Hey guys! Today, we're diving into the world of fractions and division. It might seem tricky at first, but trust me, once you get the hang of it, you'll be solving these problems like a math whiz. We're going to break down each problem step-by-step, so you can see exactly how to find the quotient (that's the answer you get when you divide) and express it in its simplest form. So, let's get started!
1. Dividing Mixed Numbers: $1 \frac{3}{4} \div \frac{2}{4}$
In this first problem, we're dealing with a mixed number ($1 \frac{3}{4}$) divided by a fraction ($\frac{2}{4}$). The first crucial step here is to convert that mixed number into an improper fraction. This makes the division process much smoother. So, how do we do that? Well, you multiply the whole number (1) by the denominator (4) and then add the numerator (3). This gives us (1 * 4) + 3 = 7. We keep the same denominator, so $1 \frac{3}{4}$ becomes $\frac{7}{4}$.
Now, our problem looks like this: $\frac7}{4} \div \frac{2}{4}$. Remember the golden rule of fraction division4}$), change the division sign to multiplication, and flip the second fraction ($\frac{2}{4}$ becomes $\frac{4}{2}$). So, our equation now is{4} \times \frac{4}{2}$.
Next, before we multiply, let's see if we can simplify (or reduce) anything. We notice that there's a 4 in the numerator of the first fraction and a 4 in the denominator of the second fraction. We can cancel these out, as 4 divided by 4 is 1. This leaves us with $\frac{7}{1} \times \frac{1}{2}$. Now we can easily multiply the numerators (7 * 1 = 7) and the denominators (1 * 2 = 2). This gives us $\frac{7}{2}$.
Finally, we need to express our answer in its simplest form. $\frac{7}{2}$ is an improper fraction (the numerator is bigger than the denominator). To convert it back to a mixed number, we divide 7 by 2. 2 goes into 7 three times (3 * 2 = 6) with a remainder of 1. So, our quotient is 3, and our remainder is 1. We write this as the mixed number $3 \frac{1}{2}$. And that's our answer in simplest form! So, $1 \frac{3}{4} \div \frac{2}{4} = 3 \frac{1}{2}$.
2. Dividing Mixed Numbers Again: $2 \frac{1}{4} \div \frac{2}{3}$
Let's tackle the second problem: $2 \frac1}{4} \div \frac{2}{3}$. Just like before, the first thing we need to do is convert the mixed number, $2 \frac{1}{4}$, into an improper fraction. We multiply the whole number (2) by the denominator (4) and add the numerator (1)4}$ becomes $\frac{9}{4}$. Now, our problem looks like this{4} \div \frac{2}{3}$.
Remember our trusty "Keep, Change, Flip" rule? Let's apply it! We keep the first fraction ($\frac9}{4}$), change the division to multiplication, and flip the second fraction ($\frac{2}{3}$ becomes $\frac{3}{2}$). Our equation is now{4} \times \frac{3}{2}$.
Before we jump into multiplying, let's check if we can simplify. In this case, there's no common factor between 9 and 2, or between 4 and 3. So, we'll just go ahead and multiply. Multiplying the numerators, we get 9 * 3 = 27. Multiplying the denominators, we get 4 * 2 = 8. This gives us $\frac{27}{8}$.
Again, we have an improper fraction, $\frac{27}{8}$. To express it in simplest form, we need to convert it to a mixed number. We divide 27 by 8. 8 goes into 27 three times (3 * 8 = 24) with a remainder of 3. So, our quotient is 3, and our remainder is 3. This means the mixed number is $3 \frac{3}{8}$. Therefore, $2 \frac{1}{4} \div \frac{2}{3} = 3 \frac{3}{8}$.
3. Dividing with Variables: $1 \frac{x}{a} \div \frac{1}{2}$
Okay, guys, let's mix things up a bit and introduce some variables! Don't worry; the process is still the same. We have the expression $1 \fracx}{a} \div \frac{1}{2}$. Just like before, we need to convert the mixed number into an improper fraction. To do this, we multiply the whole number (1) by the denominator (a) and add the numerator (x)a}$ becomes $\frac{a + x}{a}$. Now, our problem looks like this{a} \div \frac{1}{2}$.
Time for our "Keep, Change, Flip" rule! We keep the first fraction ($\fraca + x}{a}$), change the division sign to multiplication, and flip the second fraction ($\frac{1}{2}$ becomes $\frac{2}{1}$). So, the expression is now{a} \times \frac{2}{1}$.
Let's see if we can simplify anything. In this case, there are no common factors between (a + x) and 1, or between a and 2. So, we multiply straight across. Multiplying the numerators, we have (a + x) * 2, which we can write as 2(a + x). Multiplying the denominators, we have a * 1, which is just a. This gives us $\frac{2(a + x)}{a}$.
We've got our answer, but let's think about simplest form. We could distribute the 2 in the numerator, giving us $\frac{2a + 2x}{a}$. However, we can't simplify further unless we have specific values for 'a' and 'x'. So, either $\frac{2(a + x)}{a}$ or $\frac{2a + 2x}{a}$ can be considered the simplest form in this case. Thus, $1 \frac{x}{a} \div \frac{1}{2} = \frac{2(a + x)}{a}$ or $\frac{2a + 2x}{a}$.
4. A Straightforward Division: $1 \frac{2}{4} \div \frac{1}{4}$
Lastly, let's solve $1 \frac2}{4} \div \frac{1}{4}$. We start by converting the mixed number $1 \frac{2}{4}$ into an improper fraction. Multiply the whole number (1) by the denominator (4) and add the numerator (2){4}$ becomes $\frac{6}{4}$. Our problem is now $\frac{6}{4} \div \frac{1}{4}$.
Using our "Keep, Change, Flip" rule, we keep $\frac{6}{4}$, change the division to multiplication, and flip $\frac{1}{4}$ to get $\frac{4}{1}$. The expression is now $\frac{6}{4} \times \frac{4}{1}$.
Before multiplying, let's simplify! We can see a 4 in the denominator of the first fraction and a 4 in the numerator of the second fraction. These cancel each other out. This leaves us with $\frac{6}{1} \times \frac{1}{1}$. Now we simply multiply the numerators (6 * 1 = 6) and the denominators (1 * 1 = 1), which gives us $\frac{6}{1}$.
Any fraction with a denominator of 1 is just the numerator itself. So, $\frac{6}{1}$ simplifies to 6. Therefore, $1 \frac{2}{4} \div \frac{1}{4} = 6$.
Wrapping Up Fraction Division
And there you have it! We've successfully solved four different fraction division problems, including one with variables. The key takeaways here are:
- Always convert mixed numbers to improper fractions first.
- Remember the "Keep, Change, Flip" rule for division.
- Look for opportunities to simplify before multiplying.
- Express your final answer in its simplest form, usually as a mixed number if it's an improper fraction.
I hope this breakdown has helped you understand how to divide fractions with confidence. Keep practicing, and you'll become a fraction-dividing pro in no time! Let me know if you have any other questions, and happy calculating!