Complete exam-focused revision notes on reasoning, series, coding, relationships, fractions, percentages, commercial arithmetic, averages, speed and work.
Maths में speed formula रटने से नहीं, question की structure पहचानने और सही base चुनने से आती है।
Which direction does the inference travel?
Reasoning uses available statements, observations or relations to reach a conclusion. The key exam question is: What is given, what is claimed, and how strongly does it follow?
Conclusion सही लगना पर्याप्त नहीं; वह दिए गए evidence से logically follow होना चाहिए।
General rule → specific conclusion. If premises are true and the form is valid, the conclusion must be true.
All mammals breathe. Whale is a mammal. Therefore, whale breathes.
CERTAINTY IF VALIDSpecific observations → probable generalization. New evidence can strengthen or weaken the conclusion.
Several sampled metals expand on heating; infer that metals generally expand.
PROBABILITYObservation → best available explanation. The conclusion is plausible, not guaranteed.
The lawn is wet; rain is one possible best explanation, but a sprinkler may also explain it.
BEST EXPLANATIONBecause two cases share relevant features, infer a further similarity. Strength depends on the relevance and number of shared features.
Check simple operations before inventing a complicated rule
For 4, 9, 16, 25, 36:
Constant second difference signals a quadratic pattern; here terms are squares 2² to 6².
Split positions before calculating:
Generate the expected series independently, then compare. Do not force a new rule around the suspicious term.
Convert letters into positions, then look for movement
After Z, continue from A when a coding rule shifts forward.
For two letters in a word, compare:
Check both forward and reverse alphabetical order if the question allows it.
Find the stable transformation, not a lucky coincidence
| Type | What changes? | Best first move |
|---|---|---|
| Shift/substitution | Each letter replaced by another | Write positions and differences |
| Reverse/opposite | Order or alphabet mirror changes | Check reversal and sum 27 |
| Position/value | Letters converted to numbers | Test sum, product, difference |
| Conditional/symbol | Table plus special conditions | Apply base code, then condition |
| Language coding | Words mapped to arbitrary code words | Use common-word intersection |
| Renaming/chain | Known objects receive new names | Find real role first, then rename |
Apply the same rule to every letter. If shifts vary, inspect position-wise patterns.
If:
Common word learners ↔ common code mi.
A rule inferred from one pair may not be unique. Confirm against every available example and all letter positions.
एक example से कई rules बन सकते हैं; सभी दिए हुए data पर rule test करें।
Blood relations and set relations become easy when converted into structure
“How is A related to B?” और “How is B related to A?” के answers उलटे हो सकते हैं।
Make unlike forms comparable
Cancel before multiplying to reduce arithmetic.
For positive denominators, compare cross-products:
Example: 5/8 vs 9/13 → 5×13=65; 9×8=72, so 9/13 is larger.
Ratios compare; proportions declare equality
a:b compares quantities in the same units. Multiplying or dividing both terms by the same non-zero number gives an equivalent ratio.
Ratio में order महत्वपूर्ण है: 2:3 और 3:2 समान नहीं हैं।
a:b = c:d means two ratios are equal.
Outer terms are extremes; inner terms are means.
Direct: y ∝ x → y/x constant.
Inverse: y ∝ 1/x → xy constant.
Divide ₹840 in 3:4:
If A:B = 2:3 and A rises 15%, B rises 10%:
Apply change to each original term before simplifying.
Always identify the base
Ask first: “Percentage of what?” The denominator/base controls the answer.
Percentage change में पुरानी value base होती है, नई value नहीं।
Use signs: +20% then −20% → 20−20−4 = −4%, not zero.
If final value 120 follows a 20% rise:
Do not simply subtract 20% of the final value.
If price falls by x%, required consumption rise:
20% price fall → 20/80×100 = 25% consumption rise.
Keep CP, SP and MP in separate boxes
20% then 10% discount = 20+10−2 = 28%.
If two articles sell at the same SP, one at x% gain and one at x% loss:
At 20% each → overall loss = 4%.
Profit/loss percentage uses CP; discount percentage uses MP. The same rupee difference can create different percentages because the bases differ.
Simple growth is linear; compound growth grows on growth
P = principal, R = annual rate %, T = time in years.
For months use T = months/12; for days use the convention stated in the question.
For half-yearly compounding: use R/2 and 2T periods. For quarterly: R/4 and 4T.
SI is computed on original principal. CI is computed on accumulated amount.
Money received later is discounted back using a rate.
Here r is the rate per period in decimal form. Higher r or longer n lowers present value.
Approximate doubling time:
At 12% → about 6 years. It is an approximation, not the exact compound-interest equation.
Average is a balance point, not just a formula
If one value is replaced, adjust the total first; then divide by the unchanged count.
Do not take a simple average of group averages unless group sizes are equal.
Weights reflect different importance or frequency.
Average speed is total distance / total time, not usually (v₁+v₂)/2.
Mix values a and b to obtain mean m:
8% and 18% profit mixed for 14% mean → ratio = (18−14):(14−8)=4:6=2:3.
Draw the triangle, fix the units, then model relative motion
For fixed distance:
If A:B speeds = 3:4 and B takes 24 min, A takes 24×4/3 = 32 min.
Convert days into one-day rates
A in 10 days and B in 15 days:
More efficient worker takes less time.
Inlet is positive work; outlet/leak is negative.
Use LCM units when opening/closing times differ.
For the same work and equal efficiency:
If efficiency varies, include its factor too.
Calculate work completed in one full cycle, count complete cycles, then assign the remaining work to the next worker in sequence.
Useful “etc.” topics found in the supplied material
Leap year: divisible by 4, but a century year must be divisible by 400.
Odd days = remainder after complete weeks.
Smaller angle = min(θ, 360−θ).
For three sets, add singles, subtract pairwise intersections, then add the triple intersection once.
Powers repeat in cycles. Example: last digits of 2ⁿ are 2,4,8,6 (cycle length 4).
One formula wall + one mixed checkpoint
| Topic | Core relation | Trap to avoid |
|---|---|---|
| Percentage | Part/Base ×100 | Wrong base |
| Successive change | a+b+ab/100 (signed) | Simple addition only |
| Profit/Loss | Gain or loss / CP ×100 | Using SP as base |
| Discount | Discount / MP ×100 | Using CP as base |
| SI | PRT/100 | Wrong time unit |
| CI | P(1+r)ⁿ−P | Ignoring compounding period |
| Average | Sum/n | Averaging unequal group means |
| Speed | D/T | Averaging speeds blindly |
| Work | Rate = 1/time | Adding days instead of rates |
| Proportion | a:b=c:d ⇒ ad=bc | Reversing order |
These notes synthesize the supplied book pages with the complete NTA syllabus using original explanations, examples, formula layouts and interactive practice. Re-check arithmetic by method, not memory.
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