PrepShorts · Teaching notes · Class 8 Mathematics · Chapter 1, Fractions in DisguisePrepShorts

Chapter 1 · Fractions in Disguise

Why forcing every fraction onto a scale of 100 makes them comparable

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10 min.

What to assume they know

  • Equivalent fractions: that multiplying numerator and denominator by the same non-zero number leaves the fraction unchanged
  • Comparing two fractions by putting them over a common denominator
  • Multiplying a fraction by a whole number, and dividing a whole number by a whole number to get a decimal
  • Reading a number line on which the labelled points are fractions
  • Solving a one-step equation of the shape a/b = x/100 by multiplying both sides by the same number
  • That a decimal like 0.31 means thirty-one hundredths

What they should be able to do

  • State what the symbol % abbreviates, and rewrite any percentage as a fraction with denominator 100
  • Convert a fraction to a percentage by the equivalent-fraction route and by the solve-for-the-numerator route, and say why the two give the same answer
  • Convert a percentage back to a fraction, and give three further fractions equivalent to it
  • Read a bar model on which a fraction scale above and a hundredths scale below share their endpoints and their ticks
  • Rank a fraction against a percentage without computing, by comparing each to a landmark such as one half
  • Explain why a common denominator is what a comparison of two proportions requires, and why 100 is the useful choice rather than 10, 50 or 43
  • Recognise that two percentages quoted in different contexts refer to different wholes and cannot be combined

Where it usually goes wrong

  • "Per cent is a unit, like kilogram." It is a denominator. 60% water is not 60 of anything; it is 60 parts in every 100 parts, and the parts are whatever the whole was made of.
  • "Converting to a percentage changes the number." 3/4 and 75% are the same number written for two different purposes. The bar model on Part II p.2 exists to make that visible: one bar, with the fraction scale above it and the hundredths scale below, precisely because the ticks coincide.
  • "3/4 = 75% is a fact you memorise." It is one equivalent fraction out of infinitely many, and Example 1 reaches it by walking 3/4 → 6/8 → 30/40 → 75/100. A student who can only recall the pair cannot do 72/150.
  • "7/14 must be 7% or 14%." The commonest error on Q1. Neither number is a percentage until the fraction has been put over 100 — and 7/14 is a half.
  • "Any common denominator would do, so 100 is arbitrary." Mathematically true, practically false, and the chapter argues the point rather than asserting it: 100 is a power of ten, so hundredths are also two decimal places; and per ten is too coarse to separate 26.47 from 28.88.
  • "Percentages are whole numbers." 26.47%, 28.88% and 87.5% all appear in this chapter. A percentage is a fraction and inherits fractions' behaviour.
  • "Two percentages can be compared, full stop." The six findings on Part II p.5 sit on six unrelated wholes — body weight, ice-cream volume, world population, teenagers, Solar System mass, farmland. Adding or ranking them across wholes is meaningless. The next module makes this the main event.
  • "The bigger the percentage, the bigger the amount." Not until the wholes match. Deliberately plant this here so that Comparing two proportions that have different totals can take it apart.

Questions to check understanding

  • Convert a given fraction to a percentage, including one whose denominator does not divide 100 (the board sets 1/3 and 5/11 in this very exercise)
  • Convert a percentage to a fraction in lowest terms, and give two other equivalent fractions
  • Part-of-a-whole word problems in one step: m of n items are such-and-such, what percentage
  • Insert >, < or = between a fraction and a percentage with the instruction not to compute — tests landmarking against 1/2 and 1/3
  • Match positions on a marked line to approximate percentages (the running-race item's format)
  • Multiple choice with a "none of these" option, where the distractors are the raw counts rather than the percentage
  • Short reasoning: why is 100 used as the denominator, given that any common denominator would allow comparison — a competency-style question the chapter explicitly poses on Part II p.4

Examples worth working on the board

Values marked arithmetic added here are added here, worked from the chapter's own numbers; the chapter prints no answer key for Part II.

  • The opening reading of the symbol (Part II, §1.1, p.1). Two everyday lines are quoted as the hook — a sale advertising up to 50% off, and a student named Hiya scoring 83% in her board exams. Then: 25% means 25 out of every 100, and the page instantiates that on three different wholes in one breath — 25 people in every 100 people, 25 rupees in every 100 rupees, 25 marks in every 100 marks.
  • 50% of a quantity, derived rather than asserted (Part II, §1.1, p.1). For a quantity written s, the page runs 50% of s = 50 × (1/100) × s = (50/100) × s = ½ s, and calls the middle step fifty times the unit fraction of s. Two more instances follow: 20% = 20/100 = 2/10 = 1/5, and 33% = 33/100.
  • Surya's deep orange paint (Part II, §1.1, pp.1–2, Example 1). He mixes red and yellow to catch a sunset; red is 3/4 of the mixture. The page reads 3/4 as three out of every four, then re-reads it as six out of every eight, then thirty out of every forty, then seventy-five out of every hundred — the chain 3/4 = 6/8 = 30/40 = 75/100 is set in the margin. That is 75%. Left open on the page: the percentage contributed by the yellow. Arithmetic added here: 25%.
  • The same conversion twice (Part II, §1.1, p.2). Method 1 multiplies top and bottom by 25: (3 × 25)/(4 × 25) = 75/100. Method 2 writes 3/4 = x/100, multiplies both sides by 100 — the page marks that step explicitly — and gets x = (3/4) × 100 = 75. Worth pausing on: Method 1 changes how the fraction is written, Method 2 asks a question about a number. Same answer, different move.
  • The bar model (Part II, §1.1, p.2) — the page's own name for the figure, which it presents as a picture of 3/4 and 75% being one and the same. It is one bar, not two number lines: the bar's left three-quarters is filled red and its right quarter yellow. Above it runs a fractions scale marked 0, 1/4, "2/4 = 1/2" (printed as one label, side by side, over the same tick), 3/4, and 4/4 = 1; below it a hundredths scale marked 0, 25/100, 50/100, 75/100, 100/100, with (25%), (50%), (75%), (100%) directly beneath. The ticks line up vertically with the bar's divisions. The two fills are the point, not decoration: the question printed immediately beneath the bar asks the class to give the yellow share as a percentage, and the yellow quarter answers it — 25%. A redraw that reduces this to two bare scales throws away both the bar and the question.
  • Surya's prize money (Part II, §1.1, pp.2–3, Example 2). He keeps 2/5 of it for a canvas. Method 1: 2/5 = 20/50 = 40/100 = 40%. Method 2: 2/5 = x/100, so x = (2/5) × 100 = 40. Method 3 is printed as an empty frame for the student (Part II p.3): a fifths scale marked 0, 1/5, 2/5, 3/5, 4/5, 5/5 = 1, a bar beneath it whose left end reads 0% and right end reads 100%, a cyan "Savings for canvas" block covering the first two fifths inside a red-outlined whole labelled "Total prize money", and a speech bubble asking whether this is not simply 2/5 of 100. Between the 0% and 100% ends the page prints four empty boxes, one under each of the 1/5, 2/5, 3/5 and 4/5 ticks, and the printed instruction is to complete Method 3 by filling them — they take 20%, 40%, 60% and 80%. The explanation's live fill needs those four boxes or there is nothing to fill.
  • The rule the section lands on (Part II, §1.1, p.3, boxed): a fraction is measured against one unit while a percentage is measured against 100, so multiplying the fraction by 100 is the whole conversion.
  • Backwards, Example 3 (Part II, §1.1, p.3). 24% is 24/100; the page also offers 12/50, 6/25 and 48/200 as equivalent forms; generally, z% is any fraction equivalent to z/100. Note the direction of generality — a percentage names not one fraction but a whole equivalence class.
  • Figure it Out (Part II, §1.1, pp.3–4). Hand these over intact.
    • Q1, fractions to convert: 3/5, 7/14, 9/20, 72/150, 1/3, 5/11. (The last two do not terminate — see the FDP topic, The FDP trio: fraction, decimal and percentage as one object.)
    • Q2, multiple choice: Nandini has 25 marbles and 15 of them are white; the options offered are 10%, 15%, 25%, 60%, 40%, and none of these.
    • Q3: 15 out of 80 students walk to school.
    • Q4, the running-race figure: a horizontal line with Start at the left end and Finish at the right, carrying four filled circular markers labelled A, B, C, D from left to right — A roughly a third of the way along, B a little past halfway, C between B and D, D almost touching Finish. Six option boxes are printed below: 55%, 20%, 38%, 72%, 84%, 93%. Four are to be matched to the four runners' positions after 15 minutes; two are decoys.
    • Q5, insert >, < or = without calculating: 50% and 5%; 5/10 and 50%; 3/11 and 61%; 30% and 1/3.
  • The two biscuit varieties — the argument for percentages (Part II, §1.1, p.4). A factory is trying two new biscuits. Sugar is 9/34 of Variety 1 and 13/45 of Variety 2. The page concedes that this is not decidable at a glance, then re-presents the same two facts as 26.47% and 28.88% and points out that now it is. This is the single most important input in the topic: the information did not change, only its denominator did.
  • The per-100 / per-10 / per-1000 panel (Part II, §1.1, p.4, tinted box at the right margin). The same fraction three ways: 9/34 = 26.47 per cent (per 100), = 2.647 per decem (per 10), = 264.7 per mille (per 1000). Alongside it, in the running text, 31% = 31/100 = 0.31. The page's stated reasons for choosing 100: base ten makes hundredths line up with decimals; 100 is a round number that is easy to hold in the head; per 10 would be too coarse for many purposes; per 1000 and per 100,000 do get used, in statistics phrased per thousand people or per lakh.
  • Per cent before the per cent sign (Part II, §1.1, p.5). Kautilya's Arthaśhāstra, as early as the 4th century BCE, quotes monthly interest per cent in panas: a pana and a quarter is called just, five panas commercial, ten panas the rate prevailing among forests, twenty panas the rate among sea traders. Around the same period the Romans used 1/20 and 1/100 on trade and auction transactions. Fifteenth-century Italian manuscripts write xx p cento, x p cento, vii p cento — our 20%, 10% and 7%. The page also observes that the need for hundredths was felt in computation long before decimal fractions arrived.
  • "Percentages Around Us" (Part II, §1.1, p.5), six illustrated findings, each with its own picture. Hand all six over with their wholes attached: the human body is on average about 60% water by weight; ice cream is about 30–50% air by volume; 45% of the world's population watched at least part of the 2022 FIFA World Cup; over 80% of teenagers globally fail to meet the recommendation of at least one hour of daily physical activity; about 99.86% of the Solar System's mass sits in the Sun; an estimated 52% of agricultural land worldwide is degraded.

Figures to have open

  • The bar model of Part II p.2, redrawn as a schematic: one bar with its left three-quarters red and its right quarter yellow, a fractions scale above and a hundredths/percentage scale below, ticks aligned to the bar's divisions. Keep both fills — the yellow quarter answers the page's own question. This is the figure the whole topic turns on.
  • The Method 3 frame of Part II p.3, redrawn: fifths scale above a 0%–100% bar, with the part and the whole labelled. Keep it partly blank.
  • The running-race line of Part II p.4 Q4: Start–Finish axis, four markers at the printed relative positions, six option boxes beneath. Chapter's own figure; redraw rather than reproduce.
  • The per-100 / per-10 / per-1000 panel, as three lines of the same fraction. Standard schematic.
  • Six icons for the "Percentages Around Us" findings, each captioned with its number and its whole. The captions matter more than the pictures.
  • No photograph is needed anywhere in this topic.

Where this sits in the book

The book

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