The 25% Price Rise and the 20% Answer: Whose Method Counts?
Sanjib, a mathematics teacher of a Class IX section with 42 students, sets a three-mark homework question: 'The price of rice rises by 25%. By what percentage must a family reduce its consumption so that its expenditure on rice stays the same?' The correct answer is 20%. On checking he finds 11 students wrote 25%, 24 wrote 20%, and 7 left it blank. Of the 24 who are correct, 15 wrote only the answer with no working, 6 used the shortcut 25/125 × 100, and 3 set up an equation, 1.25 × (1 − x) = 1. One student, Farhan, writes: 'Reduce by 20%, because you now pay 125 rupees where you used to pay 100, so to get back from 125 to 100 you need 25 out of 125, which is 20%.' In the same 40-minute period Sanjib must return the previous day's test and begin the chapter on linear equations. The school's board-oriented marking policy gives full marks for a correct answer with correct working and zero for a correct answer with no working.
1. Diagnose the source of the 25% error and explain why Farhan's reasoning is or is not generalisable. State the single question you would put to the eleven students.
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Diagnosis: the base changes and the students have not noticed. '25% reduction' in their heads is taken as 25% of the old expenditure, but the reduction must be measured against the new price. Expenditure is price × quantity, so the quantity must fall by the fraction by which the price has risen relative to the new price.
Farhan's method is generalisable, and that is the point. He does not simply invert 25; he re-anchors on the new total (125) and asks what fraction 25 is of that. Written generally, for a rise of r per cent the required reduction is r/(100 + r) × 100. Test it: a 50% rise needs 50/150 = 33⅓% reduction; a 100% rise needs 50%. Also test the asymmetry - a 20% fall in price requires a 25% rise to restore the original, not 20% - which is the same base error seen from the other side. A method that survives both tests is a method, not a trick.
The single question: 'Twenty-five per cent of what? If rice cost ₹100 a kilo, what does the family pay now - and 25 is what fraction of that new amount?' The question forces the base to be named out loud. Asking 'which is bigger, 25% of 100 or 25% of 125?' is the fallback for students who still hesitate.
2. How would you spend the 40 minutes? Justify what you deliberately cut.
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Plan (40 min):
- 0–3 min: return tests face down, no discussion, no marks read out. This buys time and removes the emotional noise.
- 3–11 min: one bar model on the board. Bar of ₹100 becomes ₹125. The family still spends ₹100, so the shaded part to be removed is 25 out of 125. Students label the bar themselves before any formula appears.
- 11–21 min: paired work on two parallel cases - a 40% rise (answer 2/7 ≈ 28.6%) and the reverse case where the price falls 20% and students must find the required increase (25%). Circulate and collect two scripts to display.
- 21–28 min: put Farhan's reasoning and the equation 1.25(1 − x) = 1 side by side and show they are the same statement - this is the bridge into the linear-equations chapter, so the period does double duty.
- 28–35 min: exit ticket with one unfamiliar case.
- 35–40 min: set homework.
What is cut, and why: the full test discussion (return it, discuss only two items tomorrow - a full walkthrough of 15 questions would consume the period for a handful of students), and the planned 15 minutes of independent practice, which is lower value than repairing a misconception that will recur in profit-and-loss, discount, and data-handling questions. Cutting content coverage is the deliberate cost; the linear-equations chapter still begins, from Farhan's equation rather than from a definition.
3. What do you do about the fifteen students who wrote only the answer, given the school's marking policy, and what does that tell you about the validity of this homework as an assessment?
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Applying the policy mechanically here would be a mistake, and it is worth saying why in one line to the class. The policy exists to protect students in the board examination, where an unexplained answer earns nothing. The purpose of this homework was diagnostic. Awarding zero after the fact to fifteen students who were never told the rule applied to homework punishes them for a rule change, not for a mathematical error, and it teaches them that the safest strategy is to copy working from a neighbour.
What to do instead:
- Do not revise the marks. Record them, and flag the fifteen as 'evidence incomplete' rather than 'wrong'.
- In the next class, state the working rule explicitly and apply it going forward, including on the next class test - consistency from a stated date is fair; retrospection is not.
- Re-task the fifteen with the same question plus one sentence: 'Show two different methods.' Farhan's method and the equation method are both acceptable, which signals that method variety is valued.
On validity: a score built from answers-only tells you whether students got 20%, not whether they can reason about a changing base. For the 15, the assessment has produced no usable diagnostic information at all, and their 3 out of 3 is weaker evidence than Farhan's 3 out of 3. The redesign follows: any homework intended for diagnosis must carry a compulsory justification line, and only the class test should be marked on the board pattern.