Water Left In Jar: A Fraction Problem

Hey guys! Let's dive into a fun math problem today. We're going to help Paulo figure out how much water he has left in his jar after filling a glass. This is a classic fraction problem, and I'm going to break it down step by step so it's super easy to understand. So, grab your thinking caps, and let's get started!

Understanding the Problem: Paulo's Water Woes

The core of this math problem lies in fractions, those sometimes tricky but super useful numbers that represent parts of a whole. In Paulo's case, we're dealing with fractions of a liter of water. He starts with a jar that's 3/4 full, meaning it has three-quarters of a liter of water. Then, he pours out 1/6 of a liter to fill a glass. The question we need to answer is: How much water is left in the jar? This is a subtraction problem involving fractions, where we need to subtract the amount of water Paulo used (1/6 liter) from the amount he started with (3/4 liter). To solve this effectively, we need to find a common denominator for the fractions, which will allow us to perform the subtraction smoothly. Remember, the denominator is the bottom number in a fraction, representing the total number of equal parts, and the numerator is the top number, representing how many of those parts we have. Understanding this basic concept is crucial for tackling any fraction problem. So, let's move on to figuring out that common denominator and solving Paulo's water puzzle!

Key Concepts: Fractions and Subtraction

To really nail this, we need to be comfortable with fractions and how they work, as fractions represent parts of a whole. Think of a pizza cut into slices; each slice is a fraction of the whole pizza. In our case, the 'whole' is a liter of water, and we're dealing with fractions of that liter. Paulo's jar starts with 3/4 of a liter, and he uses 1/6 of a liter. Now, the tricky part is that we can't directly subtract fractions with different denominators (the bottom number). Imagine trying to subtract slices from a pizza that's been cut into both big and small slices – it just doesn't work! That's why we need to find a common denominator, a number that both 4 and 6 divide into evenly. This allows us to express both fractions in terms of the same 'size slice,' making subtraction possible. The concept of finding a common denominator is fundamental to adding and subtracting fractions, and it's a skill that will come in handy in many different math scenarios. Don't worry if it seems a bit confusing at first; we'll walk through it step by step. And, of course, once we've subtracted the fractions, we'll have our answer – the amount of water left in Paulo's jar.

Finding the Common Denominator

The secret to easily subtracting fractions, like in our water problem, is all about finding a common denominator. Think of it like this: you can't directly compare or subtract things unless they're measured in the same units. For fractions, the denominator is like the unit of measurement. So, if we want to subtract 1/6 from 3/4, we need to make sure both fractions have the same denominator. How do we do that? We need to find the least common multiple (LCM) of the denominators, which in this case are 4 and 6. One way to find the LCM is to list out the multiples of each number: Multiples of 4: 4, 8, 12, 16, 20... Multiples of 6: 6, 12, 18, 24... Notice that 12 appears in both lists! That means 12 is a common multiple of 4 and 6. It's also the least common multiple, which is exactly what we want. Now that we know our common denominator is 12, we need to convert both fractions to have this denominator. We'll do that in the next step, and you'll see how it all comes together to solve Paulo's water jar puzzle.

Step-by-Step Solution: Solving the Fraction Problem

Now, let's get into the nitty-gritty and walk through the solution step-by-step, so you can see exactly how to tackle this kind of fraction problem. Remember, Paulo started with 3/4 of a liter of water and used 1/6 of a liter. We've already figured out that the common denominator for 4 and 6 is 12. So, our next task is to convert both fractions to have a denominator of 12. This involves multiplying both the numerator (top number) and the denominator (bottom number) of each fraction by a number that will result in a denominator of 12. For the fraction 3/4, we need to multiply the denominator 4 by 3 to get 12. So, we also multiply the numerator 3 by 3, giving us 9. This means 3/4 is equivalent to 9/12. For the fraction 1/6, we need to multiply the denominator 6 by 2 to get 12. So, we also multiply the numerator 1 by 2, giving us 2. This means 1/6 is equivalent to 2/12. Now we've got our fractions in a format we can easily subtract: 9/12 - 2/12. Are you ready to see how we finish this off? Let's jump into the subtraction!

Converting Fractions to a Common Denominator

The trick of converting fractions is essential for comparing and combining them. It's like speaking the same language – you can't have a meaningful conversation if you're using different words for the same thing! In our case, we need to express both 3/4 and 1/6 in terms of twelfths, since 12 is our common denominator. So, how do we turn 3/4 into something-over-12? We ask ourselves: