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Mathematics forms the foundation for solving a variety of practical problems in daily life, especially when preparing for competitive entrance exams. Understanding the applications of basic mathematics equips you to handle questions involving numbers, money, measurement, and relationships between quantities. This chapter focuses on key concepts such as number operations, working with fractions and decimals, calculating percentages and discounts, understanding simple interest, and solving problems involving ratios and proportions.
In India, the metric system is the standard for measurements, and prices are typically given in Indian Rupees (INR). Problems will often involve these units, making the connection to real-life situations stronger and helping you build skills that are both exam-ready and practical.
We will start from the basics, gradually moving into more complex applications with clear examples, step-by-step solutions, and tips to solve problems faster and more accurately.
The concept of percentage represents a part of a whole expressed out of 100. For example, 30% means 30 parts out of 100 parts.
Converting percentages to decimal or fraction form helps simplify calculations. The relationship between these three forms is fundamental for solving problems in discounts, markups, and comparisons.
| Percentage (%) | Fraction | Decimal |
|---|---|---|
| 50% | 1/2 | 0.50 |
| 25% | 1/4 | 0.25 |
| 10% | 1/10 | 0.10 |
| 20% | 1/5 | 0.20 |
| 5% | 1/20 | 0.05 |
| 75% | 3/4 | 0.75 |
Important: To convert any percentage to a decimal, divide by 100. For example, 30% = 30/100 = 0.30.
Calculating a percentage of a given amount is done using this formula:
In real life, items often have a discount applied during sales. A discount is a reduction from the marked price (MRP) expressed as a percentage.
The discount amount is found by:
Then, the selling price (SP) after discount is:
Step 1: Calculate the discount amount using the formula:
Discount = (15/100) x 1200 = 0.15 x 1200 = Rs.180
Step 2: Find the selling price after subtracting the discount:
SP = MRP - Discount = Rs.1200 - Rs.180 = Rs.1020
Answer: The selling price of the shirt is Rs.1020
Simple interest refers to the interest earned or paid on a principal amount over a period, calculated only on the original principal, not on accumulated interest.
Three main components determine simple interest:
The formula for calculating simple interest is:
Note on units: If time is given in months, convert it to years by dividing by 12. For example, 6 months = 6/12 = 0.5 years.
Step 1: Identify the values:
Step 2: Use the simple interest formula:
\( I = \frac{P \times R \times T}{100} = \frac{50,000 \times 8 \times 3}{100} \)
\( = \frac{1,200,000}{100} = 12,000 \)
Answer: The interest Riya will pay after 3 years is Rs.12,000.
A ratio compares two quantities of the same kind by division, showing how many times one quantity contains the other.
The ratio of two numbers \( a \) and \( b \) is written as:
Proportion states that two ratios are equal:
To solve proportion problems, cross multiplication is commonly used:
If \(\frac{a}{b} = \frac{c}{d}\), then \(a \times d = b \times c\).
graph TD A[Start: Given proportion problem] --> B[Identify quantities a, b, c, d] B --> C[Set up proportion \frac{a}{b} = \frac{c}{d}] C --> D{Is an unknown value present?} D -- Yes --> E[Use cross multiplication to form equation: a x d = b x c] E --> F[Solve for unknown] D -- No --> G[Verify equality of ratios] F --> H[Answer] G --> HScaling involves multiplying or dividing quantities to maintain the same ratio.
Mixture problems often require combining substances with different proportions or concentrations, then using ratios to determine final composition.
Step 1: Identify the quantities and required concentration:
Step 2: Let the volume of solution A be \( x \) liters. Then volume of solution B will be \( 20 - x \) liters.
Step 3: Set up equation using the concept of alligation (difference method):
Difference between concentrations:
Step 4: The ratio in which Solution A and B should be mixed is:
\( A : B = 10 : 20 = 1 : 2 \)
Step 5: Since total volume is 20 liters, divide in ratio 1:2:
Volume of A = \( \frac{1}{1+2} \times 20 = \frac{1}{3} \times 20 = 6.67 \) liters
Volume of B = \( 20 - 6.67 = 13.33 \) liters
Answer: Mix approximately 6.67 liters of Solution A and 13.33 liters of Solution B to get 20 liters of 50% alcohol solution.
Step 1: Use percentage formula:
\( \text{Percentage} = \frac{\text{Marks obtained}}{\text{Total marks}} \times 100 \)
\( = \frac{450}{600} \times 100 = 0.75 \times 100 = 75\% \)
Answer: The student scored 75% marks.
Step 1: Calculate discount:
Discount = (10/100) x 40,000 = Rs.4,000
Step 2: Calculate selling price:
SP = 40,000 - 4,000 = Rs.36,000
Answer: Discount is Rs.4,000 and selling price is Rs.36,000.
Step 1: Identify known values:
Principal, \(P = 1,00,000\)
Rate, \(R = 6\%\)
Time, \(T = 5\) years
Step 2: Calculate interest using formula:
\( I = \frac{1,00,000 \times 6 \times 5}{100} = \frac{30,00,000}{100} = 30,000 \)
Answer: Rahul will earn Rs.30,000 as interest.
Step 1: Ratios given:
Flour : Sugar : Butter = 3 : 5 : 2
Sugar amount given = 900g corresponds to the part '5' in ratio.
Step 2: Find the multiplier:
Multiplier \( = \frac{900}{5} = 180 \)
Step 3: Calculate quantity of flour and butter:
Answer: For 900g sugar, use 540g flour and 360g butter.
When to use: When two ratios are equal and you need to find an unknown value.
When to use: While calculating percentages, discounts, or interest.
When to use: In almost all interest calculation problems.
When to use: For quick elimination in multiple-choice questions or rough estimations.
When to use: Time-based interest or rate problems.
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