Simplify Ratios: $6 rac{2}{3}$ To $3 rac{1}{7}$ As A Fraction

Oct 29, 2025 by Andrew McMorgan 64 views

Hey guys! Today, we're diving deep into the awesome world of ratios and how to simplify them into fractions. We're going to tackle a specific problem: writing the ratio $6 rac{2}{3}$ to $3 rac{1}{7}$ as a simplified fraction. Don't worry if mixed numbers in ratios seem a bit intimidating at first; we'll break it down step-by-step, making it super easy to understand. By the end of this, you'll be a ratio-simplifying pro!

Understanding Ratios and Fractions

Alright, let's kick things off by getting our heads around what ratios and fractions actually are. A ratio is basically a way to compare two quantities. Think of it like saying, "For every X of this, there are Y of that." We can write ratios in a few ways: using a colon (like $X:Y$ ), using the word "to" (like X to Y), or, and this is super important for our problem today, as a fraction ( $rac{X}{Y}$ ). When we express a ratio as a fraction, the first quantity becomes the numerator, and the second quantity becomes the denominator. This fractional form is incredibly useful because it allows us to use all the handy tools we have for simplifying fractions to simplify our ratios.

Now, about fractions. You guys know these: a part of a whole. A fraction has a numerator (the top number) and a denominator (the bottom number). The fundamental goal when working with fractions, whether they represent parts of a whole or a ratio between two numbers, is often to simplify them. Simplifying a fraction means finding an equivalent fraction where the numerator and denominator have no common factors other than 1. This makes the fraction easier to understand and compare. For example, $rac{2}{4}$ is a perfectly valid fraction, but we usually prefer to write it as $rac{1}{2}$ because it's simpler. We achieve this by dividing both the numerator and the denominator by their greatest common divisor (GCD).

So, when we're asked to write a ratio as a simplified fraction, we're essentially taking those two numbers being compared, putting the first one over the second one, and then simplifying that resulting fraction just like we would any other fraction. The twist in our specific problem is that the numbers we're starting with are mixed numbers. Mixed numbers, like $6 rac{2}{3}$ and $3 rac{1}{7}$ , are numbers that have a whole number part and a fractional part. Before we can do any ratio magic, we need to convert these mixed numbers into a format that's easier to work with – improper fractions.

Why improper fractions? Because improper fractions (where the numerator is greater than or equal to the denominator) are much simpler to multiply and divide, which are the operations we'll need to perform when dealing with ratios of fractions. Converting mixed numbers to improper fractions is a straightforward process. For a mixed number like $a rac{b}{c}$ , you multiply the whole number part ( $a$ ) by the denominator ( $c$ ), and then add the numerator ( $b$ ). That result becomes the new numerator, and the denominator ( $c$ ) stays the same. So, $a rac{b}{c}$ becomes $rac{(a imes c) + b}{c}$ . We'll apply this conversion to both of our numbers in the ratio, and then we'll be ready to set up our fraction and simplify it.

Understanding these core concepts – what ratios are, how they can be expressed as fractions, the process of simplifying fractions, and how to handle mixed numbers by converting them to improper fractions – lays the groundwork for solving our problem. It's all about building blocks, guys! Each concept might seem simple on its own, but when you put them together, they unlock the ability to solve more complex mathematical challenges. So, let's keep this momentum going as we move on to the actual calculation!

Converting Mixed Numbers to Improper Fractions

Alright, my math enthusiasts, the first crucial step in tackling our ratio $6 rac{2}{3}$ to $3 rac{1}{7}$ and expressing it as a simplified fraction is to get rid of those pesky mixed numbers. We need to convert both $6 rac{2}{3}$ and $3 rac{1}{7}$ into improper fractions. This is a fundamental skill, and once you've got it down, it makes working with these types of numbers a breeze. Remember, an improper fraction is just a fraction where the numerator is bigger than or equal to the denominator. It's the same value as the mixed number, just written in a different, more calculation-friendly format.

Let's start with the first number in our ratio: $6 rac{2}{3}$ . To convert this mixed number into an improper fraction, we follow a simple, consistent rule. We take the whole number part (which is 6) and multiply it by the denominator of the fractional part (which is 3). So, $6 imes 3 = 18$ . Now, we take that result (18) and add the numerator of the fractional part (which is 2). So, $18 + 2 = 20$ . This sum, 20, becomes the new numerator of our improper fraction. The denominator, however, stays exactly the same as the original fractional part, which is 3. Therefore, $6 rac{2}{3}$ is equivalent to the improper fraction $rac{20}{3}$ . Easy peasy, right?

Now, let's apply the same logic to the second number in our ratio: $3 rac{1}{7}$ . Here, the whole number part is 3, and the fractional part is $rac{1}{7}$ . First, we multiply the whole number (3) by the denominator (7). That gives us $3 imes 7 = 21$ . Next, we add the numerator of the fractional part (which is 1) to this product. So, $21 + 1 = 22$ . This sum, 22, becomes our new numerator. And just like before, the denominator remains the same as the original fractional part, which is 7. So, $3 rac{1}{7}$ is equivalent to the improper fraction $rac{22}{7}$ .

See? It's not so bad! By converting our mixed numbers into improper fractions, we now have the ratio expressed as $rac{20}{3}$ to $rac{22}{7}$ . This is a huge step forward because improper fractions are what we need to set up our ratio as a single fraction. It's like getting all your ingredients ready before you start cooking. We've transformed the numbers into a format that's ready for the next stage of our mathematical journey. This conversion process is fundamental in many areas of math, especially when you're dealing with operations like multiplication, division, or, as in our case, expressing relationships between quantities. Keeping these improper fractions in mind is key, as they will be the building blocks for the final simplification.

It's always a good idea to double-check your conversions. For $6 rac{2}{3}$ , think: how many thirds are in 6 whole ones? That's $6 imes 3 = 18$ thirds. Plus the extra $rac{2}{3}$ , gives you $18 + 2 = 20$ thirds, or $rac{20}{3}$ . For $3 rac{1}{7}$ , how many sevenths are in 3 whole ones? That's $3 imes 7 = 21$ sevenths. Plus the extra $rac{1}{7}$ , gives you $21 + 1 = 22$ sevenths, or $rac{22}{7}$ . The conversions are solid. Now we're fully prepped to construct the final fraction and get it simplified!

Setting Up the Ratio as a Single Fraction

Alright team, we've successfully converted our mixed numbers into improper fractions: $6 rac{2}{3}$ is now $rac{20}{3}$ , and $3 rac{1}{7}$ is now $rac{22}{7}$ . The next logical step in our mission to write the ratio $6 rac{2}{3}$ to $3 rac{1}{7}$ as a simplified fraction is to actually set up this ratio as a single fraction. Remember, a ratio of 'A to B' can be written as the fraction $rac{A}{B}$ . So, our ratio of $rac{20}{3}$ to $rac{22}{7}$ will be written as:

rac{ rac{20}{3}}{ rac{22}{7}}

This might look a little intimidating with fractions stacked on top of each other – it's called a complex fraction. But honestly, guys, it's just a division problem in disguise! What this notation is really telling us is to divide the numerator fraction ( $rac{20}{3}$ ) by the denominator fraction ( $rac{22}{7}$ ). So, the expression above is exactly the same as:

rac{20}{3} ext{ divided by } rac{22}{7}

Or, using the division symbol:

rac{20}{3} oldsymbol{\div} rac{22}{7}

Now, how do we divide fractions? This is where some golden rules of fraction arithmetic come into play. To divide one fraction by another, we use the technique of multiplying by the reciprocal of the second fraction (the divisor). The reciprocal of a fraction is simply that fraction flipped upside down – the numerator becomes the denominator, and the denominator becomes the numerator. So, the reciprocal of $rac{22}{7}$ is $rac{7}{22}$ .

Therefore, our division problem transforms into a multiplication problem:

rac{20}{3} imes rac{7}{22}

This is fantastic news because multiplying fractions is generally much simpler than dividing them or dealing with complex fractions directly. We are now in a position to perform the multiplication and then simplify the result. This step is crucial because it converts the complex fraction representing our ratio into a single, standard fraction that we can then simplify to its lowest terms. It's all about transforming the problem into a more manageable form. We've gone from mixed numbers to improper fractions, and now we've set up the division (or multiplication) of these fractions. The path to the final answer is getting clearer with every step!

Think of it this way: if you have a pizza cut into 3 slices, and you have 20 of those slices (that's $rac{20}{3}$ pizzas), and you want to know how many servings of size $rac{22}{7}$ (or $rac{22}{7}$ of a pizza) you can get, you're essentially asking how many times $rac{22}{7}$ fits into $rac{20}{3}$ . This is a division problem. By changing it to multiplication by the reciprocal, we're making the calculation much more direct. We're setting the stage for the final simplification, which will give us the most concise representation of our original ratio.

Performing the Multiplication

Alright, pros! We've set up our ratio as a multiplication problem: $ rac{20}{3} imes rac{7}{22} $. Now it's time to actually do the multiplying. When you multiply fractions, you simply multiply the numerators together to get the new numerator, and you multiply the denominators together to get the new denominator. It's straightforward multiplication, no weird tricks needed here.

So, for our problem, the numerators are 20 and 7. Multiplying them gives us: $ 20 imes 7 = 140 $. This will be our new numerator.

The denominators are 3 and 22. Multiplying them gives us: $ 3 imes 22 = 66 $. This will be our new denominator.

Putting it all together, our multiplied fraction is: $ rac{140}{66} $.

At this point, we have successfully combined the two fractions into a single one. This fraction, $rac{140}{66}$ , represents the simplified form of our original ratio $6 rac{2}{3}$ to $3 rac{1}{7}$ . However, the job isn't quite done yet. Remember our goal? To write the ratio as a simplified fraction. Right now, $rac{140}{66}$ is not simplified because both the numerator (140) and the denominator (66) share common factors other than 1. We need to find these common factors and divide them out to reach the simplest form.

Before we move on to the simplification step, let's quickly recap what we've done. We converted mixed numbers to improper fractions ( $rac{20}{3}$ and $rac{22}{7}$ ), set up the ratio as a division of these fractions, and then converted that division into a multiplication by the reciprocal ( $rac{20}{3} imes rac{7}{22}$ ). Finally, we performed the multiplication to get a single fraction ( $rac{140}{66}$ ). This fraction is mathematically equivalent to the original ratio, but it needs one final polish – simplification. Keep this fraction $rac{140}{66}$ in mind, as it's the immediate result of our multiplication and the starting point for our final simplification.

It's worth noting that sometimes, before you even multiply, you can simplify across the fractions. This is called