Image B, revisited (Part I, §7.1, p.160). Width and height each 20 mm less than image A's. As ratios, 60 : 40 became 40 : 20 — the same subtraction, a different claim. This is the same arithmetic as section 9 with the sign reversed, and putting the two side by side is the cleanest way to show that subtraction is not a neutral operation on a ratio.

Figures to have open

• A ratio annotated as notation: two terms, a colon, and a unit label under each term. Standard schematic, and it should stay available for the rest of the chapter's videos.
• The Neelima timeline: two ages advancing together, the constant gap drawn as a bar, the ratio evaluated at two instants. This is an added figure; the chapter gives only the numbers.
• The 0-to-1 number line of section 9, with the two arrows. An added figure and the load-bearing one — without it, "moves towards 1 : 1" is a phrase rather than a picture.
• The two builders' wall as a plan sketch with the two side lengths and the bag counts. Standard schematic.
• No textbook artwork is required. The chapter's illustrations for these examples are decorative.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 7, "Proportional Reasoning-1", §7.2 "Ratios", Part I p.161 — the definition box, the terms, and the image ratios.
• Part I p.163, §7.4 Examples 3 and 4; Part I p.164, §7.4 Example 6. These are §7.4 examples read for what a ratio claims rather than for a solving method; Example 6 in particular is the chapter's own instance of this topic's thesis, which is why it is carried here and not in the §7.4 topic.
• Part I p.165, exercise item 1, statement (iii).
• Part I pp.164–165, the "Filter Coffee!" passage, for the three mixture rates only. The table and the strength question belong to Solving a proportion problem, and the Trairasika rule of three.
• Part I p.177, exercise item 6 (Harmain). It is printed in the §7.6 exercise block but tests this topic's argument.
• Part I p.160, §7.1, for the subtraction case that starts the argument.
• The chapter's SUMMARY (Part I p.177) restates the reading of a : b and names the terms; it is the only summary line this topic needs.