Mathematical Reasoning & Aptitude
गणितीय तर्क
Unit overview: the topics below, with their summaries, are the syllabus for this unit. The comprehensive note further down covers every topic in full — explanations, definitions, worked examples, tables, exam points and practice questions — and the short revision note after it is only a quick recap. Standalone one-topic learning pages are not built yet — every topic on this page is covered in the notes and the practice questions below, and each topic card links to those sections.
Premium chapters for this unit
Number & letter series → premium chapter · free preview · ~15 min · 4 practice questions
Percentages, ratio, profit-loss → premium chapter · free preview · ~15 min · 4 practice questions
Fractions, decimals, time-work → premium chapter · free preview · ~14 min · 4 practice questions
Topics checklist — with weightage (platform estimate from past-question frequency, not an NTA guarantee)
Number & letter series
High weightageAP/GP, coding-decoding, missing numbers.
Percentages, ratio, profit-loss
High weightageEveryday maths: discount, interest, averages.
Fractions, decimals, time-work
MediumSpeed-distance-time, work, pipes.
Comprehensive unit note · ~18 min read
Mathematical reasoning in UGC NET Paper 1 is not a test of advanced mathematics. It is a test of clear, calm, everyday numeracy: can you spot a pattern, translate a word problem into numbers, and check your answer before moving on? The questions reward method over speed. A candidate who writes down one line of working, names the quantity being asked for, and tests the result against common sense will outperform a faster candidate who jumps straight to an option. This unit therefore teaches habits first and formulas second. The unit covers three linked areas. Number and letter series train pattern recognition: differences, ratios, squares, cubes, alternating patterns, and letter positions. Percentages, ratio, profit and loss train proportional thinking, the single most useful skill in the whole unit, because the same idea of a part compared with a whole reappears in discount, interest, averages, and data interpretation. Fractions, decimals, time and work train rate thinking: how much is done per unit of time, how fast something moves, and how combined rates add or subtract. A second theme runs through the unit: units and bases matter. A percentage is meaningless until you know the percentage of what. A ratio is meaningless until you know the total it divides. A speed is meaningless until distance and time use matching units. Most errors in this unit are not calculation errors at all; they are base errors, such as applying a second percentage change to the original price instead of the new price, or adding speeds that refer to different journeys.
Learning objectives
- Identify the rule in number and letter series, including difference, ratio, square, cube, alternating, and coding patterns, and find missing terms.
- Solve percentage, ratio-proportion, profit-loss, discount, simple and compound interest, and average problems with the correct base.
- Convert fluently between fractions, decimals, and percentages and use them in word problems.
- Solve time-work, pipes-and-cisterns, and speed-distance-time problems using per-unit rates.
- Check answers by estimation, unit consistency, and reverse calculation, and avoid common base and percentage-point errors.
Number & letter series
A series is a list of terms that follows a rule. Your job is to recover the rule from the terms you are given and apply it to the missing term. Start with the simplest tests, in order. First, look at differences between consecutive terms. If the differences are constant, the series is arithmetic. If the differences themselves change by a constant amount, look at second differences. Second, look at ratios: divide each term by the previous one. If the ratio is constant, the series is geometric. Third, look for squares and cubes, prime numbers, and alternating patterns in which odd-position and even-position terms follow separate rules. Many NET questions combine two ideas, for example a difference that grows by two each time, or a series that alternates between adding and multiplying.
In an arithmetic progression, the first term is usually written a and the common difference d. The nth term is a plus (n minus 1) times d, and the sum of the first n terms is n over 2 times twice a plus (n minus 1) times d. In a geometric progression with first term a and common ratio r, the nth term is a times r to the power n minus 1, and the sum of the first n terms is a times (r to the power n minus 1) divided by (r minus 1), when r is not 1.
Letter series work the same way once you convert letters to positions: A is 1, B is 2, and so on to Z is 26. Write the positions under the letters before you look for a pattern. A series such as A, C, F, J, O has positions 1, 3, 6, 10, 15, with differences 2, 3, 4, 5, so the next difference is 6 and the next position is 21, which is U. Mixed series, in which a letter and a number alternate or move together, are solved by splitting the series into its two strands and solving each strand separately. Always check the last step: does your rule reproduce every given term, not just the first two? A rule that fits two terms but fails the third is a guess, not a solution.
Coding-decoding
Coding-decoding questions give you a word coded by a rule and ask you to code or decode another word by the same rule. The most common rules are letter shifting, where each letter moves forward or backward by a fixed number of places, letter reversal, where the word is written backwards, and position-value coding, where each letter is replaced by its position number or by the sum of positions. Start by aligning the given word with its code, letter by letter. If CAT is coded as DBU, compare C with D, A with B, and T with U: every letter has moved forward by one place, so the rule is plus one. If the shifts differ from letter to letter, look for a pattern in the shifts themselves, such as plus one, plus two, plus three.
Position-value coding needs care with sums. CAT has positions 3, 1, and 20, whose sum is 24, so CAT coded as 24 fits a sum rule. Test the rule on every example given before applying it; a single example can fit several rules, and the question setter usually gives enough examples to pin the rule down. Also watch for reverse position values, where A is 26 and Z is 1, and for codes that mix letters and numbers. Write out the alphabet positions for the letters involved rather than trusting memory under time pressure, and after decoding, read the result as a word: if it is nonsense, your rule is probably wrong.
Percentages, ratio, profit-loss
A percentage is a fraction out of 100. To find x percent of a quantity, multiply the quantity by x and divide by 100. To express one quantity as a percentage of another, divide the first by the second and multiply by 100. The crucial discipline is naming the base. An increase from 80 to 100 is an increase of 20 on a base of 80, which is 25 percent, not 20 percent. A decrease from 100 to 80 is a decrease of 20 on a base of 100, which is 20 percent. The same absolute change gives different percentages because the base changed.
A ratio compares quantities of the same kind. The ratio 3 to 5 means that for every 3 parts of the first quantity there are 5 parts of the second, making 8 parts in total. To divide an amount in a ratio, first add the parts, then give each part an equal share. Proportion extends this: if a is to b as c is to d, then a times d equals b times c, and you can solve for an unknown term by cross-multiplication. Profit and loss are percentages with cost price as the base. Profit percent equals profit divided by cost price times 100, and loss percent equals loss divided by cost price times 100. Selling price equals cost price plus profit, or cost price minus loss. Discount is a percentage of the marked price, not of the cost price, and successive discounts apply one after another to the reduced price, never both to the original price. Worked examples below show each of these bases explicitly.
Interest and averages
Simple interest grows only on the principal. With principal P, annual rate R percent, and time T years, simple interest is P times R times T divided by 100, and the amount is principal plus interest. Compound interest grows on principal plus accumulated interest, so each year the base grows. The amount after n years at annual rate R percent, compounded annually, is P times (1 plus R over 100) to the power n, and compound interest is amount minus principal. The difference between simple and compound interest is zero in the first year and grows afterwards; for two years, the difference equals P times (R over 100) squared, a useful check.
An average is a total divided by a count. The average of a set of numbers is their sum divided by how many there are. The power of averages lies in working backwards: if you know an average and a count, you know the total, and a new average after adding or removing a value can be found by adjusting the total. For example, if the average of five numbers is 30, their total is 150, and replacing one number changes the total by the difference between the new and old values. Averages of groups combine by weighting: the combined average of two groups is not the simple average of their averages unless the groups are equal in size. It is the total of both groups divided by the total count. NET questions often test exactly this point, so always ask whether the groups being averaged have the same size before averaging the averages.
Fractions, decimals, time-work
A fraction names a part of a whole: the numerator counts parts and the denominator says how many equal parts make the whole. To compare fractions, convert them to a common denominator or to decimals. To find a fraction of a quantity, multiply the quantity by the numerator and divide by the denominator. Decimals are fractions with denominators 10, 100, and so on, so 0.75 is 75 over 100, which simplifies to 3 over 4. Keep fractions exact through the working and convert to a decimal only at the end, unless the question asks for a decimal answer.
Time and work problems are rate problems. If a person finishes a job in n days, that person does 1 over n of the job per day. Rates add when people work together and subtract when one agent undoes the work of another, as with a leak emptying a tank. If A does 1 over 12 of a job per day and B does 1 over 18 per day, together they do 1 over 12 plus 1 over 18, which is 5 over 36 per day, so the job takes 36 over 5 days, which is 7.2 days. Do not average the times: the combined time is always less than either individual time when both help. Pipes follow the same rule with tanks instead of jobs, and the only extra step is the sign: a filling pipe adds its rate and an emptying pipe subtracts it. If the net rate is negative, the tank empties rather than fills, and the time is found from the magnitude of the net rate.
Speed, distance and time
Speed is distance divided by time, distance is speed times time, and time is distance divided by speed. These three forms are one relationship, so any question gives two of the quantities and asks for the third, or gives a change in one and asks for its effect on another. The most common error is mismatched units: kilometres per hour must pair with kilometres and hours, while metres per second must pair with metres and seconds. To convert kilometres per hour to metres per second, multiply by 5 over 18; to convert metres per second to kilometres per hour, multiply by 18 over 5.
When the same distance is covered at two speeds, the average speed for the whole journey is not the simple average of the two speeds unless the times are equal. For equal distances, average speed is twice the product of the speeds divided by their sum. For example, going 60 kilometres per hour one way and 40 kilometres per hour back over the same distance gives an average speed of 2 times 60 times 40 divided by 100, which is 48 kilometres per hour, not 50. Relative speed also matters: two bodies moving towards each other close the gap at the sum of their speeds, while one catching up from behind closes the gap at the difference of their speeds.
Key definitions
- Arithmetic progression: A sequence in which each term after the first is obtained by adding a fixed common difference to the previous term.
- Geometric progression: A sequence in which each term after the first is obtained by multiplying the previous term by a fixed common ratio.
- Percentage: A number expressed as a fraction of 100; x percent of a quantity means x hundredths of that quantity.
- Percentage point: The arithmetic difference between two percentages, distinct from the relative percentage change.
- Ratio: A comparison of two quantities of the same kind, expressed as a to b, showing how many parts of each make up the whole.
- Proportion: An equality of two ratios; if a is to b as c is to d, then a times d equals b times c.
- Cost price: The price at which an article is bought; profit and loss percentages are calculated on cost price unless stated otherwise.
- Marked price: The price printed on an article before discount; discount is calculated on marked price.
- Simple interest: Interest calculated only on the original principal for the whole period.
- Compound interest: Interest calculated on principal plus previously accumulated interest, so the base grows each period.
- Average: The sum of a set of values divided by the number of values; the total can be recovered as average times count.
- Work rate: The fraction of a job completed per unit of time; a worker finishing a job in n days has rate 1 over n per day.
- Relative speed: The rate at which the distance between two moving bodies changes: the sum of speeds when moving towards each other, the difference when one follows the other.
Formulas
- Arithmetic progression nth term: a(n) = a + (n - 1)d; sum S(n) = n/2 x [2a + (n - 1)d]
- Geometric progression nth term: a(n) = a x r^(n - 1); sum S(n) = a x (r^n - 1)/(r - 1) for r not equal to 1
- Percentage: x percent of Q = Q x x/100; percentage change = (new value - old value)/old value x 100
- Profit percent = profit/cost price x 100; loss percent = loss/cost price x 100; selling price = cost price + profit = cost price - loss
- Discount: selling price = marked price x (1 - discount percent/100); successive discounts d1 and d2 give a single equivalent discount of d1 + d2 - d1 x d2/100 percent
- Simple interest: SI = P x R x T/100; amount = P + SI
- Compound interest amount: A = P x (1 + R/100)^n; compound interest = A - P
- Average = sum of values/number of values; combined average = (total of group 1 + total of group 2)/(count 1 + count 2)
- Speed = distance/time; distance = speed x time; time = distance/speed; km/h to m/s multiply by 5/18; m/s to km/h multiply by 18/5
- Average speed for equal distances at speeds u and v = 2uv/(u + v)
- Work: if A finishes in a days and B in b days, together they take 1/(1/a + 1/b) = ab/(a + b) days
- Pipes: net filling rate = sum of filling rates - sum of emptying rates; time to fill = 1/net rate
Worked examples
Example 1: Find the next term of the series 2, 6, 12, 20, 30, ...
Differences are 6 - 2 = 4, 12 - 6 = 6, 20 - 12 = 8, and 30 - 20 = 10. The differences increase by 2 each time, so the next difference is 12. The next term is 30 + 12 = 42.
Example 2: A price rises by 20 percent and then falls by 20 percent. What is the net percentage change?
Start with 100. After a 20 percent rise it is 100 + 20 = 120. A 20 percent fall is calculated on 120, giving a fall of 120 x 20/100 = 24. The final price is 120 - 24 = 96. The net change is 96 - 100 = -4, a fall of 4 percent, not zero, because the second percentage uses the new base.
Example 3: An article costs 800 and is sold for 960. Find the profit percent.
Profit = 960 - 800 = 160. Profit percent = 160/800 x 100 = 0.2 x 100 = 20 percent. The base is cost price, not selling price.
Example 4: Find the simple and compound interest on 10000 at 10 percent per annum for 2 years.
Simple interest = 10000 x 10 x 2/100 = 2000. For compound interest, the amount after year 1 is 10000 + 1000 = 11000, and after year 2 it is 11000 + 1100 = 12100. Compound interest = 12100 - 10000 = 2100, which is 100 more than simple interest because the second year earns interest on the first year interest of 1000.
Example 5: A finishes a job in 12 days and B in 18 days. How long do they take working together?
A does 1/12 of the job per day and B does 1/18 per day. Together they do 1/12 + 1/18 = 3/36 + 2/36 = 5/36 per day. The time for the whole job is the reciprocal, 36/5 = 7.2 days.
Example 6: A car travels 60 km at 60 km/h and returns the same 60 km at 40 km/h. Find the average speed for the whole journey.
Outward time = 60/60 = 1 hour. Return time = 60/40 = 1.5 hours. Total distance = 120 km and total time = 2.5 hours. Average speed = 120/2.5 = 48 km/h. The simple average of 60 and 40 is 50 km/h, which is wrong because more time is spent at the slower speed. The formula 2 x 60 x 40/(60 + 40) = 4800/100 = 48 km/h agrees.
Simple interest versus compound interest
| Feature | Simple interest | Compound interest |
|---|---|---|
| Base each period | Original principal only | Principal plus accumulated interest |
| Growth pattern | Equal interest each period | Interest grows each period |
| Difference after 1 year | None | None |
| Difference after 2 years | Lower by P x (R/100) squared | Higher by P x (R/100) squared |
Percentage bases that are commonly confused
| Feature | Correct base | Common error |
|---|---|---|
| Profit or loss percent | Cost price | Using selling price as the base |
| Discount percent | Marked price | Using cost price as the base |
| Second successive change | The new value after the first change | Applying both changes to the original value |
| Percentage increase | The old value | Dividing by the new value |
Rate problems at a glance
| Situation | Rate | Combined rate |
|---|---|---|
| Two workers together | 1/a and 1/b per day | 1/a + 1/b |
| Filling pipe and emptying pipe | 1/a and 1/b per hour | 1/a - 1/b |
| Two bodies towards each other | Speeds u and v | u + v |
| One body catching another | Speeds u and v | Difference of speeds |
Diagrams (described)
Series-solving flowchart, described: write the terms, compute differences, and if differences are not constant compute second differences; if neither is constant, compute ratios; if ratios are not constant, convert letters to positions and repeat, and check for alternating strands before guessing a rule.
Percentage-base diagram, described: a bar for the original value is marked as the base for the first change, then a new bar for the changed value is marked as the base for the second change, showing why successive percentages never add directly.
Work-rate diagram, described: a tank or job divided into equal daily slices, with A contributing one slice pattern and B another, so that the combined daily progress is the sum of the two slice counts and the total time is the reciprocal of the combined rate.
Exam points
- Series questions usually test differences and second differences before ratios; always compute differences first and check the rule against every term.
- Successive percentage changes are a favourite NET trap: a rise of x percent followed by a fall of x percent always gives a net fall, because the fall acts on the larger value.
- Profit and loss percentages use cost price as the base, while discount uses marked price; questions that mix the two test whether you switched bases.
- For two years, compound interest exceeds simple interest by P times (R over 100) squared; this shortcut also serves as a check on a full calculation.
- Combined work time is always shorter than each individual time when both workers help; an answer longer than either individual time signals an error.
- Average speed over equal distances uses the harmonic form 2uv/(u + v), not the simple average of the two speeds.
- Letter series require writing positions 1 to 26 under the letters; most errors come from miscounting positions, not from missing the pattern.
- In pipes problems, an emptying pipe subtracts its rate; if the net rate is negative, the tank empties and the time uses the magnitude of the net rate.
Common mistakes
- Adding successive percentage changes directly: a 20 percent rise followed by a 20 percent fall is not zero change; the fall applies to the risen value, giving a net fall of 4 percent.
- Calculating profit percent on selling price: profit percent always uses cost price as the base unless the question explicitly says otherwise.
- Averaging times for combined work: two workers taking 12 and 18 days do not take 15 days together; rates add, so the time is 7.2 days.
- Averaging speeds for equal distances: 60 km/h one way and 40 km/h back do not average to 50 km/h; the correct average speed is 48 km/h.
- Confusing percentage points with percent change: a rise from 20 percent to 25 percent is 5 percentage points but a 25 percent relative rise.
- Forgetting unit conversion: a speed in km/h cannot be multiplied by a time in minutes without converting one of them first.
- Accepting a series rule that fits only the first two terms: a valid rule must reproduce every given term before it is used to predict the next one.
- Rounding a recurring decimal early: keep 1/3 as a fraction through the working and convert to a decimal only at the end.
Practice questions from this note (original practice — not official PYQs)
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Quick revision: Keep three models in mind and everything else follows. Series are solved by differences, then ratios, then letter positions, with every rule checked against all given terms. Percentages, profit, and discount are solved by naming the base first: cost price for profit and loss, marked price for discount, and the new value for any second change. Work, pipes, and speed are solved by rates per unit of time, added when agents help and subtracted when they oppose. Key lines to carry into the exam: combined work time is the reciprocal of the sum of rates; average speed over equal distances is 2uv over u plus v; compound interest for two years exceeds simple interest by P times R over 100 squared; and a percentage is never complete until you can say percent of what.
Unit revision note (quick recap)
This unit rewards method, not memory: write the relation, use the base value correctly, and check the answer by estimation.
Learning objectives
- Solve series, ratio, percentage and time-work problems
- Use successive-percentage logic
- Estimate before calculating
Key concepts
- Percentage change: (New − Old) / Old × 100. The denominator is always the original value.
- Successive change: a% then b% gives a + b + ab/100 (with signs), not a + b.
- Time–work rate: If A finishes in n days, A does 1/n of the work per day; add rates for joint work.
Formulas
- % change = (New − Old)/Old × 100
- Net of +a% then −a% = −a²/100 %
- Combined rate = 1/n1 + 1/n2
Worked examples
- Price +20% then −20%: net = 20 − 20 − 400/100 = −4%.
- A in 10 days, B in 15 days: together 1/10 + 1/15 = 1/6 per day = 6 days.
Exam points
- Series: check differences first, then ratios.
- In profit/loss, always state the base (cost price vs selling price).
Common mistakes
- Dividing percentage change by the new value instead of the old value.
Quick revision: Series, percentages, ratio, averages, time–speed–distance, time–work — method first, arithmetic second.
Practice questions from the whole unit bank (28 in the bank)
This unit's practice bank — the note above has its own smaller practice set drawn from the same bank, so a question can appear in both lists. The list below shows the free sample or the full bank depending on your plan.
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Related tests
Unit test for this unit (28 bank questions, 34 min) · Full Mock — Paper 1 (50 questions) · Combined 150-question mock (180 min). All +2 per correct, no negative marking.
Revision summary
Series, percentages, ratio, averages, time–speed–distance, time–work — method first, arithmetic second.
After revising, take the unit test above, then check your unit-wise report on the Dashboard to find weak topics.