Calculate Remaining Pipe Length A Math Problem Solved

Hey guys! Let's tackle this math problem together. We've got a pipe that's initially $30 \frac{3}{4}$ inches long, and we're cutting five pieces from it, each measuring $4 \frac{1}{3}$ inches. The main goal here is figuring out how much pipe is left after these cuts. To solve this, we'll need to use a mix of fractions and basic arithmetic. Let's break it down step by step so it's super clear.

Understanding the Initial Length and Cuts

First off, let's look at the initial length of the pipe. It's given as $30 \frac{3}{4}$ inches. This is a mixed number, and while it's perfectly fine as is, converting it to an improper fraction can make calculations easier. Remember, an improper fraction is where the numerator (the top number) is larger than the denominator (the bottom number). So, how do we convert? We multiply the whole number part (30) by the denominator (4) and then add the numerator (3). This gives us $(30 \* 4) + 3 = 123$. We keep the same denominator, so our initial length as an improper fraction is $\frac{123}{4}$ inches. This representation will be handy when we start subtracting lengths.

Next, we need to consider the length of each cut piece. Each piece is $4 \frac{1}{3}$ inches long. Just like before, let's convert this mixed number into an improper fraction. Multiply the whole number (4) by the denominator (3) and add the numerator (1): $(4 \* 3) + 1 = 13$. Keep the denominator the same, and we get $\frac{13}{3}$ inches for each piece. Now that we know the length of each piece, we need to figure out the total length of all five pieces combined. To do this, we simply multiply the length of one piece by the number of pieces.

So, we have 5 pieces, each $\frac{13}{3}$ inches long. That means we need to calculate $5 \* \frac{13}{3}$. When multiplying a whole number by a fraction, you can think of the whole number as a fraction with a denominator of 1. So, we're really doing $\frac{5}{1} \* \frac{13}{3}$. To multiply fractions, you multiply the numerators together and the denominators together. This gives us $\frac{5 \* 13}{1 \* 3} = \frac{65}{3}$ inches. This is the total length of pipe that will be cut off. Now, we're getting closer to the solution! We know the initial length of the pipe and the total length being cut off. All that's left is to subtract to find the remaining length.

Calculating the Total Cut Length

We've already figured out that the total length cut from the pipe is $\frac{65}{3}$ inches. This was calculated by multiplying the length of each piece ($4 \frac{1}{3}$ inches, or $\frac{13}{3}$ inches as an improper fraction) by the number of pieces (5). Remember, each piece being cut is $4 \frac{1}{3}$ inches long. So, five of these pieces will total a significant portion of the original pipe. Understanding this total cut length is crucial because it's what we'll subtract from the initial length to find our answer.

Now, before we jump into the subtraction, it's a good idea to have both lengths in the same format. We have the initial length as $\frac{123}{4}$ inches and the total cut length as $\frac{65}{3}$ inches. To subtract these fractions, they need to have a common denominator. This means we need to find a number that both 4 and 3 divide into evenly. The easiest way to find this is to multiply the two denominators together: $4 \* 3 = 12$. So, our common denominator will be 12.

But we can't just change the denominators without changing the numerators as well. We need to create equivalent fractions. To convert $\frac{123}{4}$ to a fraction with a denominator of 12, we need to multiply both the numerator and the denominator by the same number. Since we're changing the denominator from 4 to 12, we've multiplied by 3. So, we also multiply the numerator by 3: $123 \* 3 = 369$. This gives us the equivalent fraction $\frac{369}{12}$.

Similarly, to convert $\frac{65}{3}$ to a fraction with a denominator of 12, we need to multiply both the numerator and the denominator by the same number. Since we're changing the denominator from 3 to 12, we've multiplied by 4. So, we also multiply the numerator by 4: $65 \* 4 = 260$. This gives us the equivalent fraction $\frac{260}{12}$. Now we're all set to subtract! We have both the initial length and the total cut length expressed as fractions with a common denominator, which makes the subtraction straightforward.

Determining the Remaining Length

Okay, guys, we're in the final stretch! We've done the groundwork, converting mixed numbers to improper fractions, calculating the total cut length, and finding a common denominator. Now, it's time to subtract the total cut length from the initial length to find out how much pipe remains. We've established that the initial length is $\frac{123}{4}$ inches, which we converted to $\frac{369}{12}$ inches. The total cut length is $\frac{65}{3}$ inches, which we converted to $\frac{260}{12}$ inches. So, the subtraction we need to perform is $\frac{369}{12} - \frac{260}{12}$.

Subtracting fractions with a common denominator is pretty simple: you just subtract the numerators and keep the denominator the same. In this case, we subtract 260 from 369, which gives us 109. So, the result of the subtraction is $\frac{109}{12}$ inches. This is the length of the pipe remaining after the five pieces have been cut. However, this is an improper fraction, and while it's perfectly correct, it's often more helpful to express it as a mixed number. This will give us a better sense of the actual length in terms of whole inches and a fraction of an inch.

To convert $\frac{109}{12}$ to a mixed number, we divide the numerator (109) by the denominator (12). 12 goes into 109 nine times (9 * 12 = 108), with a remainder of 1. This means that our mixed number will have a whole number part of 9. The remainder, 1, becomes the numerator of the fractional part, and we keep the same denominator, 12. So, $\frac{109}{12}$ is equivalent to $9 \frac{1}{12}$ inches. And that's our answer! After cutting five pieces, each $4 \frac{1}{3}$ inches long, from a pipe that was initially $30 \frac{3}{4}$ inches long, we are left with $9 \frac{1}{12}$ inches of pipe.

Final Answer

So, the final answer to our problem is $9 \frac{1}{12}$ inches. This corresponds to option B in the multiple-choice answers provided. We got here by carefully working through each step: converting mixed numbers to improper fractions, calculating the total cut length, finding a common denominator, subtracting the fractions, and finally, converting the improper fraction back to a mixed number. It might seem like a lot of steps, but breaking it down like this makes the problem much more manageable. Math problems like these are all about taking things one step at a time, guys!