Simplest form, and using it to test whether two ratios are proportional

What to assume they know

• What a ratio claims, and why it is not a difference — what a ratio claims, and its terms
• Highest common factor of two whole numbers, and how to find it
• Dividing both terms of a ratio by the same number
• Multiples and factors; recognising when two numbers still share a factor
• Multiplication by a fraction, for the family-building section
• Counting objects in a repeating picture

What they should be able to do

• Reduce a given ratio to its simplest form by dividing both terms by their HCF
• Explain why dividing by a smaller common factor does not finish the job
• Decide whether two given ratios are proportional by reducing both, and write the conclusion with the proportion symbol
• Judge a list of stated proportions as true or false, giving the reduction as the reason
• Generate several ratios proportional to a given one, and explain why the list is endless
• Fill a missing term so that a ratio is proportional to a given one, including when the missing term is not a whole number
• Obtain a ratio by counting in a repeating pattern, and identify the repeating unit before counting
• State what the simplest form of a ratio does not record

Where it usually goes wrong

• "Reducing changes the ratio." It changes the numbers and not the claim. Say the reduced ratio out loud in the "for every" form and check that it means the same thing.
• "Any common factor will do." Section 3 exists for this. Stopping short leaves a pair that still depends on which factor you happened to notice, so it is not a name for anything.
• "Same simplest form is a good sign, and to be sure you should still find the factor." It is not a sign; it is the whole test. Two ratios with the same simplest form cannot fail to be proportional.
• "Both terms must be whole numbers, so a blank always has a whole answer." The third blank of item 3 does not, and Example 7 has a factor of three sevenths.
• "3 : 2 tells you the picture is 60 mm wide." It tells you nothing about size. Two of the chapter's images are 3 : 2 and differ by a factor of three.
• "To find the ratio in a patterned wall, count the bricks you can see." The wall continues past the edge of the picture. Counting the fragment answers a question nobody asked.
• "12 : 18 and 28 : 12 look similar enough." Judgement by appearance is what reduction replaces. Every item on the list has to be reduced.
• "HCF is only for fractions." The same tool, the same reason: cancelling a shared factor.

Questions to check understanding

• Reduce a given ratio to its simplest form, showing the HCF used
• Decide whether two given ratios are proportional, and write the conclusion using the proportion symbol
• Circle the true statements in a list of proportion claims
• Supply the missing term of a proportion, including a case whose answer is not a whole number
• List three ratios proportional to a given one and explain how they were made
• Count in a repeating pattern to obtain a ratio, having first marked the repeat
• Given two ratios in simplest form, say what can and cannot be concluded about the sizes of the quantities involved
• Find the error in a reduction that divided by a common factor but not the highest

Examples worth working on the board

Values marked verified are worked out here or an added count on the printed page. The chapter works some of these and leaves most as blanks.

• Reducing the image ratios (Part I, §7.3, p.161). Worked on the page: 60 : 40 has HCF 20 and reduces to 3 : 2; 90 : 60 has HCF 30 and reduces to 3 : 2; 40 : 20 reduces to 2 : 1; 60 : 60 reduces to 1 : 1. The page does not reduce image C's 30 : 20, and asking the class to do it is the natural first exercise. Verified: it is 3 : 2, so three of the five images share one name.
• The stopping-short demonstration for section 3. Verified: dividing 60 : 40 by the common factor 2 gives 30 : 20, while dividing 90 : 60 by the common factor 30 gives 3 : 2. Two members of one family, two different answers — so a rule that says "divide by a common factor" does not give a name. Only "divide by the highest" does. This demonstration is added here; the chapter gives the recipe without arguing for it.
• Why the name is unique (the argument of section 5). Write the terms as the HCF times what is left: a = g × p and b = g × q, where p and q share nothing. Any ratio in the same family is f × g × p to f × g × q, whose highest common factor is f × g, so it reduces to p : q as well. Conversely, two ratios that reduce to the same p : q are both multiples of it, so each is a multiple of the other. The chapter states the test and does not prove it; this is an added argument and it is the reason the topic exists.
• The definition box and the symbol (Part I, §7.3, p.162). One tinted box states that ratios with equal simplest forms are in proportion, introduces the double-colon symbol, and shows it used with letters. The page then writes the two image proportions out in full using the symbol.
• Example 1 (Part I, §7.4, p.162). Are 3 : 4 and 72 : 96 proportional? The page notes 3 : 4 is already reduced, asks for the HCF of 72 and 96, gives it as 24, and reduces 72 : 96 to 3 : 4. Good demonstration numbers precisely because 24 is not obvious.
• Six proportion claims to judge (Part I, p.165, exercise item 1). The printed statements are (i) 4 : 7 :: 12 : 21, (ii) 8 : 3 :: 24 : 6, (iii) 7 : 12 :: 12 : 7, (iv) 21 : 6 :: 35 : 10, (v) 12 : 18 :: 28 : 12, (vi) 24 : 8 :: 9 : 3. The instruction is to circle the true ones. Verified as an error-check as a check: exactly three of the six are true, and one of the three false ones is false only because it reverses the terms. The reduction has to happen anyway.
• Building a family (Part I, p.165, exercise item 2). Three blanks, to be filled with ratios proportional to 4 : 9. Open-ended, and the discussion of why the answers differ across the class is the point.
• Filling blanks against 18 : 24 (Part I, p.165, exercise item 3). Four printed blanks: 3 : __, 12 : __, 20 : __, 27 : __. Verified: 18 : 24 reduces to 3 : 4, so the first, second and fourth come out whole, and the third does not — 20 is not a multiple of 3, and its partner is a mixed number. Flagging this as a check: it is not a misprint, it is the same lesson as the fractional factor in Example 7 (Part I p.164), and an explanation that quietly skips the third blank throws away the most interesting one.
• Brick wall (a) (Part I, p.166, exercise item 6a). Three courses of identical bricks in running bond, each course offset by half a brick from the one above. Coloured bricks are red and sit in a repeating downward-pointing triangle: three side by side in the top course, two in the middle course, one in the bottom course. Verified by counting on the printed page: the motif repeats every five brick-widths, so one repeat is five columns by three courses — fifteen bricks, six of them red and nine grey. The printed strip shows seventeen columns, that is three repeats plus two plain columns, so counting every brick on the page gives 33 grey to 18 red, which reduces to a different ratio from the one the wall actually has. The whole item turns on counting the repeat and not the fragment.
• Brick wall (b) (Part I, p.166, exercise item 6b). Seven courses in the same running bond, coloured bricks in a tan or ochre shade forming a chain of linked hexagonal outlines along the wall. Verified by the same count: the motif repeats every four brick-widths, so one repeat is four columns by seven courses — twenty-eight bricks, twelve coloured and sixteen grey. The printed strip shows seventeen columns, four repeats plus one column, which again makes the whole-strip count the wrong count.
• A measured ratio to reduce (Part I, p.176, exercise item 1). 600 mL of orange juice mixed with 900 mL of apple juice; the item asks for the ratio of orange to apple in simplest form. Note the order named in the question.

Figures to have open

• The two brick-wall strips of Part I p.166, redrawn. Both are essential and neither can be read from the printed text: the courses, the half-brick offset and the shape of the coloured motif all matter. Redraw as flat rectangles in two shades with the repeat outlined; do not reproduce the printed art.
• A family tree of one proportional family collapsing onto its reduced pair. An added figure and the one that carries section 5.
• The five images, or five plain rectangles standing in for them, each carrying its reduced name. Derived from the chapter's figure on Part I p.159.
• The reduction of 72 : 96 as a two-row division, with the HCF named. Standard schematic.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 7, "Proportional Reasoning-1", §7.3 "Ratios in their Simplest Form", Part I pp.161–162, including the tinted box that introduces the proportion symbol.
• Part I p.162, §7.4 Example 1 — the reduction test with HCF 24.
• Part I p.165, exercise items 1, 2 and 3.
• Part I p.166, exercise item 6, both walls. Items 4 and 5 on the same page test similarity of shape and are carried in Why some resized images look right and others look stretched.
• Part I p.176, exercise item 1.
• The chapter's SUMMARY (Part I p.177) states the proportionality test in the cross-multiplied form rather than the reduced form; the two are reconciled in Solving a proportion problem, and the Trairasika rule of three, which owns the cross-multiplication derivation on Part I p.168.