Arithmetic

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Arithmetic in CLAT quantitative techniques covers the fundamental mathematical operations and their applications in problem-solving contexts. The questions test your numerical ability, speed, and accuracy rather than advanced mathematical concepts.

Core Arithmetic Operations:

• Addition/Subtraction: Align by place value; watch borrowing/carrying
• Multiplication: Use tables up to 20; learn short multiplication tricks (e.g., $98 \times 87 = 98 \times 100 - 98 \times 13$)
• Division: Long division; divisibility rules
• Fractions: Convert mixed to improper; lowest terms; compare by cross-multiplication
• Decimals: Align decimal points before adding/subtracting; multiply by powers of 10
• Squares and Cubes: Memorise squares 1-30 and cubes 1-15

Divisibility Rules:

Divisor	Rule
2	Last digit is even
3	Sum of digits divisible by 3
4	Last two digits divisible by 4
5	Last digit is 0 or 5
6	Divisible by both 2 and 3
8	Last three digits divisible by 8
9	Sum of digits divisible by 9
11	(Sum of digits in odd positions) − (Sum of digits in even positions) is divisible by 11
12	Divisible by both 3 and 4

Percentage Basics:

• $X%$ means $X/100$
• To express $A$ as a percentage of $B$: $(A/B) \times 100$
• Percentage increase: $((B - A)/A) \times 100$ where $A$ is original
• Successive percentage change: $x% \text{ then } y% = (1 + x/100)(1 + y/100) - 1$ times original

⚡ Exam Tip (CLAT): The most common arithmetic errors in CLAT are: (1) misreading the decimal point, (2) choosing the wrong base for percentage calculation, (3) confusing “percentage points” with “percent.” For example, if a price increases from ₹100 to ₹120, that’s a 20% increase (not 20 percentage points — percentage points refer to arithmetic differences between percentages).

---

🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding of arithmetic problem-solving.

Ratio and Proportion:

• Ratio: $a:b$ means $a/b$. If $a:b = 1:2$ and $b:c = 3:4$, then $a:b:c = 3:6:8$ (make $b$ common by LCM of 2 and 3 = 6)
• Direct proportion: As $a$ increases, $b$ increases (same ratio)
• Indirect (inverse) proportion: As $a$ increases, $b$ decreases ($a_1 b_1 = a_2 b_2$)

Work and Time:

If $A$ can complete a job in $x$ days and $B$ can complete it in $y$ days:

• Combined rate of work: $1/x + 1/y = (x + y)/xy$ jobs per day
• Time to complete together: $xy/(x + y)$ days

Pipes and Cisterns:

Same as work problems, but with inflow (+) and outflow (−). If pipe A fills in $x$ hours and pipe B empties in $y$ hours, net rate $= 1/x - 1/y$ (filling).

Speed, Distance, and Time:

• $Speed = Distance/Time$
• $Time = Distance/Speed$
• $Distance = Speed \times Time$
• Average speed: If equal distances at speeds $s_1$ and $s_2$: $v_{avg} = 2s_1 s_2/(s_1 + s_2)$
• If equal times at speeds $s_1$ and $s_2$: $v_{avg} = (s_1 + s_2)/2$
• Relative speed: Moving in same direction: $|s_1 - s_2|$; opposite directions: $s_1 + s_2$

Profit and Loss:

• $Profit% = (Profit/Cost Price) \times 100$
• $Loss% = (Loss/Cost Price) \times 100$
• When discount $d%$ is offered on marked price: $Selling Price = Marked Price \times (1 - d/100)$
• Relationship: $SP = CP \times (1 + profit%/100)$ or $SP = CP \times (1 - loss%/100)$

Simple and Compound Interest:

• Simple Interest: $SI = P \times R \times T/100$; Amount $= P(1 + RT/100)$
• Compound Interest (annual): $A = P(1 + R/100)^T$; $CI = A - P$
• Compound Interest (half-yearly): Rate halves, time doubles: $A = P(1 + R/200)^{2T}$
• Compound Interest (quarterly): Rate quarters, time quadruples: $A = P(1 + R/400)^{4T}$

⚡ Common CLAT Arithmetic Error: In compound interest, don’t forget that the rate applies to the NEW amount each period (amount including previous interest), whereas in simple interest the interest is always calculated on the original principal. Over multiple periods, compound interest > simple interest when there is profit/interest.

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Mixture and Alligation:

If you have two ingredients costing $a$ and $b$ per unit, and you want a mixture costing $c$ per unit (where $a < c < b$):

• Ratio of quantities: $(b - c) : (c - a)$
• Alligation method: Write the costs in a column, write the desired mean in the middle, draw lines to get the differences as the ratio

Example: Tea at ₹80/kg and ₹120/kg to be mixed to cost ₹100/kg:

• $(120 - 100) : (100 - 80) = 20 : 20 = 1:1$

Partnership:

If $A$ invests $P_A$ for $T_A$ time and $B$ invests $P_B$ for $T_B$ time:

• Profit/loss sharing ratio $= P_A \times T_A : P_B \times T_B$
• If investment amounts change over time, calculate separately for each period

Simple Interest from Compound Interest Perspective:

The difference between Compound Interest and Simple Interest for 2 years at rate $R%$: $$CI - SI = P \times (R/100)^2$$ This is useful for finding the rate when given SI and CI for 2 years.

Time and Work — Advanced:

If $A$ is twice as efficient as $B$:

• $A$‘s 1 day’s work $= 2 \times B$‘s 1 day’s work
• If $A$ works for 3 days then $B$ finishes in $x$ more days, set up equation: $3 \times A’s rate + x \times B’s rate = 1$ (whole work)

If a work is done in $x$ days at normal rate, but $y$ workers left after $z$ days:

• Remaining work: $1 - (z/x)$
• Remaining workers: adjust ratio
• Calculate additional time needed

Boats and Streams:

• Speed of boat in still water $= u$; speed of stream $= v$
• Downstream speed $= u + v$; upstream speed $= u - v$
• To find $u$ and $v$: $u = (down + up)/2$, $v = (down - up)/2$

Calendar Problems:

• Odd days = remainder when divided by 7
• 1 ordinary year = 365 days = 52 weeks + 1 odd day
• 1 leap year = 366 days = 52 weeks + 2 odd days
• 100 years: 76 ordinary + 24 leap = 76×1 + 24×2 = 124 odd days = 124 mod 7 = 5 odd days. So 100 years from Jan 1, 2000 to Jan 1, 2100: 5 odd days.

Clock Problems:

• Minute hand moves 360° in 60 min = 6°/min
• Hour hand moves 360° in 12 hours = 0.5°/min = 30°/hour
• Relative speed: 5.5°/min
• Angle between hands: $|30H - 5.5M|$ degrees; if >180°, subtract from 360°

⚡ Advanced Tip — Unitary Method Mastery: The unitary method (finding value of 1 unit then multiplying) is the most powerful technique in arithmetic. It simplifies any ratio, proportion, or work problem. Always ask: “What is the value of 1 unit (1 day, 1 kg, 1 km, etc.)?” then scale up to the required value.

---

Content adapted based on your selected roadmap duration. Switch tiers using the pill selector above.