Decoding 70% Of [45(40+25+25)+(1/2)(40*15)]
Hey guys! Today, we're diving deep into a fascinating mathematical problem that combines percentages, arithmetic operations, and a bit of geometric thinking. We're going to break down the expression:
70% of [45(40 + 25 + 25) + (1/2)(40 * 15)]
And not just find the answer, but also understand the reasoning behind each step. So, buckle up, grab your calculators (or your mental math muscles!), and let's get started!
Dissecting the Expression: A Step-by-Step Journey
Before we even think about percentages, we need to simplify the expression inside the brackets. This is where the order of operations (PEMDAS/BODMAS) comes to our rescue. Remember, we tackle parentheses/brackets first, then exponents/orders, then multiplication and division (from left to right), and finally addition and subtraction (from left to right).
The First Term: 45(40 + 25 + 25)
- Inside the Parentheses: Let's start with the easy part. We have
40 + 25 + 25. These are simple additions, and we can add them up in any order.
So, our expression now looks like this:40 + 25 + 25 = 9045(90) + (1/2)(40 * 15) - Multiplication: Now we have
45(90). This means 45 multiplied by 90. This is where our multiplication skills shine. You can do this manually, use a calculator, or even break it down further (like 45 * 9 * 10 if that helps!). The result is:
So, the first term simplifies to 4050. This is a significant chunk of the puzzle solved!45 * 90 = 4050
The Second Term: (1/2)(40 * 15)
- Inside the Parentheses: We have
40 * 15. Again, we need to perform the multiplication within the parentheses first. Let's do it:
Now our expression looks like this:40 * 15 = 6004050 + (1/2)(600) - Multiplication by a Fraction: We have
(1/2)(600). This means one-half of 600, which is the same as dividing 600 by 2. Easy peasy!
The second term simplifies to 300. We're on a roll!(1/2) * 600 = 300
Combining the Terms: Addition to the Rescue
Now we have simplified the two main parts of the expression. We're left with:
4050 + 300
This is a simple addition problem. Let's add them up:
4050 + 300 = 4350
So, the expression inside the brackets simplifies to 4350. We're almost there!
The Percentage Power-Up: Finding 70% of 4350
Now for the final step: finding 70% of 4350. Remember, a percentage is just a fraction out of 100. So, 70% is the same as 70/100.
Converting Percentage to Decimal (Optional but Helpful)
To make the calculation easier, we can convert the percentage to a decimal. To do this, we divide the percentage by 100:
70 / 100 = 0.70
So, 70% is equal to 0.70.
The Multiplication Magic: 0. 70 * 4350
To find 70% of 4350, we multiply 0.70 by 4350:
0.70 * 4350 = 3045
Therefore, 70% of 4350 is 3045.
The Final Answer: Drumroll Please…
So, after all that mathematical maneuvering, we've arrived at the answer!
70% of [45(40 + 25 + 25) + (1/2)(40 * 15)] = 3045
Breaking Down Linda's Expression: A Geometric Perspective
Now, let's imagine this expression represents something in the real world. The prompt mentions "The net Linda drew is correct. The first term of Linda's expression...". This hints at a geometric interpretation. Let's try to decipher what Linda might have been up to.
Visualizing the Terms: Shapes and Areas
The expression strongly suggests we're dealing with areas, likely of rectangles and triangles.
- The First Term: 45(40 + 25 + 25) or 45 * 90: This term can be interpreted as the area of a rectangle. Think of it as a rectangle with a width of 45 and a length of (40 + 25 + 25) which equals 90. So, we have a rectangle with dimensions 45 x 90.
- The Second Term: (1/2)(40 * 15): This term looks suspiciously like the formula for the area of a triangle! Remember, the area of a triangle is (1/2) * base * height. Here, we have a triangle with a base of 40 and a height of 15.
Linda's Net: Putting the Pieces Together
So, Linda's net likely represents a composite shape made up of a rectangle (45 x 90) and a triangle (base 40, height 15). The entire expression then calculates the total area of this composite shape.
The 70% Twist: A Proportion of the Area
Finally, the 70% comes into play. It means Linda is interested in 70% of the total area of the shape represented by her net. Maybe she's calculating the area she needs to paint, or the amount of material required to cover a certain portion of the shape.
Linda's Reasoning: Connecting the Dots
So, what's the first term of Linda's expression representing? It's the area of the rectangle within her net. This is a crucial piece of information! Without the rectangle, her shape wouldn't be complete, and the total area calculation would be incorrect.
Linda's net is a clever way to combine geometric shapes and use mathematical expressions to calculate areas. By breaking down the expression step-by-step and visualizing the shapes, we can fully understand the meaning behind the numbers.
Conclusion: Math is More Than Just Numbers
Guys, we've successfully cracked this mathematical puzzle! We not only found the answer but also explored the reasoning behind each step. We saw how the order of operations is crucial, how percentages work, and how mathematical expressions can represent real-world concepts like area.
Remember, math isn't just about crunching numbers; it's about understanding the logic, patterns, and connections that make the world around us tick. Keep exploring, keep questioning, and keep those mathematical muscles flexed!
#SEO Keywords Value of Expression, Percentage Calculation, Order of Operations, PEMDAS/BODMAS, Area of Rectangle, Area of Triangle, Geometric Shapes, Mathematical Reasoning, Problem Solving, Step-by-Step Solution
