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Mathematics forms the backbone of many competitive exams, and the ability to perform calculations quickly and accurately is essential. At the heart of these calculations lie the basic operations on numbers: addition, subtraction, multiplication, and division. Mastering these operations allows you to handle more complex topics such as fractions, decimals, percentages, ratios, and interest calculations.
This section will guide you through these concepts step-by-step, using metric units and the Indian Rupee (INR) currency to keep examples relevant and relatable. You will also learn how to apply these calculations to real-life scenarios, like shopping discounts or bank interest, enabling you to tackle questions efficiently and with confidence.
The four fundamental number operations are addition, subtraction, multiplication, and division. These are the basic processes by which we combine or separate quantities.
Addition means combining two or more quantities to find the total. Subtraction means taking away one quantity from another to find the difference.
Multiplication is repeated addition; for example, 4 x 3 means adding 4 three times (4 + 4 + 4). Division is the opposite of multiplication; it means splitting a quantity into equal parts.
Understanding the inverse relationships between these operations is key to simplifying problems:
graph LR Addition -->|Inverse| Subtraction Multiplication -->|Inverse| Division Add[Addition: 10 + 5 = 15] Sub[Subtraction: 15 - 5 = 10] Mul[Multiplication: 4 x 3 = 12] Div[Division: 12 / 3 = 4] Add --> Add Sub --> Sub Mul --> Mul Div --> Div Add --> Sub Mul --> Div
This flowchart shows how addition and subtraction undo each other, and the same for multiplication and division. Recognizing these pairs can help you check your answers and solve problems faster.
Key Concept: Addition and subtraction are inverse operations; so are multiplication and division. Use this fact to verify your calculations.
Step 1: Add the weights in kilograms:
3.5 kg + 2.75 kg = 6.25 kg
Step 2: Add the amounts in INR:
Rs.150 + Rs.235.50 = Rs.385.50
Answer: Total weight is 6.25 kg, and total money is Rs.385.50.
Fractions are numbers that represent parts of a whole, written as one number over another, like \(\frac{3}{4}\). Decimals are another way to express part of a whole using the place value system, like 0.75.
Understanding how to convert between fractions and decimals, perform operations on them, and compare their sizes is crucial for many problems.
| Fraction | Decimal |
|---|---|
| \(\frac{1}{2}\) | 0.5 |
| \(\frac{1}{4}\) | 0.25 |
| \(\frac{3}{4}\) | 0.75 |
| \(\frac{1}{5}\) | 0.2 |
| \(\frac{2}{5}\) | 0.4 |
To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a fraction, write the decimal over a power of 10 depending on the number of digits after the decimal point, then simplify.
Step 1: Convert \(\frac{3}{5}\) to decimal.
\(\frac{3}{5} = 3 \div 5 = 0.6\)
Step 2: Add the decimals.
0.6 + 0.4 = 1.0
Answer: The sum is 1.0 (which is also 1).
Percentage means "per hundred" and represents a part of a whole divided into 100 equal parts. For example, 25% means 25 out of 100.
Percentages connect closely to fractions and decimals and are widely used in problems involving discounts, taxes, interest rates, and ratios.
This pie chart shows 25% of a whole, which is a quarter of the circle. Understanding the visual helps grasp how percentages describe parts of a whole.
Step 1: Calculate the discount amount.
Discount = \(\frac{20}{100} \times 1200 = 240\) INR
Step 2: Subtract the discount from the original price.
Discounted price = Rs.1200 - Rs.240 = Rs.960
Answer: The jacket will cost Rs.960 after the discount.
A ratio compares two quantities showing how many times one value contains or is contained within the other, such as 3:2. A proportion states that two ratios are equal.
Ratios allow us to scale quantities and solve mixture problems, a common type of competitive exam question.
graph TD A[Identify Quantities] --> B[Set Ratio] B --> C[Write Proportion Equation] C --> D[Solve for Unknown] D --> E[Interpret Result]
This flowchart shows the logical steps to solve mixture and ratio problems: identify, set ratio, write proportion, solve, and interpret.
Step 1: Let the quantity of 20% solution be \(x\) litres.
Step 2: Total volume after mixing is 15 litres:
5 + \(x\) = 15 -> \(x\) = 10 litres
Step 3: Calculate total alcohol volume from each solution:
Alcohol from 40% solution = 40% of 5 = \(0.40 \times 5 = 2\) litres
Alcohol from 20% solution = 20% of \(x\) = \(0.20 \times x = 0.20x\) litres
Step 4: Total alcohol in final 15 litres at 30% concentration:
30% of 15 = \(0.30 \times 15 = 4.5\) litres
Step 5: Write equation and solve for \(x\):
\(2 + 0.20x = 4.5\)
0.20x = 4.5 - 2 = 2.5
\(x = \frac{2.5}{0.20} = 12.5\) litres
Note: We earlier found \(x=10\) litres from total volume, but alcohol quantity calculation gave 12.5 litres. This indicates a contradiction.
Correct Step 2: Since total volume = 15 litres, \(x = 15 - 5 = 10\) litres. So we re-check alcohol amount:
2 + 0.20 x 10 = 2 + 2 = 4 litres, which is less than 4.5 litres needed. So the problem's constraints mean either volume or percentage is off.
Recheck problem assumption: The question likely asks: "Find \(x\)" given the final concentration, implying that total is 15 litres, so volume constraint is fixed.
Step 6: Let's set \(x\) as unknown and total volume as 5 + \(x\):
Concentration formula:
\(\frac{(40\% \times 5) + (20\% \times x)}{5 + x} = 30\%\)
\(\frac{2 + 0.2x}{5 + x} = 0.30\)
Multiply both sides by \(5 + x\):
2 + 0.2x = 0.30(5 + x) = 1.5 + 0.30x
Simplify:
2 - 1.5 = 0.30x - 0.2x
0.5 = 0.10x -> \(x = \frac{0.5}{0.10} = 5\) litres
Answer: Mix 5 litres of 20% solution with 5 litres of 40% solution to get 10 litres of 30% solution.
Step 1: Align decimal points to add.
7.2 + 3.8 = 11.0 kg
Answer: Total weight is 11 kg.
Step 1: Subtract the bill amount from payment.
700 - 560 = Rs.140
Answer: Balance returned is Rs.140.
Step 1: Ignore decimals and multiply 45 by 23.
45 x 23 = (45x20) + (45x3) = 900 + 135 = 1035
Step 2: Count total decimal places (1 in 4.5 and 1 in 2.3, total 2).
Step 3: Place the decimal point two places from right in 1035.
Result = 10.35
Answer: 4.5 x 2.3 = 10.35
Step 1: Calculate the discount amount.
Discount = \(\frac{12}{100} \times 15000 = Rs.1800\)
Step 2: Subtract discount from original price.
Discounted price = Rs.15000 - Rs.1800 = Rs.13200
Answer: The smartphone costs Rs.13,200 after discount.
Step 1: Total parts in ratio = 2 + 3 = 5 parts
Step 2: Weight of expensive tea = \( \frac{2}{5} \times 5 = 2 \) kg
Weight of cheaper tea = \( \frac{3}{5} \times 5 = 3 \) kg
Step 3: Calculate cost of each tea quantity:
Cost_expensive = 2 kg x Rs.240 = Rs.480
Cost_cheaper = 3 kg x Rs.150 = Rs.450
Step 4: Total cost = Rs.480 + Rs.450 = Rs.930
Step 5: Cost per kg of mixture = Total cost / Total weight
= Rs.930 / 5 = Rs.186 per kg
Answer: Cost of the tea mixture is Rs.186 per kg.
When to use: When comparing sizes of two or more fractions quickly.
When to use: While solving ratio and proportion questions in competitive exams.
When to use: To quickly calculate discounts, interest, or conversions without a calculator.
When to use: Multiplying large numbers or decimals mentally during exams.
When to use: Problems involving monetary calculations or measurements to keep track of units.
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