Fractions And Oranges Solving Math Problems Step By Step

Hey there, math enthusiasts! Today, we're diving into a couple of interesting problems that involve fractions. Fractions, those tricky little numbers, often seem daunting, but trust me, they're actually quite fun to work with once you get the hang of them. We'll tackle a problem about Fred sharing his oranges and then move on to arranging fractions in the correct order. So, grab your thinking caps, and let's get started!

Let's start with our fruity dilemma. The question is this Fred has some oranges. He gives away $\frac{1}{3}$ of them, and his sister gives away $\frac{1}{9}$ of them. How many oranges did Fred keep for himself? This problem is a classic example of how fractions are used in everyday situations. To solve it, we need to break it down step by step and think about what each fraction represents.

First, let's consider Fred's oranges. The problem tells us that Fred gives away $\frac{1}{3}$ of his oranges. This means that if we divide Fred's total number of oranges into three equal parts, he gives away one of those parts. The question, however, does not provide the number of oranges Fred initially had. To tackle this, we will assume that Fred initially has a specific number of oranges so that we can follow the steps to solving this problem. Suppose Fred starts with 9 oranges.

Now, let’s calculate how many oranges Fred gave away. He gave away $\frac{1}{3}$ of his 9 oranges. To find $\frac{1}{3}$ of 9, we multiply the fraction by the total number of oranges: $\frac{1}{3} \times 9 = 3$. So, Fred gave away 3 oranges.

Next, Fred’s sister comes into the picture. She gives away $\frac{1}{9}$ of Fred's oranges. It's crucial here to understand that the $\frac{1}{9}$ refers to the total number of oranges Fred initially had, which is 9. So, Fred's sister gives away $\frac{1}{9}$ of 9 oranges. To calculate this, we again multiply the fraction by the total: $\frac{1}{9} \times 9 = 1$. Therefore, Fred's sister gave away 1 orange.

Now we know that Fred gave away 3 oranges, and his sister gave away 1 orange. To find out how many oranges Fred kept for himself, we need to subtract the number of oranges given away from the initial number of oranges. Fred started with 9 oranges, gave away 3, and his sister gave away 1, totaling 4 oranges given away. So, we subtract 4 from 9: $9 - 3 - 1 = 5$. Thus, Fred kept 5 oranges for himself.

But what if Fred had a different number of oranges initially? The key here is that we assumed Fred had 9 oranges to make the calculations straightforward, as both $\frac{1}{3}$ and $\frac{1}{9}$ of 9 result in whole numbers. If Fred had, say, 10 oranges, we would run into a bit of a problem because $\frac{1}{3}$ of 10 is not a whole number. This highlights an important aspect of fraction problems: the numbers often need to be chosen carefully to avoid remainders or decimals, unless the problem specifically asks for them.

To generalize, if Fred starts with 'x' oranges, he gives away $\frac{1}{3}x$ oranges, and his sister gives away $\frac{1}{9}x$ oranges. The number of oranges Fred keeps is then $x - \frac{1}{3}x - \frac{1}{9}x$. To simplify this expression, we need to find a common denominator for the fractions, which in this case is 9. So, we rewrite the expression as $x - \frac{3}{9}x - \frac{1}{9}x$. Combining the fractions, we get $x - \frac{4}{9}x$. This means Fred keeps $\frac{5}{9}$ of his original number of oranges. If we know the initial number of oranges, we can easily calculate how many he kept.

In summary, to solve this type of problem, you need to identify the fractions, understand what they represent in the context of the problem, and perform the necessary calculations (multiplication and subtraction in this case) to arrive at the solution. Remember, it's all about breaking the problem down into smaller, manageable steps.

Moving on, let's tackle the second part of our mathematical adventure: arranging fractions in order of size. We have the fractions $\frac{2}{3}$ and $\frac{4}{5}$. Our task is to arrange these fractions from largest to smallest. This might seem simple, but it's a fundamental skill in understanding fractions and their relative values. Let's explore a couple of methods to accomplish this.

One straightforward way to compare fractions is to find a common denominator. This means converting the fractions so that they have the same denominator, which makes it easy to compare their numerators. Think of it like this: if you're comparing slices of pizza, it's easier to tell which slice is bigger if all the pizzas are cut into the same number of slices.

To find a common denominator for $\frac{2}{3}$ and $\frac{4}{5}$, we need to find the least common multiple (LCM) of the denominators 3 and 5. The LCM of 3 and 5 is 15. So, we'll convert both fractions to have a denominator of 15.

To convert $\frac{2}{3}$ to a fraction with a denominator of 15, we need to multiply both the numerator and the denominator by the same number. In this case, we multiply by 5 because $3 \times 5 = 15$. So, $\frac{2}{3}$ becomes $\frac{2 \times 5}{3 \times 5} = \frac{10}{15}$.

Similarly, to convert $\frac{4}{5}$ to a fraction with a denominator of 15, we multiply both the numerator and the denominator by 3 because $5 \times 3 = 15$. So, $\frac{4}{5}$ becomes $\frac{4 \times 3}{5 \times 3} = \frac{12}{15}$.

Now that we have both fractions with the same denominator, we can easily compare them. We have $\frac{10}{15}$ and $\frac{12}{15}$. Since 12 is greater than 10, $\frac{12}{15}$ is greater than $\frac{10}{15}$. Therefore, $\frac{4}{5}$ is larger than $\frac{2}{3}$.

Another way to compare fractions is to convert them to decimals. This method is particularly useful when you have a calculator handy, or when the fractions are difficult to convert to a common denominator. To convert a fraction to a decimal, you simply divide the numerator by the denominator.

Let's convert $\frac{2}{3}$ to a decimal. Dividing 2 by 3, we get approximately 0.667. Similarly, let's convert $\frac{4}{5}$ to a decimal. Dividing 4 by 5, we get 0.8. Now it's easy to see which is larger: 0.8 is greater than 0.667, so $\frac{4}{5}$ is larger than $\frac{2}{3}$.

A third method, which is more of a mental math trick, involves cross-multiplication. To compare two fractions $\frac{a}{b}$ and $\frac{c}{d}$, you multiply the numerator of the first fraction by the denominator of the second fraction (ad) and the numerator of the second fraction by the denominator of the first fraction (bc). If ad > bc, then $\frac{a}{b} > \frac{c}{d}$. If ad < bc, then $\frac{a}{b} < \frac{c}{d}$.

Applying this to our fractions, we multiply 2 by 5 (which gives us 10) and 4 by 3 (which gives us 12). Since 12 is greater than 10, $\frac{4}{5}$ is greater than $\frac{2}{3}$. This method is quick and doesn't require finding a common denominator or converting to decimals.

In conclusion, arranging fractions in order of size is a fundamental mathematical skill that can be approached in several ways. Whether you prefer finding a common denominator, converting to decimals, or using the cross-multiplication trick, the key is to understand the relative values of fractions and how they compare to one another.

So, there you have it! We've tackled two different types of fraction problems today. In the first, we explored a real-world scenario of sharing oranges and used fractions to determine how many oranges Fred kept. We saw how important it is to understand what each fraction represents and to perform the correct operations. In the second problem, we looked at different methods for arranging fractions in order of size, highlighting the importance of understanding relative values and choosing the most efficient method for comparison.

Fractions are a fundamental part of mathematics, and mastering them opens doors to more advanced concepts. Whether you're sharing oranges, comparing quantities, or solving complex equations, a solid understanding of fractions is essential. Keep practicing, and you'll become a fraction whiz in no time! Remember, math is not just about getting the right answer; it's about understanding the process and the underlying concepts. Keep exploring, keep questioning, and most importantly, keep having fun with math!