Arithmetic in the CDS paper is not a single topic — it is a cluster of inter-related techniques built on one core idea: relationships between quantities. Whether a question asks you about profit on articles, mixing milk with water, or a train crossing a platform, every single one reduces to setting up a ratio, a proportion, or a percentage relationship and solving cleanly.
Think of CDS Arithmetic like a Swiss Army knife. Each blade — profit/loss, ratio-proportion, percentages, averages, simple interest, time-speed-distance — looks different on the surface, but the handle is identical: assign a variable (usually x), write one or two equations, solve. The exam rewards the candidate who recognises which blade to pull, not the one who tries every blade in sequence.
Here is an analogy that works in the exam hall. Imagine you are a supply officer managing rations for a battalion. If you are told "12 boxes of rations cost the same as selling 9 boxes earns you", that is a profit/loss relationship. If you are told "the battalion has 5 parts wheat to every 3 parts rice, and you add 16 kg more rice", that is a ratio-proportion question with an additive change. The underlying arithmetic is the same domestic mathematics you learnt in Class 8-10 — the CDS just dresses it in slightly more formal language.
The practical implication: do not over-prepare any single sub-topic. Prepare the unifying skill — translating a sentence into an equation — and all sub-topics become variations of the same drill. This also means that if you see an unfamiliar framing in the exam, the equation-setup instinct will carry you through even without prior exposure to that exact question type.
One more thing before the deep dive: CDS Arithmetic questions almost never require computations beyond two-digit multiplication. If your working is getting messy, you have set up the wrong equation. Restart.
The foundational relationship is:
The most common CDS variant is the "CP of m articles = SP of n articles" format. The cleanest approach: assume CP of each article = Re. 1.
m articles = Rs. mn articles = Rs. m, so SP of each = m/n((m/n) - 1) / 1 × 100 = (m - n) / n × 100This is a formula worth internalising: when CP of m = SP of n, Profit% = (m-n)/n × 100. For the classic case m=12, n=9: profit% = 3/9 × 100 = 33.33%. That took four seconds.
When a ratio changes because something is added to one component, the structure is always:
Assign the original components as 5x and 3x (or whatever the original ratio dictates). Write the new ratio equation, cross-multiply, solve for x. The trick is: never go back to re-read the question after setting up variables — trust the equation.
For age problems, the structure is the same. Present ages 3x and 4x, future ages 3x+6 and 4x+6. One equation, one unknown.
Two templates cover 90% of CDS percentage questions:
Template 1 — "p% of x added to/subtracted from c = x":
Write: (p/100)x ± c = x, isolate x.
Template 2 — Successive percentages: If a value first increases by a% then by b%, the net change is a + b + ab/100 percent. This avoids double multiplication in many questions.
The total-difference method is faster than working with averages directly. If average of n items = A, total = nA. When a sub-group is removed, the remaining total = remaining count × new average. The excluded group's total is just the difference.
Key check: if the removed group's average comes out lower than the overall average, that makes sense (removing weaker scorers raises the average). If it comes out higher, re-examine your subtraction.
For train problems:
This formula covers crossing a platform, crossing a bridge, or passing a stationary object (platform length = 0 for the last case). After getting speed in m/s, convert to km/hr by multiplying by 18/5.
A common trap: forgetting to add the train's own length when it crosses another moving train. When two trains move in the same direction, relative speed = difference of speeds. Opposite directions, relative speed = sum of speeds.
To find R: R = (SI × 100) / (P × T). This is mechanical. The only way CDS makes this hard is by embedding it in compound interest or by giving you unusual time units (months instead of years). Always convert time to years before substituting.
If three consecutive integers are x, x+1, x+2, their sum = 3x+3. For consecutive even/odd integers: x, x+2, x+4, sum = 3x+6. These give you x in one step. Largest number = x+4, middle = x+2.
When "CP of m articles = SP of n articles", the profit % is directly (m - n) / n × 100. No need to assign Re. 1 and compute SP per article separately.
Example: CP of 12 = SP of 9 → Profit% = (12-9)/9 × 100 = 33.33%.
Standard method (assigning Re.1, finding SP, computing profit): ~45 seconds. This formula: ~8 seconds. The formula works equally for loss when n > m: Loss% = (n-m)/n × 100.
Multiply m/s by 18/5 to get km/hr. Divide km/hr by 18/5 (i.e., multiply by 5/18) to get m/s. Treat this as a reflex — the moment you see a train problem answer in m/s, immediately write × 18/5 without thinking.
For 20 m/s: 20 × 18/5 = 20 × 3.6 = 72 km/hr. Computation takes 4 seconds if you remember that 18/5 = 3.6.
Never compute the new average from scratch. Use: excluded total = original total − remaining total.
Formula: Excluded avg = (n₁ × A₁ − n₂ × A₂) / (n₁ − n₂), where n₁ = original count, A₁ = original avg, n₂ = remaining count, A₂ = new avg.
Example (40 students avg 75, remove 5 so remaining 35 avg 78): Excluded total = 40×75 − 35×78 = 3000 − 2730 = 270. Excluded avg = 270/5 = 54. Three arithmetic steps instead of setting up a weighted average equation from scratch — saves ~30 seconds.
In mixture problems where only one component changes, the unchanged component is your anchor. If milk stays at 5x throughout and only water changes, you never need to touch milk again after writing 5x. All work happens on the water side.
This prevents the common error of accidentally adding to both components. For a problem where 16 L water is added: new water = 3x + 16, new ratio equation is 5x/(3x+16) = 5/7. One equation, no ambiguity. Standard approach of writing two separate equations for milk and water: ~60 seconds. Anchor method: ~25 seconds.
For any "p% of number plus/minus c = number" question, rewrite as (1 − p/100)x = c (or (1 + p/100)x = c for addition on the right side).
Example: "25% of x added to 75 = x" → x − 0.25x = 75 → 0.75x = 75 → x = 100. You never write the intermediate step "0.25x + 75 = x, therefore 0.75x = 75" — you go directly from the sentence to "0.75x = 75". Saves one line and prevents sign errors. From sentence to answer: ~12 seconds.
Read the question and identify the domain first — profit/loss, ratio, average, TSD, SI, percentage, or consecutive numbers. Never start writing until you know which domain you are in.
Decision tree in the exam hall:
Does the question mention CP/SP or profit/loss? → Use CP-SP article formula if "CP of m = SP of n". Otherwise assign CP = Re. 1 or Rs. 100.
Does it mention a ratio that changes after adding/removing something? → Assign original components as multiples of x. Write one equation for the new ratio. Cross-multiply. Solve.
Does it mention average with a sub-group removed or added? → Total-difference method. Total 1 − Total 2 = excluded total. Divide by sub-group size.
Does it mention a train crossing something? → Distance = train length + obstacle length. Speed = distance/time. Convert m/s to km/hr via ×18/5.
Does it mention simple interest? → R = (SI × 100) / (P × T). Verify time is in years.
Is it a percentage question with "p% of x ± c = x"? → Rewrite as (1 ± p/100)x = c. Solve directly.
Consecutive integers/evens/odds? → Sum = 3x + 0/3/6. Solve for x. Largest = x+4.
If working exceeds 4-5 lines, stop and re-read — the setup is wrong.
Why this question: The CP-SP ratio format is the single most common profit/loss structure in CDS. Mastering it removes all computation.
Solving path: Identify this as a "CP of m = SP of n" pattern. Apply formula directly: Profit% = (12−9)/9 × 100 = 3/9 × 100 = 33.33%. Match to option B. Elapsed time under 10 seconds once the pattern is recognised.
Why this question: Ratio-with-additive-change is tested in mixtures, alloys, and population problems. The anchor method is essential.
Solving path: Original: milk = 5x, water = 3x. After adding 16 L water: new ratio = 5x:(3x+16) = 5:7. Cross-multiply: 35x = 15x + 80 → 20x = 80 → x = 4. Original milk = 5(4) = 20 L. Option A.
Why this question: Age ratio problems appear in almost every CDS paper. The structure here is transferable to partnership and work problems.
Solving path: Present ages: 3x and 4x. After 6 years: (3x+6)/(4x+6) = 4/5. Cross-multiply: 15x+30 = 16x+24 → x = 6. A's present age = 3(6) = 18 years. Option A.
Why this question: This is a pure time-speed-distance question using the train-crossing formula. The conversion to km/hr is the most commonly dropped step.
Solving path: Total distance = 150 + 250 = 400 m. Time = 20 s. Speed = 400/20 = 20 m/s. Convert: 20 × 18/5 = 72 km/hr. Option C. Note: if you forget to add the train's own length (150 m), you get 250/20 = 12.5 m/s = 45 km/hr, which is not even in the options — a built-in trap check.
Why this question: The percentage-identity template ("p% of x equals or combines with a constant to give x") recurs across CDS papers in different disguises.
Solving path: 0.25x + 75 = x → 0.75x = 75 → x = 100. Then 40% of 100 = 40. Option A. The question asks for 40% of the number, not the number itself — a deliberate distractor if you stop at x = 100 and mark option B (50).
Forgetting the train's own length. In every crossing problem, the train must travel its own length plus the obstacle's length. Students who only use the platform/bridge length get a speed that's too low and won't match any option — but may still guess wrong rather than recheck.
Treating percentage increase and percentage point increase as the same. "The rate rose from 6% to 8%" is a 2 percentage point increase, not a 33% increase. CDS occasionally uses both phrasings in the same paragraph to test this distinction.
Stopping at x instead of answering what is asked. In age problems and percentage problems, the question almost always asks for 3x or 40% of x, not x itself. Mark the question's target quantity before you start solving so you don't stop one step early.
Cross-multiplying with the wrong orientation. When setting up a/b = c/d, students occasionally write ad = bc correctly but then make sign errors when terms move across the equality. Write the cross-multiplication step explicitly rather than doing it mentally.
Using n = 40 for the removed sub-group instead of 5. In average problems, the excluded count is total − remaining, not total. This mistake inflates the denominator and gives a lower average that may still match a wrong option.
Not converting time to years in SI problems. If the problem states "18 months" and you substitute T = 18 instead of T = 1.5, your rate comes out as 1/12 of the correct value. Always write the units next to T before substituting.