Ribbon Length Calculation: Solve The Math Problem!

Hey guys! Let's dive into this cool math problem about ribbons. It's a classic example of how we can use algebra to solve everyday puzzles. We've got three ribbons, and their lengths are related in a specific way. Our mission, should we choose to accept it, is to figure out exactly how long each ribbon is. So, grab your thinking caps, and let's get started!

Understanding the Problem

Okay, first things first, let's break down what we know. Understanding the problem is the most important step. We know that the total length of the three ribbons combined is 86 cm. That's our big picture. But, we also have some clues about how the lengths of the ribbons compare to each other. The intermediate ribbon (the one in the middle) is 15 cm longer than the shortest ribbon, and it's also 20 cm shorter than the longest ribbon. These relationships are key to solving the puzzle. We need to translate these words into math. Think of it like a detective trying to decipher clues – each sentence gives us a little more information, and it all fits together to reveal the answer.

The challenge here is to take this word problem and turn it into something we can actually work with mathematically. We need to represent the unknown lengths of the ribbons with variables (like x, y, and z) and then create equations that show the relationships between those variables. This is where algebra comes in handy! It allows us to manipulate symbols and equations to find the values we're looking for. So, let's put on our algebraic thinking hats and get ready to translate those ribbon clues into mathematical expressions!

Setting Up the Equations

Now, let's get to the equations setup, the fun part where we turn words into math! We need to give each ribbon a name. Let's call the shortest ribbon 'x', the intermediate ribbon 'y', and the longest ribbon 'z'. These variables are going to represent the unknown lengths we're trying to find. Remember, in algebra, variables are like placeholders – they stand in for numbers we don't know yet.

Next, we'll use the information given in the problem to create equations that relate these variables. We know the total length is 86 cm, so our first equation is super straightforward: x + y + z = 86. This equation simply states that if you add the lengths of all three ribbons together, you get 86 cm. Easy peasy!

But, we have more clues! The problem tells us that the intermediate ribbon (y) is 15 cm longer than the shortest ribbon (x). We can write this as an equation: y = x + 15. This equation shows the relationship between the intermediate and shortest ribbons. Similarly, the intermediate ribbon (y) is 20 cm shorter than the longest ribbon (z), which we can write as: y = z - 20. This equation connects the intermediate and longest ribbons. Now, we have three equations that represent all the information given in the problem. These equations are our tools for solving the puzzle!

• Equation 1: x + y + z = 86 (Total length)
• Equation 2: y = x + 15 (Intermediate vs. Shortest)
• Equation 3: y = z - 20 (Intermediate vs. Longest)

Solving the System of Equations

Alright, time to put on our equation-solving hats! We've got our three equations, and now we need to figure out how to use them to find the values of x, y, and z. This is where our algebraic skills really shine. There are a few different ways we could tackle this, but one common method is called substitution. Substitution involves solving one equation for a variable and then plugging that expression into another equation. This helps us reduce the number of variables and simplify the system.

Looking at our equations, we notice that Equations 2 and 3 both have 'y' isolated on one side. This makes substitution a great option! We can use these equations to express both 'x' and 'z' in terms of 'y'. Let's rearrange Equation 2 to solve for 'x': x = y - 15. Now we have 'x' in terms of 'y'. Similarly, let's rearrange Equation 3 to solve for 'z': z = y + 20. Now we have 'z' in terms of 'y' as well.

Now, the magic happens! We can substitute these expressions for 'x' and 'z' into our first equation (x + y + z = 86). This will give us an equation with only 'y' as the variable. So, we replace 'x' with (y - 15) and 'z' with (y + 20), resulting in the equation: (y - 15) + y + (y + 20) = 86. See how we've eliminated 'x' and 'z' and now have a single equation with just 'y'? This is a huge step towards solving for the lengths of the ribbons! Now, let's simplify this equation and solve for 'y'.

Calculating the Ribbon Lengths

Okay, let's roll up our sleeves and calculate those ribbon lengths! We're picking up right where we left off, with our simplified equation: (y - 15) + y + (y + 20) = 86. The first thing we need to do is combine like terms. We have three 'y' terms, so y + y + y becomes 3y. Then, we have -15 and +20, which combine to +5. So, our equation now looks like this: 3y + 5 = 86. We're getting closer!

Next, we want to isolate the 'y' term. To do that, we need to subtract 5 from both sides of the equation. This keeps the equation balanced and moves us closer to solving for 'y'. Subtracting 5 from both sides gives us: 3y = 81. Now we're in the home stretch!

Finally, to solve for 'y', we need to divide both sides of the equation by 3. This will give us the value of 'y', which represents the length of the intermediate ribbon. Dividing both sides by 3, we get: y = 27. Woohoo! We've found the length of the intermediate ribbon! It's 27 cm long.

But we're not done yet! We still need to find the lengths of the shortest and longest ribbons (x and z). Remember, we already have equations that relate x and z to y. We can use these equations and the value we just found for y to calculate x and z. Let's do it!

Finding the Shortest and Longest Ribbon Lengths

Now, let's track down the lengths of the shortest and longest ribbons. We already know the intermediate ribbon (y) is 27 cm. We also have our trusty equations from earlier that connect y to x and z. Remember Equation 2: y = x + 15? This equation tells us that the intermediate ribbon is 15 cm longer than the shortest ribbon. So, to find the length of the shortest ribbon (x), we just need to subtract 15 from the length of the intermediate ribbon (y).

So, x = y - 15 = 27 - 15 = 12 cm. Awesome! The shortest ribbon is 12 cm long. We're on a roll!

Now, let's find the length of the longest ribbon (z). We have Equation 3: y = z - 20. This equation tells us that the intermediate ribbon is 20 cm shorter than the longest ribbon. So, to find the length of the longest ribbon (z), we need to add 20 to the length of the intermediate ribbon (y).

So, z = y + 20 = 27 + 20 = 47 cm. Fantastic! The longest ribbon is 47 cm long.

We've done it! We've successfully calculated the lengths of all three ribbons: the shortest ribbon is 12 cm, the intermediate ribbon is 27 cm, and the longest ribbon is 47 cm. High fives all around!

Checking Our Solution

Before we declare victory, let's do a quick solution check to make sure our answers make sense. It's always a good idea to double-check your work, especially in math problems. We want to be absolutely sure that we haven't made any silly mistakes along the way.

First, let's make sure the total length of the ribbons adds up to 86 cm, as the problem stated. We have 12 cm (shortest) + 27 cm (intermediate) + 47 cm (longest). Adding those together, we get 86 cm. Perfect! So, our lengths add up correctly.

Next, let's check the relationships between the ribbon lengths. The intermediate ribbon (27 cm) should be 15 cm longer than the shortest ribbon (12 cm). Is it? Yes, 27 - 12 = 15. Check!

The intermediate ribbon (27 cm) should also be 20 cm shorter than the longest ribbon (47 cm). Is it? Yes, 47 - 27 = 20. Check!

All the conditions of the problem are satisfied. We've confirmed that our solution is correct! We can confidently say that the lengths of the three ribbons are 12 cm, 27 cm, and 47 cm. Great job, everyone!

Conclusion

So, there you have it! We've successfully unraveled the mystery of the ribbons. We started with a word problem, translated it into algebraic equations, solved those equations, and then double-checked our solution to make sure it was correct. This whole process highlights the power of algebra in solving real-world problems.

The key takeaway here is that breaking down complex problems into smaller, manageable steps is crucial. We identified the unknowns, set up equations to represent the given information, and then used algebraic techniques to solve for those unknowns. And remember, checking your work is just as important as finding the solution itself!

This ribbon problem is a great example of how math isn't just about numbers and formulas – it's about logical thinking, problem-solving, and the ability to see relationships between things. So, the next time you encounter a tricky problem, remember the steps we used here: understand the problem, set up equations, solve the system, and check your solution. You've got this!