Computing a percentage in your head by splitting it

What to assume they know

• Why forcing every fraction onto a scale of 100 makes them comparable — that a percentage is a fraction with denominator 100
• Multiplying and dividing a whole number by 10 and by 100
• Halving numbers, including numbers that halve to a decimal
• The distributive law over addition, in the form a·k + b·k = (a+b)·k
• Reading a proportion written as a : b :: c : d, and the same statement written as two equal fractions
• Converting a ratio of two parts into the fraction each part is of the total

What they should be able to do

• Write "y% of a quantity" as (y/100) × quantity for a named or unnamed quantity
• Recognise 25% as one quarter, 50% as one half, 10% as one tenth, and use the fraction rather than the percentage when it is easier
• State and justify that 20% of a value is twice 10% of the same value, and that the answers for two percentages add when the percentages add
• Build 15%, 40%, 55%, 70%, 75% and 90% of a value from 10%, 5%, 25% and 50% without writing anything down
• Estimate a percentage before computing it, and use the estimate to catch a wrong answer
• Convert a two-part ratio into a percentage of the total, and then into a quantity
• Solve the inverse problem: given that a stated percentage of an unknown equals a known amount, recover the whole and any other percentage of it

Where it usually goes wrong

• "You have to divide by 100 every time." Only if you insist on the percentage form. 25% of 40 and a quarter of 40 are the same product, and the chapter argues this on Part II p.7 rather than stating it.
• "15% has to be computed from scratch." 15% = 10% + 5%, and 5% is half of 10%. Students who do not know that percentages of a number add will reach for long multiplication every time.
• "Percentages add for any two percentages of anything." They add only when they are percentages of the same quantity — that is the shared factor y/100. 20% of a plus 5% of b is not 25% of anything. This is the seed of the whole trouble in Tricky percentages: why a 50% margin followed by a 50% discount leaves a 25% loss, and it is worth planting cleanly here.
• "40% of a journey being 92 km means the journey is 92 × 40." The commonest inverse error. The bar model on Part II p.10 is the corrective: 92 km is a piece of the bar, and the whole bar is 230 km.
• "Once I have the whole, I am done." Example 5 asks for the remaining distance. Methods 3 and 4 differ from Methods 1 and 2 in exactly this last step, and marks are lost there.
• "2 : 7 means millet is 2/7 of the porridge." It is 2/9. The ratio's two numbers are parts, and the whole is their sum. Section 8 has to say this out loud.
• "An estimate is what you do when you cannot do it properly." The chapter takes the opposite position, and its own margin working on 2/9 shows why: the brackets 20%–25% would catch a slipped decimal point in 22.22%.
• "A bigger percentage always means a bigger amount." Q7 is built to break this: it sets 10% of a day against 1% of a week, and the larger percentage of the smaller whole is the one that wins — 2.4 hours against 1.68. A percentage carries no information at all about the size of its whole, so two percentages cannot be compared as quantities until both wholes are named.

Questions to check understanding

• Find a stated percentage of a stated quantity, including a quantity in time or mass units that must be converted first
• Fill a grid of several percentages of several values under a no-writing instruction
• Compare two percentage-of-a-quantity expressions without computing (the board's own format, six pairs at a time)
• Given one percentage of an unknown, produce two or three others — the scaling chain of Q5
• The three-way fill-in: a is what per cent of b; what is p% of b; a is p% of what
• Convert a two- or three-part ratio to percentages, then to quantities of a stated total
• Rate problems where a percentage of a job takes a stated time, with the constant-rate assumption to be identified
• Show that x% of y equals y% of x, algebraically

Examples worth working on the board

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

• The general statement (Part II, §1.2, p.7, tinted box at the top): y% of a value — 80 is the instance used — is (y/100) × 80; and 45% of a value z is (45/100) × z. Both forms matter. The first varies the percentage on a fixed quantity, which is what sections 4–6 exploit; the second fixes the percentage and varies the quantity.
• 25% is a quarter, argued not asserted (Part II, §1.2, p.7). The page asks whether 25% of 40 is the same as a quarter of 40, and answers by noting that 25 is a quarter of 100 — set as 25/100 = 1/4 — so (25/100) × 40 and (1/4) × 40 are the same product. This is the template for every conversion in section 6.
• The free-hand table (Part II, §1.2, p.7). Read from the printed page. Seven column headings across the top: 100, 200, 50, 80, 10, 35, 287. Four row labels down the side: 25%, 10%, 20%, 5%. Exactly one interior cell is filled — 25% of 100, printed as 25. Every other cell of the 4 × 7 grid is blank, and the top-left corner cell is blank too. The instruction is to fill it without pen and paper.
• The doubling observation (Part II, §1.2, p.7, flagged Math Talk). 20% of a value is double 10% of the same value, and the page gives the reason: 20 parts in every 100 is twice 10 parts in every 100. The follow-up asks for 40% of each of the seven table values mentally.
• The splitting identity (Part II, §1.2, p.7). The page notices that (20% of y) + (5% of y) = 25% of y, and then verifies it as a general property rather than a coincidence, displaying ((20/100) × y) + ((5/100) × y) = ((25/100) × y). The next instruction (Part II p.8) is to get 15% of the seven table values from this. Then a Math Talk asks how one would mentally handle 75%, 90%, 70% and 55%. The argument: the common factor is y/100. Percentages of one number add because 20/100 + 5/100 = 25/100, and that is the only reason. It follows immediately that 90% = 100% − 10%, 70% = 50% + 20%, 55% = 50% + 5%, and 15% = 10% + 5%.
• Estimating before computing (Part II, §1.2, p.9, italic box, plus a Note to the Teacher on the same page). The chapter's position is that estimating first sharpens number sense and cuts mistakes, and that an exact value is often not needed at all. The Note to the Teacher asks that estimating be built into the problem-solving routine rather than added afterwards.
• The millet kanji, Example 4 (Part II, §1.2, p.9). The ratio of millet to water for boiling is 2 : 7, so millet to mixture is 2 : 9 — in one unit of mixture, millet is 2/9 and water is 7/9. A bar model divides a bar into a part of 2 and a part of 7 above a total of 9. Three estimating remarks are printed in the margin before any calculation: half of 9 is 4.5, so 2 is clearly under 50%; half of 4.5 is 2.25, so 2/9 is under 25%; 10% of 9 is 0.9 and 20% of 9 is 1.8, so 2/9 lands between 20% and 25%. Then (2/9) × 100 = 22.22% millet, and water is 100 − 22.22 = 77.78%. Since 100 ml of mixture holds 22.22 ml of millet, 500 ml holds 5 × 22.22 = 111.1 ml. The section closes with the general statement that any ratio can be turned into a fraction first, and from there into a percentage.
• Zubin's A grade, Example 3 (Part II, §1.2, pp.8–9). The test is out of 75; an A grade needs 80% or above; what is the least mark. Three routes are printed side by side:
  • fraction multiplication, (80/100) × 75 = (4/5) × 75 = 60;
  • decimal multiplication, 0.8 × 75 = 60;
  • out-of-a-hundred reasoning: out of 100 the minimum is 80, so out of 75 it is 75 × (80/100) = 60. A double number line accompanies it: an upper scale marked 0, 15, 30, 45, 60, 75 labelled Total marks, with the 60 tick labelled as the minimum for an A grade; a lower scale marked 0, 0.2, 0.4, 0.6, 0.8, 1.0 with (20%), (40%), (60%), (80%), (100%) beneath.
• The cyclist, Example 5 (Part II, §1.2, p.10). He rides from Delhi to Agra and has completed 40% of the journey, which is 92 km; how much further to Agra. A bar model is drawn first: Delhi at the left, Agra at the right, the left portion marked 92 km and 40%, the remainder marked with a question mark and the far end 100%. The student is told to estimate before solving. Four methods are printed:
  • Method 1 — halve the known percentage: 40% is 92, so 20% is 46, so 60% is 92 + 46 = 138 km. (This is section 5's identity doing the work.)
  • Method 2 — proportion: 40 : 92 :: 60 : r, written 40/92 = 60/r, giving r = (60 × 92)/40 = 138.
  • Method 3 — find the whole first: 40/100 = 92/d gives d = (92 × 100)/40 = 230 km, so the remainder is 230 − 92 = 138 km.
  • Method 4 — algebra on the remainder: with x the distance left, the total is x + 92 and (40/100)(x + 92) = 92, so x + 92 = (92 × 100)/40 = 230 and x = 138. The italic box beneath says a rough diagram makes the situation easier to think about. Note: Method 1 needs no equation at all, and it is the one a student can do while walking. That is the point of the section.
• Madhu and Madhav's biscuits, Example 1 (Part II, §1.2, p.6) — used here for its computation; its comparison argument belongs to Comparing two proportions that have different totals. Madhu ate 120 g of biscuits that are 25% sugar; Madhav ate 95 g of biscuits that are 35% sugar. The page states the first as a proportion with a blank, 25 : 100 :: ___ : 120, then shows three routes: unitary — 25 g of sugar per 100 g of biscuit means 5 g per 20 g, hence 30 g per 120 g; proportion — 25/100 = s/120 gives s = (25/100) × 120 = 30; and a double number line with 0, 25%, 50%, 75%, 100% above and 0, 30 g, 60 g, 90 g, 120 g below, the part labelled Sugar. For Madhav the numbers are awkward, so the page goes through one gram: 100 g holds 35 g, so 1 g holds 35/100 g, so 95 g holds (35/100) × 95 = 33.25 g — equivalently 35/100 = s/95.
• Figure it Out items that belong to this topic (Part II, §1.2, pp.12–14).
  • Q2: find the value and draw the bar model for 25% of 160, 16% of 250, 62% of 360, 140% of 40, 1% of 1 hour, 7% of 10 kg. (Two of these are unit conversions in disguise — an hour is 60 minutes, 10 kg is 10000 g.)
  • Q4: insert >, < or =, visualising or estimating and computing only to check — 50% of 510 against 50% of 515; 37% of 148 against 73% of 148; 29% of 43 against 92% of 110; 30% of 40 against 40% of 50; 45% of 200 against 10% of 490; 30% of 80 against 24% of 64.
  • Q5, scaling within one unknown: given 30% of k is 70, find 60%, 90% and 120% of k; given 100% of m is 215, find 10%, 1% and 6% of m; given 90% of n is 270, find 9%, 18% and 100% of n. Then the student is asked to invent two more of the same kind.
  • Q6, the three directions of one relation: 3 is what per cent of 300; what is 40% of 4; 40 is 80% of what.
  • Q7 pits 10% of one day against 1% of one week and asks which is the longer stretch, then asks the student to invent similar questions. Arithmetic added here: 10% of a day is 2.4 hours and 1% of a week is 1.68 hours, so the day wins; keep that off screen until the class has estimated.
  • Q9: workers pick berries in 20% of a coffee plantation in 18 days; how long for the whole plantation, assuming the rate of work is unchanged — and the item explicitly asks why that assumption is needed.
  • Q10: a badminton session split warm-up : play : cool-down as 10% : 80% : 10%, for a 90-minute training.
  • Q12: a halwa recipe for 4 people in the proportions rava 40%, sugar 40%, ghee 20% — (i) the proportions for 8 people, (ii) the weight of each ingredient if the ingredients together weigh 2 kg.
• The commuting item (Part II, §1.3 exercises, p.29, Q7). Printed in the later exercise set but it belongs to this argument: 5% of 40 against 40% of 5; 25% of 12 against 12% of 25; 15% of 60 against 60% of 15; what do you notice; and then make the general statement and justify it algebraically by comparing x% of y with y% of x. The reason: both are xy/100, so the two are equal for every x and y. This is the cheapest mental trick in the chapter — swap the numbers when one of them is friendlier.

Figures to have open

• The free-hand grid of Part II p.7, redrawn with the seven column headings and four row labels and only the one filled cell. A pre-filled grid destroys the exercise.
• The Delhi–Agra bar model of Part II p.10. Chapter's own figure and the spine of section 10. Redraw as a schematic.
• The double number line of Part II p.9 (0–75 marks above, 0–1.0 with percentages below), redrawn with the 60 tick marked.
• The millet bar model of Part II p.9: a bar split 2 and 7 over a total of 9. Standard schematic.
• The 0–120 g sugar double number line of Part II p.6, redrawn.
• An anchor-percentage map for section 6. This is an added figure; the chapter has nothing like it.

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", Part II pp.6–10 and the exercise set at Part II pp.12–14. Two unnumbered bold subheadings inside §1.2 may be named: "Free-hand Computations" (Part II p.7) and "The FDP Trio — Fractions, Decimals, and Percentages" (Part II p.8, which belongs to The FDP trio: fraction, decimal and percentage as one object).
• Note to the Teacher, Part II p.9 — estimating to be built into the problem-solving process.
• Part II p.29 Q7, the x% of y item, is printed inside §1.3's exercise set but is handed to this topic; Tricky percentages: why a 50% margin followed by a 50% discount leaves a 25% loss notes the handover.
• The chapter's SUMMARY, Part II p.31, records that the exact number can be found when a percentage of the total is given, that ratios convert to percentages, and that rough diagrams, estimation and mental computation were the tools used.