Solve These Math Problems! Show Your Work!

Hey everyone! I've got six math problems that I'd love for you all to tackle. This isn't just about getting the answers; I want to see how you got there. Please show your work, step-by-step! I'm going to be reporting any silly answers, so please, let's keep it legit. Let's dive in and get those brain juices flowing! Remember, understanding how to solve the problem is way more important than just the final answer. Show your work, and let's learn together!

Problem 1: The Age Riddle

The Problem:

• A father is 30 years old. His son is 5 years old. In how many years will the father be twice as old as his son? Show your work.

Alright, let's kick things off with a classic age riddle. These types of problems are super common in math and are a great way to practice your algebra skills. The key here is to set up an equation that represents the relationship between the father's age and the son's age at a future point in time. We'll need to consider how both their ages increase over the years. Don't let the wordiness throw you off; we'll break it down into simple steps.

To start, let's define a variable. Let 'x' represent the number of years from now when the father will be twice as old as his son. This is the unknown quantity we're trying to find. Next, we translate the given information into mathematical expressions. The father's current age is 30, so in 'x' years, his age will be 30 + x. Similarly, the son's current age is 5, and in 'x' years, his age will be 5 + x.

Now, we can set up the equation. The problem states that the father's age will be twice the son's age. So, we can write this as: 30 + x = 2 * (5 + x). See? We've turned words into a clear mathematical statement. We're one step closer to cracking the code. Now comes the fun part - solving the equation!

To solve the equation, we first need to distribute the 2 on the right side: 30 + x = 10 + 2x. Then, we want to get all the 'x' terms on one side and the constants on the other. Subtract 'x' from both sides: 30 = 10 + x. Finally, subtract 10 from both sides to isolate 'x': 20 = x. This means in 20 years, the father will be twice as old as his son. We’re basically playing detective with numbers, and it's a blast! We've found our answer and we've shown how we got there. Always double-check your work by plugging the answer back into the original problem to make sure it makes sense. Good job, everyone!

Problem 2: The Fraction Challenge

The Problem:

• Simplify the following fraction: 12/24. Show your work.

Okay, time for some fraction fun! Simplifying fractions is a fundamental skill in math, and it's crucial for understanding more complex concepts later on. The goal is to reduce the fraction to its simplest form, where the numerator and denominator have no common factors other than 1. Basically, we're trying to find the most straightforward way to represent the same value. Think of it as decluttering a room; you're making it more organized and easier to understand.

To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. In this case, the fraction is 12/24. We need to find the GCD of 12 and 24. Well, you can start by listing out the factors of each number. For 12, the factors are 1, 2, 3, 4, 6, and 12. For 24, the factors are 1, 2, 3, 4, 6, 8, 12, and 24. The greatest common factor here is 12.

Once you've found the GCD, divide both the numerator and the denominator by it. In our example, divide both 12 and 24 by 12. 12 divided by 12 equals 1, and 24 divided by 12 equals 2. Therefore, the simplified fraction is 1/2. Simplifying fractions is about making them easier to work with and understand. By reducing them to their lowest terms, you gain a clearer picture of the relationship between the numerator and the denominator. It makes future calculations and comparisons easier. So, in a nutshell, you're just making things simpler, not necessarily changing the value. It's similar to converting units; the actual amount doesn't change, just the way it's presented. Great job, everyone!

Problem 3: The Geometry Game

The Problem:

• A rectangle has a length of 10 cm and a width of 5 cm. What is the area of the rectangle? Show your work.

Let's switch gears and dive into the world of geometry. Calculating the area of a rectangle is a basic, yet fundamental, concept. This is used everywhere from measuring rooms to designing gardens. The area of a rectangle represents the space enclosed within its boundaries. To find the area, we use a very straightforward formula. Let's put our thinking caps on and get to work!

The formula for the area of a rectangle is: Area = Length x Width. It's as simple as that! You just multiply the length of the rectangle by its width to get the area. In this problem, we are given the length as 10 cm and the width as 5 cm. Now, let's plug those values into the formula. Area = 10 cm * 5 cm. When we multiply those two numbers, we get 50 square centimeters. Remember to include the units of measurement, which are square centimeters (cm²), because we're dealing with area, which is a two-dimensional measurement.

So, the area of the rectangle is 50 cm². This tells us that the rectangle covers 50 square centimeters of space. The area calculation is super useful for solving all sorts of practical problems. It helps us figure out how much material we need to cover a surface, like carpeting a room or painting a wall. It's a practical skill that you'll use throughout life. Also, it's good to know the formula and how to apply it. You will be able to tackle many other area and volume problems with this. Great work!

Problem 4: The Percentage Puzzle

The Problem:

• What is 25% of 80? Show your work.

Now let's jump into percentages. Percentages are incredibly useful in everyday life. Whether you're calculating a discount, figuring out interest rates, or understanding statistics, they're always there. Finding the percentage of a number is a common task, so let's make sure we know how to do it. It's really not that complicated once you grasp the concept.

To find a percentage of a number, you can convert the percentage to a decimal and then multiply it by the number. The process is pretty straightforward. Here, we have to find 25% of 80. First, let's convert 25% into a decimal. You do this by dividing the percentage by 100. So, 25 / 100 = 0.25. Next, multiply the decimal by the number: 0.25 * 80. This will give us the answer. Multiplying 0.25 by 80 gives us 20. That means 25% of 80 is 20. Easy peasy, right?

Percentage calculations are essential for a variety of real-world applications. Think about sales and discounts; that's all about percentages. When you see a