The FDP trio: fraction, decimal and percentage as one object

What to assume they know

• Why forcing every fraction onto a scale of 100 makes them comparable — that a percentage is a fraction over 100
• Computing a percentage in your head by splitting it — that y% of a quantity is (y/100) × quantity
• Place value in decimals: tenths, hundredths, and what the digits after the point are worth
• Converting a fraction to a decimal by division, and a terminating decimal back to a fraction
• Reducing a fraction to lowest terms
• Multiplying a decimal by a whole number

What they should be able to do

• Write any percentage as a fraction over 100, as that fraction in lowest terms, and as a decimal, and go in the reverse direction from any of the three
• Explain why finding 50% of a quantity and multiplying it by 0.5 are the same operation, not two methods with the same answer
• Complete a percentage / fraction / decimal table across values from 1% to 100%
• Say why the conversions are one step in base ten, and what would change in a system that was not base ten
• Identify which fractions give a decimal that stops and which do not, and explain why the chapter's percentages for those fractions are rounded
• Read a bar model of equal cells as a fraction, a percentage and a quantity at once
• Estimate a fraction as a percentage quickly enough to play a five-second game
• Choose the most convenient of the three notations for a given calculation, and justify the choice

Where it usually goes wrong

• "0.5% and 0.5 are the same." They differ by a factor of 100. 0.5 is 50%; 0.5% is 0.005. This is the single most expensive slip in the whole percentage topic and it should be shown side by side, not just warned about.
• "To find 50% of something you must convert to a fraction first." The whole point of Example 2 is that the conversion has already happened — 1/2, 0.5 and 50% are one multiplier.
• "1% must be 0.1, because 10% is 0.1." Two of the eight table columns are designed to catch this: 10% and 1%. Fill them adjacently.
• "43% has a nice fraction." It has 43/100, and that is already lowest terms. Students hunting for a "nicer" form conclude they have made a mistake.
• "1/3 = 33%, exactly." 33% is 33/100, which is not one third. The chapter itself prints 33.33% for one third on Part II p.27 and 33% as a plain example of a percentage on Part II p.1 — an explanation must not let those two collide silently.
• "The book's 26.47% is wrong because my calculator says 26.470588." Neither is wrong; one is rounded. Section 6 has to make rounding a visible, deliberate act rather than an error.
• "The three notations are equally good and the choice does not matter." It matters constantly: 0.8 × 75 is easier than (80/100) × 75, but a quarter of 40 is easier than 0.25 × 40. Section 11 should be a decision, not a summary.
• "A percentage of a bar tells you the quantity." Only with the end label. In Q1(i) the same five-cell bar is annotated 20% under one cell on the left and 60 across four cells on the right — and that 60 means 60 only because the end label is 75. Note that no cell is shaded in either drawing, and 80% is never printed: four cells of five is 80%, but if the explanation wants that step it has to derive it.

Questions to check understanding

• Complete a percentage / fraction / decimal table, both directions, including 1% and 100% as the trap columns
• Reduce a percentage to a fraction in lowest terms
• Multiply a quantity by a percentage expressed as a decimal, and check by the fraction route
• Identify from a list which fractions give a decimal that stops (a Class 8 reasoning item, and the reason the board can set 5/11 without expecting an exact percentage)
• Read a bar model of equal cells and report the fraction, the percentage and the quantity
• Estimate the percentage equivalent of an awkward fraction under time pressure
• Extend a pattern of fractions to percentages and describe the trend in words — the bull item's format, which is where "increases but never reaches" gets examined

Examples worth working on the board

Values marked arithmetic added here are added here; the chapter prints no answer key for Part II.

• Example 2, the identity that opens the section (Part II, §1.2, p.8). The page starts from 50% of a value being found by multiplying by 1/2, then asks whether multiplying by 0.5 gives the same answer. Yes — and the reason is set as one display line: 50% = 50/100 = 1/2 = 0.5/1 = 0.5. A tinted box beside it checks the claim on one number: 50% of 24 = 12, and 0.5 × 24 = 12. The immediate follow-up asks which decimal does the same job for 10%.
• The FDP table (Part II, §1.2, p.8). Read from the printed page. Three row labels down the left: Per cent, Fraction, Decimal. Eight columns, the Per cent row filled in throughout: 50%, 100%, 25%, 75%, 10%, 1%, 5%, 43%. In the Fraction row only the first cell is filled, with 50/100. In the Decimal row only the first cell is filled, with 0.5. The other fourteen cells are blank. Arithmetic added here as a check, not for the student yet before the students have tried it: 100/100 = 1; 25/100 = 1/4 = 0.25; 75/100 = 3/4 = 0.75; 10/100 = 1/10 = 0.1; 1/100 = 0.01; 5/100 = 1/20 = 0.05; 43/100 = 0.43.
• The base-ten reason, stated in §1.1 (Part II, §1.1, p.4). The chapter's own justification for choosing 100 includes exactly this: because the number system is base 10, denominators of 10, 100 and 1000 sit comfortably with decimals, and it gives 31% = 31/100 = 0.31 as the instance. It then says that moving among the three notations becomes quick and intuitive. Section 4: a hundredth is the second decimal place, so reading a percentage as a decimal is a re-labelling of the same digits, with no computation at all.
• The percentages in this chapter that had to be rounded. Collected from across the chapter as evidence for section 6, all printed: 26.47% and 28.88% for 9/34 and 13/45 (Part II p.4); 22.22% and 77.78% for 2/9 and 7/9 (Part II p.9); 14.28% for 50/350 (Part II p.17); 43.3% for 130/300 (Part II p.17); 33.33% for 250/750 (Part II p.18) and for 1/3 (Part II p.27); 66.66% among the options of an exercise (Part II p.19). Set against these, the values the chapter prints exactly: 75%, 40%, 24%, 84%, 87.5%, 66%, 37.5%, 120%, 250%, 133.1%. The argument: a fraction gives a decimal that stops exactly when its denominator, after cancelling to lowest terms, is built only from 2s and 5s — the factors of ten. The test has to be applied to the reduced denominator, which is the whole subtlety: 4, 5, 8, 20, 25, 40, 50 pass; 3, 7, 9, 11, 30, 34, 45 do not. So 130/300 reduces to 13/30, and 30 keeps a factor of 3, which is exactly why the chapter has to print 43.3% rounded. By contrast 72/150 and 99/150 reduce to 12/25 and 33/50, whose denominators are clean, so those terminate (0.48 and 0.66) even though 150 itself carries a 3. This is the reason the chapter's decimal percentages sometimes end in a rounded digit, and an explanation that recomputes them without saying so will look as if it is correcting the book.
• The exercise on bar models, Q1 (Part II, §1.2, pp.12–13). The annotations themselves do extract — 100%, 20%, 60, 75,?, 90 and 140 are all in the text layer. What needs the printed page is the cell counts and the arrow spans, which is what was read from the image and from the printed page. Three pairs of bars. Each pair shows the same bar twice: once with a percentage annotation, once with quantities.
  • (i), printed already worked. Left bar: five equal cells, an arrow above the whole bar reading 100%, an arrow under the first cell alone reading 20%. Right bar: five equal cells, 100% above the whole, an arrow beneath spanning the first four cells reading 60, and 75 printed at the right-hand end of the bar.
  • (ii): ten equal cells. Left bar has 100% above and a short arrow under the first cell reading ?. Right bar has 100% above, an arrow beneath spanning the first six cells reading ?, and 90 at the right-hand end.
  • (iii): four equal cells. Left bar, 100% above, ? under the first cell. Right bar, 100% above, an arrow beneath spanning the first three cells reading ?, and 140 at the right-hand end. Arithmetic added here: (ii) one cell of ten is 10%, and six cells of 90 is 54; (iii) one cell of four is 25%, and three cells of 140 is 105. This exercise is the reason section 7 exists: the cell count fixes the fraction, the fraction fixes the percentage, and the end label fixes the quantity — one picture, three notations.
• The Activity: "How Close Can You Get?" (Part II, §1.2, p.8, boxed). Played in pairs. Each player chooses a number; call them a and b. The two numbers are shared. Both players estimate the percentage equivalent of the fraction a/b, with a < b, and must announce an answer by a fixed time — five seconds is the suggested limit. Whoever is closest takes the round; ten rounds make a game. This is the topic's assessment in disguise: it can only be played by someone who holds the anchor conversions in memory.
• Surya's paint, with a quantity this time (Part II, §1.2 exercises, p.13, Q3). He made 60 ml of deep orange paint and red was 3/4 of it; how much red. Note the loop closing: Example 1 back on Part II p.1 turned 3/4 into 75% with no quantity attached at all; this item attaches one. Arithmetic added here: 45 ml.
• Mariam's bull (Part II, §1.2 exercises, p.13, Q8, flagged Math Talk). The data as printed: on the first day the bull is offered 2 units of fodder and eats 1; the next day 3 units and eats 2; the day after, 4 units and eats 3; the pattern continues, and on the 99th day it is offered 100 units and eats 99. Represent all of these as percentages, share the work around the class, and say what you notice. Arithmetic added here, and the thing the item is fishing for: the fractions eaten are 1/2, 2/3, 3/4, 4/5, …, 99/100, so the percentages are 50%, 66.67%, 75%, 80%, …, 99%. They rise every single day and never reach 100%, because exactly one unit is always left — and one unit out of n+1 is a share that keeps shrinking without ever becoming nothing. A single fixed amount left over looks like a smaller and smaller percentage as the whole grows. That is the observation, and it is a genuinely good one for a Class 8 classroom.
• The SUMMARY panel (Part II, p.31). Among the summary points, the fraction–decimal–percentage equivalence is illustrated with 4/10 = 0.4 = 40%, drawn as a 0%-to-100% bar with a mark at 40%, and with 40/100 = 0.4 and 100/100 = 1 printed under the two positions.

Figures to have open

• The FDP grid of Part II p.8, redrawn with the two filled cells and the rest blank. Do not pre-fill it.
• The three pairs of bar models of Part II pp.12–13 Q1, redrawn to the printed cell counts and span lengths (5 cells with a 4-cell span; 10 cells with a 6-cell span; 4 cells with a 3-cell span). These are the chapter's own figures and the counts are load-bearing — read them off this brief, not off the text layer.
• A place-value strip for 0.31 showing the hundredths place, annotated three ways. Standard schematic.
• A rising graph of the bull's percentages with a horizontal 100% asymptote line. This is an added figure; the chapter asks for the observation and draws nothing.
• The SUMMARY bar of Part II p.31 with 40% marked. Optional, but it is the chapter's own closing image of this exact idea.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 1, "Fractions in Disguise", §1.2 "Percentage of Some Quantity", the unnumbered bold subheading "The FDP Trio — Fractions, Decimals, and Percentages", Part II p.8, with the boxed Activity on the same page. Exercise items Part II pp.12–13 no. 1 (parts (i) and (ii) on p.12, part (iii) on p.13); Part II p.13 nos. 3 and 8.
• The base-ten justification and 31% = 0.31 are at Part II p.4, inside §1.1.
• The chapter's SUMMARY, Part II p.31, third bullet, states the two-way convertibility and gives 4/10 = 0.4 = 40%.
• Rounded percentages used as evidence in section 6 are at Part II pp.4, 9, 17, 18, 19 and 27.