Rosario's Spending Problem Solve Math Word Problems Easily

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Hey guys! Let's break down this math problem together. It's about Rosario, who's spending money, and we need to figure out how much she started with. It's a classic kind of problem that involves fractions and figuring out what's left after each expense. So, let's get started and solve this thing!

Understanding the Problem

Okay, so here's what we know: Rosario spent one-third of her money on clothes. Then, she spent three-quarters of what was left on food. And after all that spending, she had 300 left. Our mission is to find out how much money Rosario received initially. This kind of problem is great because it makes us think step-by-step. We need to work backward to get to the answer. Think of it like unraveling a mystery, where each clue leads us closer to the solution. We'll be using fractions and a bit of algebra to crack this one, so stay with me!

Setting up the Equations

Alright, let's get our math hats on and start setting up the equations. This is where we translate the words of the problem into mathematical expressions. It might seem a bit tricky at first, but once we break it down, it'll make sense, I promise!

Let's use 'X' to represent the total amount of money Rosario received. This is what we're trying to find. Now, let's follow her spending step by step:

  1. Clothes: Rosario spent 1/3 of her money on clothes. So, the amount she spent on clothes is (1/3) * X. This means that after buying clothes, she had X - (1/3)X left. We can simplify this to (2/3)X. So far, so good!
  2. Food: Next up, she spent 3/4 of the remaining money on food. The remaining money, as we figured out, is (2/3)X. So, the amount she spent on food is (3/4) * (2/3)X. If we multiply those fractions, we get (6/12)X, which simplifies to (1/2)X. So, she spent half of the original amount on food.
  3. What's Left: After buying clothes and food, Rosario had 300 left. This means that the money she had after buying clothes, minus the money she spent on food, equals 300. We can write this as an equation: (2/3)X - (1/2)X = 300. This is the key equation we'll use to solve the problem!

So, we've successfully translated the word problem into a mathematical equation. Now comes the fun part – solving for X. This will tell us exactly how much money Rosario received. Hang in there; we're getting closer to the solution!

Solving the Equation

Okay, time to put on our equation-solving hats! We've got our equation all set up: (2/3)X - (1/2)X = 300. The next step is to solve for X, which will tell us how much money Rosario initially received. Don't worry, we'll take it step by step.

First, we need to combine the terms with X. To do that, we need a common denominator for the fractions 2/3 and 1/2. The least common denominator for 3 and 2 is 6. So, let's convert the fractions:

  • (2/3)X becomes (4/6)X
  • (1/2)X becomes (3/6)X

Now, our equation looks like this: (4/6)X - (3/6)X = 300. We can now subtract the fractions: (4/6 - 3/6)X = (1/6)X.

So, our equation simplifies to (1/6)X = 300. We're almost there! To isolate X and find its value, we need to get rid of the (1/6) factor. The easiest way to do this is to multiply both sides of the equation by 6:

  • 6 * (1/6)X = 6 * 300
  • This simplifies to X = 1800

Woohoo! We've solved for X! This means Rosario initially received 1800. That wasn't so bad, right? We took a seemingly complex problem, broke it down into smaller, manageable steps, and solved it. Now, let's make sure our answer makes sense in the context of the original problem. We'll do that by checking our work.

Checking the Solution

Alright, we've got our answer – Rosario received 1800. But before we celebrate, it's always a good idea to double-check our work. Let's plug our answer back into the problem and see if it all adds up. This step is super important because it helps us catch any mistakes and ensures we've got the correct solution.

Here's how we'll check: we'll go through Rosario's spending step by step, using our answer (1800) and see if we end up with 300 left at the end.

  1. Clothes: Rosario spent 1/3 of her money on clothes. So, she spent (1/3) * 1800 = 600 on clothes. After buying clothes, she had 1800 - 600 = 1200 left.
  2. Food: Next, she spent 3/4 of the remaining money on food. That's (3/4) * 1200. To calculate this, we can multiply 3 by 1200 to get 3600, and then divide by 4: 3600 / 4 = 900. So, she spent 900 on food.
  3. What's Left: After buying clothes and food, Rosario had 1200 - 900 = 300 left. This matches the information given in the problem! Yay!

Since our calculations match the problem's conditions, we can be confident that our solution is correct. Rosario initially received 1800. We did it! Checking our work not only confirms our answer but also helps us understand the problem better. Now, let's wrap things up with a neat conclusion.

Final Answer

So, after all the calculations and checks, we've arrived at the final answer: Rosario received 1800. We successfully navigated through the fractions and the word problem to find our solution. Remember, the key to solving these kinds of problems is to break them down into smaller, manageable steps and work through them one at a time. And don't forget to check your work to make sure everything adds up!

This problem showed us how to apply math to real-life situations. Figuring out expenses and how much money you have left is a super useful skill. Keep practicing these kinds of problems, and you'll become a math whiz in no time!

How much money did Rosario receive initially if she spent one-third on clothes, three-quarters of the remainder on food, and had 300 left?

Rosario's Spending Problem Solve Math Word Problems Easily