Calculate The Perimeter Of A Rectangle With A Length Of 68 Cm, Knowing That The Width Is Half The Length. Three Friends Are Born In Three Consecutive Years. They Are 93 Years Old Together. How Old Is Each One?

This article tackles two intriguing mathematical problems. First, we will delve into calculating the perimeter of a rectangle, given its length and the relationship between its length and width. Then, we will embark on a journey to determine the ages of three friends born in consecutive years, knowing their combined age. These problems not only test your mathematical prowess but also highlight the practical applications of geometry and algebra in everyday scenarios. So, let's put on our thinking caps and unravel these challenges step by step.

Calculating the Perimeter of a Rectangle

The first challenge involves finding the perimeter of a rectangle. The perimeter of any shape is the total distance around its outside. For a rectangle, this means adding up the lengths of all four sides. A rectangle has two pairs of equal sides: the length and the width. The formula for the perimeter (P) of a rectangle is: P = 2 * (length + width). To effectively calculate the perimeter of a rectangle, we need to understand the relationship between its length and width. In this particular problem, we are given that the length of the rectangle is 68 cm, and the width is half of the length. This information is crucial because it allows us to determine both dimensions of the rectangle, which are essential for calculating the perimeter. Let's break down the process step by step. First, we identify the given information: the length of the rectangle is 68 cm, and the width is half the length. Next, we calculate the width by dividing the length by 2: Width = Length / 2 = 68 cm / 2 = 34 cm. Now that we have both the length (68 cm) and the width (34 cm), we can plug these values into the perimeter formula: P = 2 * (length + width) = 2 * (68 cm + 34 cm). We simplify the expression inside the parentheses: 68 cm + 34 cm = 102 cm. Finally, we multiply the sum by 2 to find the perimeter: P = 2 * 102 cm = 204 cm. Therefore, the perimeter of the rectangle is 204 cm. This comprehensive calculation demonstrates how to apply the formula for the perimeter of a rectangle when given the length and the relationship between the length and width. Understanding these steps allows for the efficient solution of similar problems in the future, reinforcing the practical application of geometric principles.

Determining the Ages of Three Friends

The second part of our challenge focuses on a different type of mathematical problem – determining the ages of three friends. This problem falls under the realm of algebra, where we use variables to represent unknown quantities and form equations to solve for them. The key information we have is that the three friends were born in three consecutive years and their combined age is 93 years. To solve this, we need to represent their ages algebraically. Let's denote the age of the youngest friend as 'x'. Since the friends were born in consecutive years, the age of the second friend would be 'x + 1', and the age of the oldest friend would be 'x + 2'. The problem states that the sum of their ages is 93 years. Therefore, we can write the equation as: x + (x + 1) + (x + 2) = 93. This equation is the foundation for solving the problem. Now, we need to simplify and solve for 'x'. Combining like terms, we get: 3x + 3 = 93. Next, we subtract 3 from both sides of the equation to isolate the term with 'x': 3x = 90. Finally, we divide both sides by 3 to find the value of 'x': x = 30. This means the youngest friend is 30 years old. Now that we have the age of the youngest friend, we can easily find the ages of the other two friends. The second friend is 'x + 1' years old, so they are 30 + 1 = 31 years old. The oldest friend is 'x + 2' years old, so they are 30 + 2 = 32 years old. Therefore, the ages of the three friends are 30, 31, and 32 years old. This solution demonstrates how to use algebraic equations to solve real-world problems involving ages and consecutive numbers. The ability to translate word problems into algebraic expressions is a valuable skill in mathematics and problem-solving.

In conclusion, we have successfully tackled two distinct mathematical challenges: calculating the perimeter of a rectangle and determining the ages of three friends. The first problem showcased the application of geometric formulas and the importance of understanding relationships between different dimensions of a shape. By using the formula for the perimeter of a rectangle and the given relationship between length and width, we were able to accurately calculate the perimeter. The second problem highlighted the power of algebraic equations in solving problems involving unknown quantities. By representing the ages of the friends as variables and forming an equation based on their combined age, we were able to determine their individual ages. These examples demonstrate the versatility of mathematics in solving real-world problems. Whether it's calculating dimensions or deciphering age relationships, mathematical principles provide a framework for logical thinking and problem-solving. Mastering these concepts not only enhances mathematical skills but also develops critical thinking abilities that are valuable in various aspects of life.