Math Problems: Equations & Solutions

Hey guys! Let's dive into some cool math problems. We'll be converting word problems into equations and then solving them. This is super useful, whether you're brushing up on your algebra skills or just trying to help your kid with their homework. Let's get started!

Converting Equations and Solving Them: Problem a

Converting equations and solving them is the name of the game here. We are going to work on turning word problems into mathematical equations, and then finding the solutions. This is where the magic happens, so pay close attention.

The first problem is this: "The half of a number increased by 8 gives as a result the triple of that number divided by 2." Whoa, that sounds complicated, right? Don't sweat it. Let's break it down step by step to convert and solve the equation. We're going to translate the sentence into math terms. First, we need to pick a variable. Let's call our unknown number "x". Now we can translate parts of the problem like this:

• "The half of a number": This becomes x / 2, or (1/2)x, or even 0.5x.
• "Increased by 8": This means we'll add 8. So far, we have (1/2)x + 8.
• "Gives as a result": This is where the equals sign (=) comes in. We're saying that everything before this phrase equals what comes after it.
• "The triple of that number": This is 3x.
• "Divided by 2": This means we need to divide the 3x by 2, which gives us (3x) / 2, or (3/2)x, or 1.5x.

Putting it all together, we get the equation: (1/2)x + 8 = (3/2)x. Now comes the fun part: solving the equation! We want to get "x" all by itself on one side of the equation. To do that, we'll follow these steps:

1. Subtract (1/2)x from both sides. This gives us: 8 = (3/2)x - (1/2)x.
2. Simplify the right side: (3/2)x - (1/2)x = (2/2)x = x. So now we have 8 = x.

That's it! We found the solution. The number we were looking for is x = 8. Let's check our answer to make sure it's correct. If the number is 8, the half of it is 4. If we add 8 to that, we get 12. The triple of 8 is 24, and dividing it by 2 gives 12. So, 12 = 12. It checks out! We successfully converted the equation and solved it!

Plumbing Problem Solution: Finding the Hours Worked

Okay, let's move on to the second problem. This one's about a plumber, a bit more real-world, right? This time, we'll figure out how to calculate the hours worked based on the information provided. Here's the situation: "A plumber charges $48,000 per visit, plus $10,000 per hour worked. If a customer paid $120,000, approximately how many hours did the plumber work?"

We need to find out the hours worked. Let's use "h" to represent the number of hours. Here's how we'll break it down into an equation:

• The plumber's total charge is made up of a fixed cost (the visit fee) and a variable cost (the hourly rate).
• The fixed cost is $48,000. This is a constant; it doesn't change.
• The variable cost is $10,000 per hour, which we can write as 10,000h.
• The total cost is $120,000. So we have: 48,000 + 10,000h = 120,000.

Now, let's solve for "h"! Our goal is to isolate the variable “h”. Follow these steps:

1. Subtract $48,000 from both sides of the equation: 10,000h = 120,000 - 48,000.
2. Simplify the right side: 10,000h = 72,000.
3. Divide both sides by $10,000 to solve for h: h = 72,000 / 10,000.
4. Simplify: h = 7.2.

So, the plumber worked approximately 7.2 hours. If you want, you could round that to about 7 hours and 12 minutes. That makes sense, right? Now let’s double-check our work. The plumber charges $48,000 for the visit, and $10,000 x 7.2 hours = $72,000. $72,000 + $48,000 = $120,000, which is exactly the amount the customer paid. We got it!

Tips for Solving Word Problems

Alright, let’s wrap up with some tips to solve word problems and become math problem-solving rockstars! Remember, practice makes perfect. The more you work with these types of problems, the easier they'll become. Here are some key tips:

• Read the problem carefully: Understand what the problem is asking. Sometimes you need to read it more than once.
• Identify the unknowns: Figure out what you need to find. These are your variables.
• Translate the words into math: This is the most important step. Break down the sentences and turn them into mathematical expressions.
• Write the equation: Once you’ve translated, put everything together. Remember the equals sign!
• Solve the equation: Use your algebra skills to isolate the variable and find the answer.
• Check your answer: Always make sure your answer makes sense in the context of the problem. Plug it back into the original problem to see if it works.

More Practice Makes Perfect

Here are some extra practice problems you can try on your own:

1. Age Problem: Sarah is twice as old as John. In five years, Sarah will be 10 years older than John. How old are Sarah and John now?
2. Geometry Problem: The length of a rectangle is 5 cm more than its width. The perimeter of the rectangle is 30 cm. Find the length and width of the rectangle.
3. Money Problem: Maria has $20 more than twice the amount of money that John has. Together, they have $170. How much money does each person have?

Keep practicing, and don't be afraid to ask for help if you get stuck! Math can be super rewarding when you get the hang of it, and these skills are useful in so many aspects of life. You've got this!

Keywords to Remember

To summarize, here's a quick rundown of the keywords to remember when tackling these equation problems:

• Variable: A symbol, usually a letter, that represents an unknown number (like 'x' or 'h').
• Equation: A mathematical statement that shows two expressions are equal (e.g., 2x + 3 = 7).
• Solve: To find the value of the variable that makes the equation true.
• Isolate: To get the variable by itself on one side of the equation.

By following these steps and remembering these keywords, you'll be well on your way to becoming a word problem-solving pro! Keep up the great work, and happy calculating, folks!