Comparing Time, Calculating Cost, And Evaluating Expressions In Math

In the realm of mathematics, comparing fractions often arises in everyday situations, such as determining who took longer to complete a task. In this context, let's delve into the problem of Ricky and Renu, who spent different amounts of time finishing their homework. Ricky completed her homework in 5/12 of an hour, while Renu took 3/4 of an hour. To ascertain who took a longer time, we need to compare these two fractions. The main goal here is to determine which fraction represents a larger portion of an hour. To effectively compare the fractions 5/12 and 3/4, we must first find a common denominator. This involves identifying the least common multiple (LCM) of the denominators, which in this case are 12 and 4. The multiples of 4 are 4, 8, 12, 16, and so on, while the multiples of 12 are 12, 24, 36, and so on. The least common multiple of 4 and 12 is 12. Therefore, we will convert both fractions to have a denominator of 12. The fraction 5/12 already has the desired denominator, so we don't need to change it. For the fraction 3/4, we need to multiply both the numerator and the denominator by the same number to get an equivalent fraction with a denominator of 12. To convert the denominator 4 to 12, we multiply it by 3. Thus, we also multiply the numerator 3 by 3, resulting in the equivalent fraction (3 * 3) / (4 * 3) = 9/12. Now that both fractions have the same denominator, we can easily compare them. We have 5/12 and 9/12. Comparing the numerators, we see that 9 is greater than 5. Therefore, 9/12 is greater than 5/12. This means that Renu took 9/12 of an hour, while Ricky took 5/12 of an hour. Since 9/12 is greater than 5/12, Renu took a longer time to complete her homework. This problem highlights the importance of understanding fractions and how to compare them. By finding a common denominator, we can easily determine which fraction is larger, allowing us to solve real-world problems involving time, quantities, and proportions. Understanding these concepts is crucial for developing strong mathematical skills and applying them in various contexts. Moreover, this type of problem-solving enhances critical thinking and analytical abilities, which are valuable in many aspects of life. In conclusion, by converting fractions to a common denominator, we efficiently compared the time taken by Ricky and Renu to complete their homework, revealing that Renu took longer. This underscores the practical application of fractions in everyday scenarios.

Calculating the Cost of Cloth Per Meter

In the world of commerce, determining the cost of goods per unit is a fundamental calculation. Consider this scenario a piece of cloth measuring 3 1/2 meters in length costs ₹128 1/3. Our task is to find the cost of 1 meter of the cloth. This involves dividing the total cost by the total length to find the price per meter. This type of problem often arises in real-life situations, such as when purchasing fabric, materials for construction, or any other item sold by length or weight. Understanding how to calculate the cost per unit is essential for making informed purchasing decisions and managing budgets effectively. To solve this problem, we need to divide the total cost, ₹128 1/3, by the total length, 3 1/2 meters. First, we need to convert these mixed fractions into improper fractions. A mixed fraction consists of a whole number and a proper fraction, while an improper fraction has a numerator that is greater than or equal to its denominator. To convert 128 1/3 into an improper fraction, we multiply the whole number (128) by the denominator (3) and then add the numerator (1). This result becomes the new numerator, and the denominator remains the same. So, 128 1/3 = (128 * 3 + 1) / 3 = (384 + 1) / 3 = 385/3. Similarly, we convert 3 1/2 into an improper fraction. Multiplying the whole number (3) by the denominator (2) and adding the numerator (1), we get 3 1/2 = (3 * 2 + 1) / 2 = (6 + 1) / 2 = 7/2. Now we need to divide the total cost (385/3) by the total length (7/2). Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. The reciprocal of 7/2 is 2/7. Therefore, we have: Cost per meter = (385/3) ÷ (7/2) = (385/3) * (2/7). To multiply fractions, we multiply the numerators together and the denominators together. So, (385/3) * (2/7) = (385 * 2) / (3 * 7) = 770/21. Now we simplify the fraction 770/21. Both 770 and 21 are divisible by 7. Dividing 770 by 7, we get 110. Dividing 21 by 7, we get 3. So, the simplified fraction is 110/3. To express this improper fraction as a mixed fraction, we divide 110 by 3. The quotient is 36, and the remainder is 2. Therefore, 110/3 = 36 2/3. This means the cost of 1 meter of cloth is ₹36 2/3. This problem illustrates how mathematical concepts such as fractions, mixed fractions, improper fractions, and division are applied in practical situations. By converting mixed fractions to improper fractions, finding reciprocals, and performing multiplication and division, we can accurately calculate the cost per unit. Understanding these calculations is essential for various real-world applications, from personal finance to business operations. Moreover, the ability to solve such problems enhances mathematical proficiency and problem-solving skills, which are valuable assets in both academic and professional pursuits. In summary, the cost of 1 meter of cloth is calculated by dividing the total cost by the total length, converting mixed fractions to improper fractions, and simplifying the result, leading to the answer of ₹36 2/3.

Evaluating Mathematical Expressions

Mathematical expressions often require evaluation, which involves simplifying and finding the numerical value of a given expression. Evaluating expressions is a fundamental skill in mathematics and is crucial for solving various problems in algebra, calculus, and other mathematical disciplines. The process of evaluation often involves applying the order of operations, simplifying fractions, and performing arithmetic operations. In this section, we will explore the general principles of evaluating mathematical expressions, focusing on how to approach different types of problems and the common techniques used to simplify and solve them. When evaluating a mathematical expression, it is essential to follow the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order ensures that the expression is simplified correctly, leading to the correct numerical value. The order of operations dictates that we first simplify any expressions within parentheses or brackets. Then, we evaluate exponents, followed by multiplication and division (from left to right), and finally, addition and subtraction (from left to right). Consider a simple example to illustrate this order: Evaluate the expression 2 + 3 * 4. Following PEMDAS, we perform the multiplication before the addition. So, 3 * 4 = 12, and then 2 + 12 = 14. Thus, the value of the expression is 14. If we were to add before multiplying, we would get 2 + 3 = 5, and then 5 * 4 = 20, which is incorrect. Another key aspect of evaluating expressions involves simplifying fractions. This may include reducing fractions to their simplest form or performing operations with fractions. To reduce a fraction, we find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD. For example, to reduce the fraction 12/18, we find that the GCD of 12 and 18 is 6. Dividing both the numerator and denominator by 6, we get 12/6 = 2 and 18/6 = 3. So, the reduced fraction is 2/3. Performing operations with fractions requires a common denominator for addition and subtraction. To add or subtract fractions, we find the least common multiple (LCM) of the denominators and convert each fraction to an equivalent fraction with the LCM as the denominator. For multiplication, we multiply the numerators together and the denominators together. For division, we multiply by the reciprocal of the divisor. For instance, to add 1/4 + 2/3, we find the LCM of 4 and 3, which is 12. We convert 1/4 to 3/12 and 2/3 to 8/12. Then, we add the numerators: 3/12 + 8/12 = 11/12. This process of evaluating mathematical expressions is not only confined to simple arithmetic problems. It extends to more complex algebraic expressions, where variables and operations are combined. Algebraic expressions often require simplification by combining like terms and applying the distributive property. The distributive property states that a(b + c) = ab + ac. For example, to simplify the expression 2(x + 3), we distribute the 2 to both terms inside the parentheses: 2(x + 3) = 2x + 6. Evaluating mathematical expressions is a fundamental skill that underpins many areas of mathematics and its applications. By understanding and applying the order of operations, simplifying fractions, and using algebraic techniques, we can solve a wide range of problems accurately and efficiently. This skill is essential for success in mathematics and related fields, providing a solid foundation for more advanced concepts and problem-solving strategies. In conclusion, evaluating expressions involves following a systematic approach, such as PEMDAS, and employing techniques to simplify fractions and algebraic terms, ensuring accurate results.

In summary, mathematical problem-solving involves comparing fractions to determine relative values, calculating unit costs through division and fraction manipulation, and evaluating expressions using order of operations and simplification techniques. These skills are fundamental for both academic success and practical application in everyday scenarios.