Finding The Breadth Of A Room Calculating Painting Costs

Hey guys! Ever stumbled upon a math problem that seems like a real head-scratcher? Well, today we're diving deep into a fascinating question about a room's dimensions and the cost of painting it. It's like we're playing detective, piecing together clues to uncover the room's hidden breadth. So, grab your thinking caps, and let's embark on this mathematical adventure together!

The Intriguing Problem

Let's break down the problem we're tackling. Imagine a room with a height of 3 3/5 meters and a length of 6 2/3 meters. Now, we know it costs Rs. 980 to paint all four walls of this room at a rate of Rs. 12.50 per square meter. Our mission, should we choose to accept it, is to find the breadth of this room. Sounds like a puzzle, right? But don't worry, we'll crack it step by step!

Decoding the Dimensions: Height and Length

Before we jump into calculations, let's make sure we're crystal clear on the room's height and length. The height is given as 3 3/5 meters. To make things easier to work with, let's convert this mixed fraction into an improper fraction. Remember how? We multiply the whole number (3) by the denominator (5) and add the numerator (3), then put the result over the original denominator. So, 3 3/5 becomes (3 * 5 + 3) / 5 = 18/5 meters. Got it?

Next up, the length is 6 2/3 meters. Let's do the same conversion trick. 6 2/3 transforms into (6 * 3 + 2) / 3 = 20/3 meters. Fantastic! Now we have the height and length in a fraction format that's much more friendly for our calculations.

Calculating the Area of the Walls: The Key to Unlocking Breadth

The crux of this problem lies in understanding how the area of the walls relates to the cost of painting. We know the total cost (Rs. 980) and the cost per square meter (Rs. 12.50). To find the total area painted, we simply divide the total cost by the cost per square meter. So, the total area is 980 / 12.50 = 78.4 square meters. This is the combined area of all four walls. Remember, we're dealing with a rectangular room, so the walls are made up of rectangles. Therefore, to calculate the area, we need the perimeter of the room and the height.

Now, let's think about the walls. A rectangular room has two walls with the same length and two walls with the same breadth. The area of each wall is simply its length multiplied by its height. Since we know the total area of all four walls, we can set up an equation to help us find the breadth. This is where the magic happens!

Let's denote the breadth of the room as 'b'. The area of the two longer walls would be 2 * (length * height), and the area of the two shorter walls would be 2 * (breadth * height). Adding these together gives us the total area of the four walls. So, our equation looks like this: 2 * (length * height) + 2 * (breadth * height) = total area. This equation is the key to solving our problem.

Plugging in the Values: The Equation Comes to Life

Now comes the exciting part – plugging in the values we already know into our equation. We have the length (20/3 meters), the height (18/5 meters), and the total area (78.4 square meters). Let's substitute these values into our equation: 2 * ((20/3) * (18/5)) + 2 * (b * (18/5)) = 78.4. This might look a bit intimidating, but don't worry, we'll simplify it step by step.

First, let's simplify the first term: (20/3) * (18/5). We can multiply the numerators and the denominators: (20 * 18) / (3 * 5) = 360 / 15. Now, we can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 15. So, 360 / 15 = 24. Therefore, the first term in our equation becomes 2 * 24 = 48.

Now our equation looks like this: 48 + 2 * (b * (18/5)) = 78.4. Let's simplify the second term a bit. 2 * (b * (18/5)) can be rewritten as (36/5) * b. So, our equation is now: 48 + (36/5) * b = 78.4. We're getting closer to solving for 'b'!

Isolating the Breadth: The Final Countdown

Our next goal is to isolate 'b' on one side of the equation. To do this, we need to get rid of the 48 on the left side. We can subtract 48 from both sides of the equation: 48 + (36/5) * b - 48 = 78.4 - 48. This simplifies to (36/5) * b = 30.4.

Now, we need to get rid of the (36/5) that's multiplying 'b'. We can do this by multiplying both sides of the equation by the reciprocal of (36/5), which is (5/36): ((36/5) * b) * (5/36) = 30.4 * (5/36). On the left side, the (36/5) and (5/36) cancel each other out, leaving us with just 'b'. On the right side, we have 30.4 * (5/36). Let's calculate that.

30. 4 * (5/36) = (30.4 * 5) / 36 = 152 / 36. Now, we can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4. So, 152 / 36 = 38 / 9. This is the value of 'b'! However, let’s convert this improper fraction to a mixed fraction, which gives us 4 rac{2}{9} meters.

The Grand Reveal: The Breadth Unveiled

Drumroll, please! We've finally cracked the code and found the breadth of the room. The breadth, 'b', is equal to 38/9 meters, or 4 rac{2}{9} meters. Congratulations, guys! We've successfully navigated this mathematical maze and emerged victorious. This problem shows how we can use basic geometry and algebra to solve real-world problems. Remember, math isn't just about numbers and equations; it's about problem-solving and critical thinking. So, keep those thinking caps on, and let's conquer the next mathematical challenge!

Problem in Simple Words: Unraveling the Question

Let's simplify this question, guys. Imagine a room that's 3 rac{3}{5} meters high and 6 rac{2}{3} meters long. If it costs Rs. 980 to paint the four walls at a rate of Rs. 12.50 per square meter, what is the breadth of the room? In simpler terms, we're trying to figure out how wide the room is based on the cost of painting its walls. It's like a real-life puzzle, and we're the detectives solving it!