Arithmetic
Practice MCQsArithmetic is the foundation of elementary mathematics. It deals with numbers, operations, fractions, decimals, square roots, percentages, ratio, proportion, profit and loss, interest, time and work, time and distance, H.C.F., L.C.M., divisibility, prime numbers, factorisation, Euclidean algorithm, and logarithms.
Arithmetic is the foundation of elementary mathematics. It deals with numbers, operations, fractions, decimals, square roots, percentages, ratio, proportion, profit and loss, interest, time and work, time and distance, H.C.F., L.C.M., divisibility, prime numbers, factorisation, Euclidean algorithm, and logarithms. A strong command of Arithmetic is essential for almost every competitive examination.
What is Arithmetic?
Arithmetic is the branch of mathematics that studies numbers and basic operations on them. It includes addition, subtraction, multiplication, division, fractions, decimals, square roots, percentages, ratios, averages, and applications of numbers in practical problems.
In competitive exams, Arithmetic questions are usually direct, practical, and calculation-based. They test speed, accuracy, formula knowledge, number sense, and the ability to interpret word problems correctly.
| Area | What It Covers | Exam Focus |
|---|---|---|
| Number System | Natural numbers, integers, rational numbers, real numbers. | Classification, properties, operations. |
| Basic Operations | Addition, subtraction, multiplication, division, square roots, decimals. | Simplification and fast calculation. |
| Applied Arithmetic | Percentage, profit and loss, interest, ratio, proportion, variation. | Word problems and commercial calculations. |
| Time-Based Problems | Time and distance, time and work, unitary method. | Speed, efficiency, work rate. |
| Number Theory | Prime numbers, factors, multiples, divisibility, H.C.F., L.C.M. | Divisibility tests, Euclidean algorithm. |
| Logarithms | Common logarithms, laws of logarithms, logarithmic tables. | Simplification and table-based calculation. |
“Arithmetic is the foundation on which speed, accuracy, and mathematical confidence are built.”
Key points
- Understand types of numbers clearly.
- Master the four fundamental operations.
- Practice decimals, fractions, and square roots.
- Learn percentage, ratio, and proportion.
- Apply formulas in interest, profit, loss, work, and speed.
- Memorize divisibility tests.
- Use factorisation for H.C.F. and L.C.M.
- Understand common logarithms and their laws.
Number System
The number system is the basic language of Arithmetic. Different types of numbers are used for counting, measuring, comparing, and solving practical problems.
| Type of Number | Meaning | Examples |
|---|---|---|
| Natural Numbers | Counting numbers starting from 1. | \(1,2,3,4,\ldots\) |
| Whole Numbers | Natural numbers along with zero. | \(0,1,2,3,4,\ldots\) |
| Integers | Positive numbers, negative numbers, and zero. | \(\ldots,-3,-2,-1,0,1,2,3,\ldots\) |
| Rational Numbers | Numbers that can be written as \(\frac{p}{q}\), where \(q \neq 0\). | \(\frac{2}{3}, -\frac{5}{7}, 4, 0.75\) |
| Irrational Numbers | Numbers that cannot be written as \(\frac{p}{q}\). | \(\sqrt{2}, \sqrt{3}, \pi\) |
| Real Numbers | All rational and irrational numbers together. | \(-5, 0, \frac{3}{4}, \sqrt{5}, \pi\) |
Formula and Method Bank
Part \(= \frac{\text{Percentage} \times \text{Whole}}{100}\)
Loss \(= CP-SP\)
Profit % \(= \frac{\text{Profit}}{CP} \times 100\)
Amount \(=P+SI\)
\(CI=A-P\)
Distance \(=\text{Speed} \times \text{Time}\)
Total work \(=\text{Rate} \times \text{Time}\)
If \(a:b=c:d\), then \(ad=bc\)
\(\log\left(\frac{a}{b}\right)=\log a-\log b\)
\(\log(a^n)=n\log a\)
Tip: Most Arithmetic questions become simple after identifying whether they are based on percentage, ratio, speed, work, interest, or number theory.
Arithmetic Problem-Solving Guide
Arithmetic questions become easier when the student first identifies the type of numerical relationship involved. The table below helps select the correct method quickly.
| Question Type | What to Use | Typical Clue |
|---|---|---|
| Number classification | Number system rules | Natural, whole, integer, rational, irrational, real |
| Simplification | BODMAS / order of operations | Brackets, powers, roots, fractions, decimals |
| Percentage problem | \(\frac{\text{Part}}{\text{Whole}}\times 100\) | Percent, increase, decrease, comparison |
| Profit and loss | Profit, loss, discount formulas | Cost price, selling price, marked price |
| Interest problem | Simple interest or compound interest | Principal, rate, time, amount |
| Ratio and proportion | Common variable method | Share, divide, compare, parts of a total |
| Variation problem | Direct or inverse variation | More quantity gives more result, or more workers take less time |
| Time and distance | \(\text{Speed}=\frac{\text{Distance}}{\text{Time}}\) | Speed, distance, time, trains, relative speed |
| Time and work | Work-rate method | Days, efficiency, workers, pipes, combined work |
| H.C.F. and L.C.M. | Prime factorisation or Euclidean algorithm | Factors, multiples, bells, remainders, common divisor |
| Divisibility | Divisibility tests | Check divisibility by 2, 3, 4, 5, 9, 11 |
| Logarithms | Product, quotient and power laws | \(\log(ab)\), \(\log\left(\frac{a}{b}\right)\), \(\log(a^n)\) |
Fundamental Operations and Decimal Fractions
Arithmetic begins with four fundamental operations: addition, subtraction, multiplication, and division. These operations are used with whole numbers, integers, fractions, decimals, percentages, and square roots.
| Operation | Meaning | Example |
|---|---|---|
| Addition | Combining two or more quantities. | \(25+35=60\) |
| Subtraction | Finding the difference between quantities. | \(80-45=35\) |
| Multiplication | Repeated addition or scaling. | \(12 \times 5=60\) |
| Division | Sharing or grouping equally. | \(72 \div 8=9\) |
| Square Root | A number which, when multiplied by itself, gives the given number. | \(\sqrt{49}=7\) |
| Decimal Fraction | A fraction written using decimal point. | \(0.75=\frac{75}{100}=\frac{3}{4}\) |
Applied Arithmetic
Applied Arithmetic includes practical topics used in everyday calculations and competitive examinations. These topics usually appear as word problems.
Unitary Method
Find the value of one unit first, then calculate the value of required units.
- Direct variation
- Indirect variation
- Cost and quantity
- Work and wages
Time and Distance
Problems involving speed, distance, and time.
- Speed
- Average speed
- Relative speed
- Trains and races
Time and Work
Problems based on work done by people, machines, or pipes.
- Work rate
- Combined work
- Efficiency
- Pipes and cisterns
Commercial Arithmetic
Practical calculations involving money and percentage.
- Profit and loss
- Discount
- Simple interest
- Compound interest
Percentage, Interest, Profit and Loss
| Topic | Core Formula | Use |
|---|---|---|
| Percentage | \(\frac{\text{Part}}{\text{Whole}} \times 100\) | Marks, population, increase, decrease, comparison. |
| Simple Interest | \(SI=\frac{P \times R \times T}{100}\) | Interest on principal for fixed rate and time. |
| Compound Interest | \(A=P\left(1+\frac{R}{100}\right)^T\) | Interest calculated on amount after every period. |
| Profit | \(Profit=SP-CP\) | When selling price is greater than cost price. |
| Loss | \(Loss=CP-SP\) | When cost price is greater than selling price. |
| Discount | \(Discount=Marked\ Price-Selling\ Price\) | Used in shop and price-based questions. |
Ratio, Proportion and Variation
Ratio compares two quantities of the same kind. Proportion shows equality of two ratios. Variation shows how one quantity changes with another quantity.
| Concept | Meaning | Example |
|---|---|---|
| Ratio | Comparison of two quantities by division. | \(a:b=\frac{a}{b}\) |
| Proportion | Equality of two ratios. | If \(a:b=c:d\), then \(ad=bc\) |
| Direct Variation | Both quantities increase or decrease together. | More items cost more money. |
| Inverse Variation | One quantity increases while the other decreases. | More workers take less time. |
| Compound Variation | A quantity depends on two or more quantities. | Work depends on workers and time. |
Elementary Number Theory
Elementary Number Theory studies properties of whole numbers and integers. It includes divisibility, prime numbers, composite numbers, factors, multiples, H.C.F., L.C.M., division algorithm, factorisation theorem, and Euclidean algorithm.
| Concept | Meaning | Example |
|---|---|---|
| Division Algorithm | For integers \(a\) and \(b\), \(a=bq+r\), where \(0 \leq r < b\). | \(23=5 \times 4+3\) |
| Prime Number | A number greater than 1 having exactly two factors: 1 and itself. | 2, 3, 5, 7, 11 |
| Composite Number | A number greater than 1 having more than two factors. | 4, 6, 8, 9, 10 |
| Factor | A number that divides another number exactly. | 3 is a factor of 12. |
| Multiple | A number obtained by multiplying a given number by an integer. | 12, 24, 36 are multiples of 12. |
| Factorisation Theorem | Every composite number can be expressed as a product of prime factors. | \(60=2^2 \times 3 \times 5\) |
Divisibility Tests
Divisibility tests help determine whether a number is divisible by another number without performing full division. These tests are very useful in simplification, factors, H.C.F., L.C.M., and number system questions.
| Divisible By | Test | Example |
|---|---|---|
| 2 | Last digit is even. | 248 is divisible by 2. |
| 3 | Sum of digits is divisible by 3. | 123: \(1+2+3=6\), so divisible by 3. |
| 4 | Last two digits are divisible by 4. | 1316: last two digits 16, so divisible by 4. |
| 5 | Last digit is 0 or 5. | 275 is divisible by 5. |
| 9 | Sum of digits is divisible by 9. | 729: \(7+2+9=18\), so divisible by 9. |
| 11 | Difference between sum of alternate digits is 0 or divisible by 11. | 121: \((1+1)-2=0\), so divisible by 11. |
H.C.F., L.C.M. and Euclidean Algorithm
H.C.F. is the highest common factor of two or more numbers. L.C.M. is the least common multiple of two or more numbers. These are important in fractions, divisibility, time cycles, bells, remainders, and number theory problems.
| Concept | Meaning | Method |
|---|---|---|
| H.C.F. | Greatest number that divides all given numbers exactly. | Use prime factorisation or Euclidean algorithm. |
| L.C.M. | Smallest number exactly divisible by all given numbers. | Use prime factorisation or division method. |
| Prime Factorisation Method | Write each number as product of prime factors. | Useful for small and medium numbers. |
| Euclidean Algorithm | Repeated division method for finding H.C.F. | Useful for larger numbers. |
| Relation | For two numbers: Product of numbers = H.C.F. × L.C.M. | \(a \times b = HCF \times LCM\) |
Logarithms to Base 10
Common logarithms are logarithms to base 10. They are written as \(\log x\) when the base is understood to be 10. Logarithms are useful for simplifying multiplication, division, powers, roots, and calculations involving large numbers.
| Law | Formula | Meaning |
|---|---|---|
| Product Law | \(\log(ab)=\log a+\log b\) | Multiplication becomes addition. |
| Quotient Law | \(\log\left(\frac{a}{b}\right)=\log a-\log b\) | Division becomes subtraction. |
| Power Law | \(\log(a^n)=n\log a\) | Powers become multiplication. |
| Log of 1 | \(\log 1=0\) | Because \(10^0=1\). |
| Log of 10 | \(\log 10=1\) | Because \(10^1=10\). |
| Use of Tables | Characteristic + Mantissa | Used to find approximate logarithmic values. |
Step-by-Step Solving Method
| Step | Action | Example Focus |
|---|---|---|
| Step 1 | Identify the Arithmetic topic involved. | Percentage, ratio, H.C.F., L.C.M., speed, work, interest, logarithm. |
| Step 2 | Write the given values clearly. | Principal, rate, time, speed, distance, cost price, selling price. |
| Step 3 | Select the correct formula or method. | Use SI formula, speed formula, Euclidean algorithm, or divisibility test. |
| Step 4 | Substitute values carefully and simplify. | Keep units consistent and avoid sign mistakes. |
| Step 5 | Check whether the answer satisfies the question. | Verify with original condition, units, and practical meaning. |
Solved Examples
| Question | Method | Answer |
|---|---|---|
| Find 25% of 360. |
\(25\%=\frac{25}{100}=\frac{1}{4}\) \(25\%\) of 360 \(=360 \div 4=90\) |
90 |
| A person travels 180 km in 3 hours. Find the speed. |
Speed \(=\frac{\text{Distance}}{\text{Time}}\) Speed \(=\frac{180}{3}=60\) |
60 km/h |
| A can complete a work in 10 days. What is A's one-day work? | One-day work \(=\frac{1}{10}\) | \(\frac{1}{10}\) |
| Find simple interest on ₹5000 at 8% per annum for 2 years. |
\(SI=\frac{P \times R \times T}{100}\) \(SI=\frac{5000 \times 8 \times 2}{100}=800\) |
₹800 |
| An article is bought for ₹400 and sold for ₹500. Find profit percent. |
Profit \(=500-400=100\) Profit % \(=\frac{100}{400} \times 100=25\%\) |
25% |
| Find the H.C.F. of 24 and 36. |
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 Highest common factor is 12. |
12 |
| Check whether 729 is divisible by 9. |
Sum of digits \(=7+2+9=18\) Since 18 is divisible by 9, 729 is divisible by 9. |
Yes |
| Simplify \(\log(1000)\) to base 10. |
\(1000=10^3\) \(\log(1000)=\log(10^3)=3\log 10=3\) |
3 |
Note: Always check whether the problem is asking for value, percentage, rate, time, distance, H.C.F., L.C.M., divisibility, or logarithmic simplification.
Common Traps and Shortcuts
Common Traps
- Confusing natural numbers with whole numbers.
- Forgetting that 1 is neither prime nor composite.
- Using wrong base in logarithm questions.
- Calculating profit percentage on selling price instead of cost price.
- Mixing units in speed, time, and distance problems.
- Confusing direct variation with inverse variation.
- Using L.C.M. when H.C.F. is required, or vice versa.
- Forgetting decimal place movement in decimal multiplication or division.
- Applying divisibility tests incorrectly for large numbers.
Useful Shortcuts
- Use \(25\%=\frac{1}{4}\), \(50\%=\frac{1}{2}\), \(75\%=\frac{3}{4}\).
- Use divisibility tests before factorisation.
- Use Euclidean algorithm for large H.C.F. questions.
- For ratio problems, assume parts as \(x\), \(2x\), \(3x\), etc.
- For time and work, take total work as L.C.M. of given times.
- For speed questions, keep units consistent before calculation.
- For compound interest, calculate amount first and then subtract principal.
- For logarithms, convert multiplication into addition and division into subtraction.
Practice
A) Multiple Choice Questions
-
Which of the following is a natural number?
0 -3 5 \(\frac{2}{3}\)
-
Find 20% of 450.
80 90 100 110
-
Which number is divisible by 11?
121 123 125 127
-
The H.C.F. of 18 and 24 is:
3 6 12 72
-
\(\log 100\) to base 10 is:
1 2 10 100
B) Solve the Higher-Order Problems
- A car travels 240 km in 4 hours. Find its speed. (Hint: Speed = Distance / Time.)
- Find simple interest on ₹8000 at 10% per annum for 3 years. (Hint: Use \(SI=\frac{P \times R \times T}{100}\).)
- Find the L.C.M. of 12 and 18. (Hint: Use prime factorisation.)
- A shopkeeper buys an item for ₹600 and sells it for ₹750. Find the profit percentage. (Hint: Profit % = Profit / Cost Price × 100.)
- Use Euclidean algorithm to find the H.C.F. of 252 and 105. (Hint: Repeatedly divide larger number by smaller number.)
C) Match the Concept with the Correct Rule
| Concept | Correct Rule / Meaning |
|---|---|
| Natural Numbers | Counting numbers starting from 1 |
| Percentage | Part divided by whole multiplied by 100 |
| Prime Number | Number greater than 1 with exactly two factors |
| H.C.F. | Greatest common factor of given numbers |
| L.C.M. | Smallest common multiple of given numbers |
| Simple Interest | \(\frac{P \times R \times T}{100}\) |
| Log Product Law | \(\log(ab)=\log a+\log b\) |
| Euclidean Algorithm | Repeated division method to find H.C.F. |
Arithmetic Reminder
Arithmetic is the most practical area of elementary mathematics. It includes number system, operations, decimal fractions, percentages, unitary method, ratio and proportion, variation, time and distance, time and work, interest, profit and loss, divisibility, H.C.F., L.C.M., Euclidean algorithm, and logarithms. Regular practice improves both accuracy and speed.
Task: Create five Arithmetic questions using one question each from percentage, time and distance, profit and loss, H.C.F./L.C.M., and logarithms.
Show Suggested Answers
Multiple Choice
-
5
Natural numbers are counting numbers starting from 1. Therefore, 5 is a natural number. -
90
\(20\%\) of 450 \(=\frac{20}{100} \times 450=90\). -
121
For 121, \((1+1)-2=0\), so it is divisible by 11. -
6
Factors of 18: 1, 2, 3, 6, 9, 18.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
H.C.F. = 6. -
2
\(100=10^2\), so \(\log 100=2\).
Higher-Order Problems
-
Distance = 240 km, Time = 4 hours.
Speed \(=\frac{240}{4}=60\).
Answer = 60 km/h. -
\(P=8000\), \(R=10\), \(T=3\).
\(SI=\frac{8000 \times 10 \times 3}{100}=2400\).
Answer = ₹2400. -
\(12=2^2 \times 3\).
\(18=2 \times 3^2\).
L.C.M. \(=2^2 \times 3^2=36\).
Answer = 36. -
Cost Price = ₹600, Selling Price = ₹750.
Profit \(=750-600=150\).
Profit % \(=\frac{150}{600} \times 100=25\%\).
Answer = 25%. -
\(252=105 \times 2+42\)
\(105=42 \times 2+21\)
\(42=21 \times 2+0\)
Last non-zero remainder is 21.
Answer = 21.
Concept Matching
- Natural Numbers → Counting numbers starting from 1
- Percentage → Part divided by whole multiplied by 100
- Prime Number → Number greater than 1 with exactly two factors
- H.C.F. → Greatest common factor of given numbers
- L.C.M. → Smallest common multiple of given numbers
- Simple Interest → \(\frac{P \times R \times T}{100}\)
- Log Product Law → \(\log(ab)=\log a+\log b\)
- Euclidean Algorithm → Repeated division method to find H.C.F.
Clue Explanation
Arithmetic questions are solved by identifying the correct numerical relationship. Use basic operations for simplification, percentage formulas for comparison, speed formulas for movement, work rates for work problems, prime factorisation for H.C.F. and L.C.M., and logarithm laws for simplifying products, quotients, and powers.
Exam tips
- Revise multiplication tables and squares regularly.
- Use divisibility tests before factorisation.
- Convert percentages into fractions for faster calculation.
- Keep units consistent in time and distance problems.
- Use total work as L.C.M. in time and work questions.
- Calculate profit and loss percentage on cost price.
- Use Euclidean algorithm for larger H.C.F. problems.
- Remember that common logarithms are to base 10.