Study Guide · Chapter 13
4. Common Mistakes Aspirants Make
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- Inverting the ratio direction: Writing Quantity(cheaper) : Quantity(dearer) = (M-C_1):(C_2-M) instead of the correct (C_2-M):(M-C_1). Fix: always remember the cheaper ingredient’s share comes from the dearer difference (top-right minus middle), and vice versa — the diagram criss-crosses, it doesn’t go straight down.
- Mixing units before subtracting: Working with Rs 8.20 as “8.20” in one line and switching to paise in another, causing an inconsistent ratio. Fix: convert everything to the same unit (all rupees or all paise) before doing any subtraction.
- Misidentifying x and P in the replacement formula: Using the original full capacity where the withdrawn quantity should go, or using percentage removed as if it were the absolute quantity removed. Fix: x is always the actual volume withdrawn each time, and P is the total, constant vessel volume — write both down explicitly before plugging into the formula.
- Applying P(1-x/P)^n when the quantity removed changes each time: The formula assumes the same x is removed at every step. If a question says “first 10 L is removed, then 15 L is removed,” this formula cannot be used directly — you must compute step by step.
- Trying to solve a three-ingredient ratio problem with a single criss-cross line: A single alligation diagram gives only a two-way split. With three unknown quantities you need either a second given condition (e.g., a fixed ratio between two of the three) or you must convert to actual quantities and add, as shown in Section 2.7.
- Averaging ratios directly in “mixture of mixtures” problems without checking equal volumes: As covered in Trick 9 — this is one of the highest-frequency errors in this chapter. Always default to computing actual quantities.
- Using simple alligation (arithmetic mean) for average speed when distances are equal, not times: When a journey is split into two equal distances covered at speeds s_1 and s_2, the correct average speed is the harmonic mean, (2s_1s_2)/(s_1+s_2) — NOT the alligation/arithmetic-mean approach, which only applies when the split is by equal or given time, not by equal distance. Confusing these two cases is one of the most common silent errors in this topic (see Question B14).
- Forgetting that alligation’s mean price is always a cost price, not a selling price: In profit/loss-linked mixture questions, plugging the selling price straight into the criss-cross diagram without first converting it to cost price via the profit/loss percentage gives a wrong ratio entirely.
- Not reducing the final ratio to lowest terms: Leaving an answer as 12:8 instead of simplifying to 3:2 can cause an exact match to be missed among MCQ options.
- In “quantity withdrawn and replaced” problems with a mixture (not pure liquid) being withdrawn, forgetting that the fraction removed applies uniformly to both components: When you remove x litres of an already-diluted mixture, you remove liquid AND water in the same ratio as currently present — students sometimes wrongly assume only the “liquid part” is being drawn off. For instance, in a mixture with milk:water = 4:3 (total 7 units), removing 21 L of the mixture removes milk and water strictly in the ratio 4:3 — that is, 21×(4)/(7)=12 L milk and 21×(3)/(7)=9 L water — never all-milk or all-water.
- Treating the criss-cross ratio as the final quantity answer without checking what the question actually asks for: Alligation gives you a ratio, not absolute quantities. If the question gives a total volume or weight, you must still convert the ratio into actual amounts (as in Example 2.1.4) — a large fraction of silly errors on this topic come from stopping one step too early.
- Assuming the mean price/mean concentration must be exactly halfway between the two given values: This is only true when the two ingredients are mixed in equal quantities (Trick 2). In general, the mean price lies closer to whichever ingredient is present in the larger quantity — students sometimes wrongly assume a 50-50 midpoint by default.
- Sign confusion when one “ingredient” has a cost of zero (water, chalk, sand, etc.): Some aspirants hesitate over how to subtract when C_1=0, mistakenly treating it as “no contribution to the ratio.” Remember zero is still a valid price — plug it into the criss-cross diagram exactly like any other number, as done throughout Section 2.5.
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