Maths Pears Problem: Fractions & Remainders

Hey guys, welcome back to Plastik Magazine! Today, we're diving into a super fun math problem that's perfect for flexing those brain muscles. We've got a scenario involving pears, fractions, and a bit of subtraction. So, grab your thinking caps, and let's break down this pear-fectly puzzling question!

The Pear-fect Setup

Alright, imagine this: you've got a box, and in this box, there are exactly 48 pears. Sounds like a good start, right? Now, two of our friends, Elizabeth and Joyce, come along and decide to take some of these delicious fruits. Elizabeth, being a fan of one-third, snatches $\frac{1}{3}$ of the pears. Joyce, who's a bit more conservative with her pear-taking, opts for $\frac{1}{4}$ of the pears. Our mission, should we choose to accept it (and we totally should, because math!), is to figure out what's going on with these pears. Specifically, we need to answer two main questions: (a) How many pears did Elizabeth actually take? and (b) What fraction of the pears is still chilling in the box after Elizabeth and Joyce have had their share?

This problem is a classic example of how fractions are used in everyday situations, even if it's just about counting fruit. It tests our understanding of fractions as parts of a whole and our ability to perform calculations with them. When we talk about taking $\frac{1}{3}$ of 48 pears, we're essentially dividing the total number of pears into three equal groups and then taking one of those groups. Similarly, taking $\frac{1}{4}$ means dividing the total into four equal groups and taking one. The tricky part, and where the real learning happens, is when we need to figure out the remaining fraction. This involves combining the fractions taken, subtracting them from the whole (which is represented by 1), or calculating the number of pears taken by each person and then subtracting that from the total. Both methods should lead us to the same answer, and exploring both can deepen our understanding of fraction manipulation. So, let's get down to the nitty-gritty and solve this step-by-step.

Solving for Elizabeth's Pear Haul

First up, let's tackle part (a): How many pears are taken by Elizabeth? We know the total number of pears is 48, and Elizabeth takes $\frac{1}{3}$ of them. To find out how many pears this is, we need to calculate $\frac{1}{3}$ of 48. In mathematics, the word 'of' in this context usually means multiplication. So, we're looking for $\frac{1}{3} \times 48$.

There are a couple of ways to do this calculation, guys. One way is to multiply 48 by 1 (which is just 48) and then divide the result by 3. So, $48 \div 3$. If you divide 48 by 3, you get 16. Therefore, Elizabeth takes 16 pears. Easy peasy, right?

Another way to think about it is to divide the total number of pears (48) into three equal groups first. If you divide 48 by 3, each group would have 16 pears. Since Elizabeth takes one of these groups ($\frac{1}{3}$), she takes 16 pears. See? Both methods confirm our answer. It's always good to have multiple ways to solve a problem; it helps reinforce the concept and ensures you're on the right track. So, to recap, Elizabeth, with her $\frac{1}{3}$ share, walked away with a very respectable 16 pears from the box. This is a fundamental step, and understanding how to calculate a fraction of a whole number is crucial for tackling more complex problems down the line. Keep this number in mind, as it will be essential for the next part of our puzzle.

Joyce's Pear Portion

Now, let's consider Joyce. She takes $\frac{1}{4}$ of the pears. Using the same logic as we did for Elizabeth, we need to calculate $\frac{1}{4}$ of 48. This means we'll be doing the calculation $\frac{1}{4} \times 48$.

Similar to before, we can multiply 48 by 1 and then divide by 4. So, $48 \div 4$. If you do that division, you'll find that $48 \div 4 = 12$. Therefore, Joyce takes 12 pears.

Alternatively, we can divide the total 48 pears into four equal groups. Each group would contain $48 \div 4 = 12$ pears. Since Joyce takes one of these groups ($\frac{1}{4}$), she takes 12 pears. This confirms our answer again. So, Joyce helped herself to 12 pears. It's interesting to note that Elizabeth took more pears than Joyce, which makes sense because $\frac{1}{3}$ is a larger fraction than $\frac{1}{4}$. This comparison helps us build an intuitive understanding of fraction sizes. Now we know how many pears each friend took individually. The next step is to figure out what's left in the box, which brings us to part (b) of the question.

Calculating the Remaining Fraction

Part (b) asks: What fraction of the pears remains in the basket? To answer this, we have a couple of approaches, and honestly, both are super valid and teach us different things about fractions.

Method 1: Using the Number of Pears

We already know Elizabeth took 16 pears and Joyce took 12 pears. So, the total number of pears taken by both of them is $16 + 12$. That adds up to 28 pears.

Since we started with 48 pears, the number of pears remaining in the box is the total number of pears minus the number of pears taken. So, $48 - 28$. This gives us 20 pears remaining.

Now, the question asks for the fraction of pears remaining. To find this fraction, we put the number of remaining pears over the original total number of pears. So, the fraction remaining is $\frac{20}{48}$.

This fraction can be simplified! Both 20 and 48 are divisible by 4. If we divide both the numerator and the denominator by 4, we get $\frac{20 \div 4}{48 \div 4} = \frac{5}{12}$. So, $\frac{5}{12}$ of the pears remain in the basket. This method is very concrete because it deals with actual numbers of pears.

Method 2: Using Fractions Directly

This method is a bit more abstract but equally powerful. We know Elizabeth took $\frac{1}{3}$ of the pears and Joyce took $\frac{1}{4}$ of the pears. To find the total fraction of pears taken, we need to add these fractions: $\frac{1}{3} + \frac{1}{4}$.

To add fractions, they need to have a common denominator. The least common multiple (LCM) of 3 and 4 is 12. So, we convert both fractions to have a denominator of 12:

• $\frac{1}{3}$ is equivalent to $\frac{1 \times 4}{3 \times 4} = \frac{4}{12}$
• $\frac{1}{4}$ is equivalent to $\frac{1 \times 3}{4 \times 3} = \frac{3}{12}$

Now we can add them: $\frac{4}{12} + \frac{3}{12} = \frac{4+3}{12} = \frac{7}{12}$.

So, a total of $\frac{7}{12}$ of the pears were taken.

The whole box of pears represents 1 (or $\frac{12}{12}$ when using our common denominator). To find the fraction of pears remaining, we subtract the fraction taken from the whole: $1 - \frac{7}{12}$.

This is the same as $\frac{12}{12} - \frac{7}{12} = \frac{12-7}{12} = \frac{5}{12}$.

Thus, $\frac{5}{12}$ of the pears remain in the basket.

See, guys? Both methods give us the same answer, $\frac{5}{12}$. This is a fantastic way to check your work and to see how different mathematical approaches can lead to the same correct conclusion. It solidifies our understanding of fraction arithmetic and its application.

Final Pear-fect Answer

So, to wrap it all up:

(a) How many pears are taken by Elizabeth?

Elizabeth took 16 pears. This was calculated by finding $\frac{1}{3}$ of the total 48 pears ($\frac{1}{3} \times 48 = 16$).

(b) What fraction of the pears remains in the basket?

There are $\frac{5}{12}$ of the pears remaining in the basket. This was found by either calculating the number of remaining pears (20) and dividing by the total (48), then simplifying, or by adding the fractions taken ($\frac{1}{3} + \frac{1}{4} = \frac{7}{12}$) and subtracting from the whole ($1 - \frac{7}{12} = \frac{5}{12}$).

Keep practicing these kinds of problems, because the more you do, the more comfortable you'll get with fractions and word problems. It's all about breaking them down into smaller, manageable steps. What other math problems have you guys been tackling? Let us know in the comments below!

Until next time, stay curious and keep calculating!