Math Word Problems: Solving Length And Weight Questions

Let's dive into some math word problems, guys! We're going to tackle a couple of questions involving length and weight, perfect for sharpening those problem-solving skills. Get ready to put on your thinking caps and break these down step by step. We will dissect these word problems with clear, step-by-step solutions. Whether you're a student looking to improve your math grade or just someone who enjoys a good mental workout, this is for you.

1. Emmet's Bookshelf Project: Measuring the Remaining Board

So, our first problem revolves around Emmet, who's quite the handyman! He starts with an 8-foot board and cuts out two bookshelves. Each of these bookshelves measures 2 1/3 feet. The burning question is: after Emmet's done with his cutting, how much of the board is left? This is a classic subtraction problem with a fraction twist, so let's break it down together, making sure we understand each step. First, we need to determine the total length of the board Emmet used for the bookshelves. Since he cut two bookshelves, each measuring 2 1/3 feet, we need to multiply the length of one bookshelf by 2. This will give us the total length of the board used. Let's convert the mixed number 2 1/3 into an improper fraction to make the multiplication easier. To do this, we multiply the whole number (2) by the denominator (3) and add the numerator (1), keeping the same denominator. So, 2 1/3 becomes (2 * 3 + 1) / 3 = 7/3 feet. Now we can multiply 7/3 by 2, which is the same as 2/1. Multiplying fractions involves multiplying the numerators and the denominators separately. So, (7/3) * (2/1) = (7 * 2) / (3 * 1) = 14/3 feet. This means Emmet used 14/3 feet of the board for the bookshelves. To figure out how much of the board remained, we need to subtract the length used (14/3 feet) from the original length of the board (8 feet). To subtract fractions, we need a common denominator. Let's convert 8 feet into a fraction with a denominator of 3. We can do this by multiplying 8 by 3/3, which doesn't change the value but gives us the desired denominator. So, 8 = 8/1 = (8 * 3) / (1 * 3) = 24/3 feet. Now we can subtract: 24/3 feet - 14/3 feet. Subtracting fractions with the same denominator is straightforward: we subtract the numerators and keep the denominator. So, 24/3 - 14/3 = (24 - 14) / 3 = 10/3 feet. Finally, let's convert the improper fraction 10/3 back into a mixed number to make it easier to understand. To do this, we divide the numerator (10) by the denominator (3). 10 divided by 3 is 3 with a remainder of 1. So, 10/3 is equal to 3 whole units and 1/3 remaining, which we write as 3 1/3 feet. Therefore, after Emmet cut the bookshelves, 3 1/3 feet of the board remained. Isn't it satisfying to solve these kinds of problems?

The key here was to break down the problem into smaller, manageable steps: converting mixed numbers to improper fractions, multiplying fractions, finding a common denominator for subtraction, and converting back to a mixed number. Remember, practice makes perfect, so keep tackling these kinds of problems!

2. Lynne's Fruitful Haul: Weighing the Grapefruit

Next up, we have Lynne, who went on a fruit-buying spree! She ended up with a bag of grapefruit, 1 5/8 pounds of apples, and 2 3/16 pounds of bananas. The total weight of all her fruity purchases? A hefty 7 1/2 pounds. Our mission, should we choose to accept it (and we do!), is to figure out how much the grapefruit weighed. This problem is a bit like a puzzle where we know the total and some of the parts, and we need to find the missing piece. Let's approach it strategically! First, we need to find the combined weight of the apples and bananas. This will allow us to subtract that weight from the total weight to find the weight of the grapefruit. We have the weights of the apples (1 5/8 pounds) and the bananas (2 3/16 pounds). To add these, we first need to convert them into improper fractions. For the apples: 1 5/8 becomes (1 * 8 + 5) / 8 = 13/8 pounds. For the bananas: 2 3/16 becomes (2 * 16 + 3) / 16 = 35/16 pounds. Now we need a common denominator to add these fractions. The smallest common denominator for 8 and 16 is 16. So, we need to convert 13/8 into an equivalent fraction with a denominator of 16. We can do this by multiplying both the numerator and the denominator by 2: (13/8) * (2/2) = 26/16 pounds. Now we can add the weights of the apples and bananas: 26/16 pounds + 35/16 pounds. Adding fractions with the same denominator is simple: we add the numerators and keep the denominator. So, 26/16 + 35/16 = (26 + 35) / 16 = 61/16 pounds. This is the combined weight of the apples and bananas. Now, to find the weight of the grapefruit, we need to subtract the combined weight of the apples and bananas (61/16 pounds) from the total weight of all the fruit (7 1/2 pounds). First, let's convert 7 1/2 into an improper fraction: 7 1/2 becomes (7 * 2 + 1) / 2 = 15/2 pounds. Next, we need a common denominator to subtract 61/16 from 15/2. The smallest common denominator for 2 and 16 is 16. So, we need to convert 15/2 into an equivalent fraction with a denominator of 16. We can do this by multiplying both the numerator and the denominator by 8: (15/2) * (8/8) = 120/16 pounds. Now we can subtract: 120/16 pounds - 61/16 pounds. Subtracting fractions with the same denominator involves subtracting the numerators and keeping the denominator. So, 120/16 - 61/16 = (120 - 61) / 16 = 59/16 pounds. Finally, let's convert the improper fraction 59/16 back into a mixed number. 59 divided by 16 is 3 with a remainder of 11. So, 59/16 is equal to 3 whole units and 11/16 remaining, which we write as 3 11/16 pounds. Therefore, the bag of grapefruit weighed 3 11/16 pounds. Another puzzle solved! Great job, guys!

This problem highlighted the importance of working with fractions – converting mixed numbers, finding common denominators, adding, and subtracting. It also demonstrated how breaking a complex problem into smaller steps makes it much easier to solve. Remember, patience and a systematic approach are key in math!

Wrapping Up: Mastering Word Problems

So, there you have it! We've successfully navigated two different word problems involving fractions, one about measuring length and the other about calculating weight. These types of problems might seem daunting at first, but as we've seen, they become much more manageable when we break them down into smaller, logical steps. By practicing these techniques, you'll build confidence in your problem-solving abilities and tackle any math challenge that comes your way.

Remember, guys, the key to mastering word problems is:

• Read Carefully: Understand the question and identify the key information.
• Plan Your Approach: Decide what operations you need to perform and in what order.
• Break It Down: Divide the problem into smaller, manageable steps.
• Show Your Work: This helps you keep track of your calculations and makes it easier to spot any errors.
• Check Your Answer: Does your answer make sense in the context of the problem?

Keep practicing, stay curious, and you'll be a math whiz in no time! And always remember, it's okay to ask for help when you need it. Math is a journey, and we're all learning together.