Math Problems Solved: 7th Grade Edition

Hey guys! Let's dive into some cool math problems perfect for 7th graders. We're going to break down each problem step-by-step, making sure you understand the 'how' and 'why' behind the solutions. Ready to flex those brain muscles? Let's get started!

Math Exercises for 7th Grade: Calculations and Solutions

Alright, let's tackle these math exercises one by one. I'll provide detailed solutions so you can follow along and ace these types of problems. Remember, the key is practice! The more you work through these, the better you'll become. So, grab your pencils and let's go!

Problem 21: Calculations with Radicals and Fractions

This section focuses on performing calculations involving square roots and fractions. We'll use the rules of radicals to simplify expressions and the rules of fraction arithmetic to arrive at the solutions. Let's see the questions first:

• a) 3√2 : √2
• b) 25√2 : √200
• c) 70/30 : (-35/3)
• d) (√21) : (√7)
• e) (0.3/70) : (-10/10) : (-2/7)
• f) 6.(6)√30 : 0.2/2 : (-2/3)

Detailed Solutions for Problem 21

Let's break down each part of Problem 21. We'll work through the steps methodically. Remember, understanding each step is vital for future problems. Let's go!

• a) 3√2 : √2: First, remember that dividing by a number is the same as multiplying by its reciprocal. When we have 3√2 : √2, we can rewrite this as (3√2) / √2. The √2 in the numerator and denominator cancel out, so we are left with 3. Simple as that!

• b) 25√2 : √200: This one requires a bit more work. First, simplify √200. √200 can be rewritten as √(100 * 2), which simplifies to 10√2. Now, the problem becomes 25√2 : 10√2, or (25√2) / (10√2). The √2 in the numerator and denominator cancel out, leaving us with 25/10. Simplify this fraction to get 5/2, or 2.5.

• c) 70/30 : (-35/3): Here, we're dividing fractions. First, simplify 70/30 to 7/3. Then, dividing by -35/3 is the same as multiplying by -3/35. So, we have (7/3) * (-3/35). Multiply the numerators (7 * -3 = -21) and the denominators (3 * 35 = 105) to get -21/105. Simplify this fraction to -1/5.

• d) (√21) : (√7): Using the property of radicals, (√21) / (√7) can be simplified as √(21/7). Then, 21/7 equals 3, so the answer is √3.

• e) (0.3/70) : (-10/10) : (-2/7): First, calculate 0.3/70, which is approximately 0.00428. Next, calculate -10/10, which equals -1. The problem is now 0.00428 : (-1) : (-2/7). Dividing by -1 changes the sign, so it becomes 0.00428 : (-2/7). Then, the division is the same as multiplication by the inverse. So, the question is now 0.00428 * (-7/2). Therefore the approximate result is -0.015.

• f) 6.(6)√30 : 0.2/2 : (-2/3): First, let's deal with the repeating decimal. 6.(6) can be written as 6 + 2/3 = 20/3. Then 0.2/2 is 0.1. So the problem is (20/3)√30 : 0.1 : (-2/3). This becomes (20/3) * √30 : (1/10) : (-2/3), or (20√30/3) * (10) * (-3/2), or (200√30/3) * (-3/2) = -100√30. That is the exact result. You could approximate to a value to two decimal places.

Problem 22: More Calculations with Radicals and Fractions

This set of problems will continue to build on the skills from the previous section. We'll be working with combining radicals and fractions. The main idea is to first simplify each term as much as possible, then perform the operations. The questions are:

• a) (5√6) - (2/15) : (15√5)
• b) (3/12)(-√8) : (6/6)
• c) (-2√6) + (√8) : (√12)

Detailed Solutions for Problem 22

Let's break down each part of Problem 22 in detail. It's crucial to follow each step carefully. Remember, with practice, these steps become more intuitive. So, let's start!

• a) (5√6) - (2/15) : (15√5): This one has a subtraction and a division. Remember to follow the order of operations (PEMDAS/BODMAS). First, let's deal with the division: (2/15) : (15√5) = 2 / (15 * 15√5) = 2 / (225√5). To rationalize the denominator, we multiply the numerator and denominator by √5: (2√5) / (225 * 5) = (2√5) / 1125. Then, we subtract this from (5√6), but these are not like terms, so we are going to get an approximation value. Therefore, this is approximately 12.247 - 0.00397 = 12.243. That is the approximate answer.

• b) (3/12)(-√8) : (6/6): First, simplify: 3/12 simplifies to 1/4. Also, √8 can be simplified to 2√2. And 6/6 simplifies to 1. So, we now have (1/4) * (-2√2) : 1, which simplifies to (-2√2)/4, or -√2/2. Therefore the answer is approximately -0.707.

• c) (-2√6) + (√8) : (√12): Here, we'll first focus on the division: (√8) : (√12) is the same as √(8/12), which simplifies to √(2/3). Now we have (-2√6) + √(2/3). Because we cannot add unlike terms, we are going to find an approximation. Simplifying gives us (-2 * 2.449) + (0.816). This calculation is approximately: -4.898 + 0.816 = -4.082. The approximate answer is -4.082.

Conclusion: Mastering 7th Grade Math

Alright, folks, we've walked through some pretty cool math problems! You've seen how to simplify radicals, work with fractions, and apply the order of operations. Remember, practice makes perfect. The more you work on these types of problems, the easier they'll become. If you're struggling with a concept, don't worry – just go back and review the steps. Keep up the awesome work, and you'll be acing those math tests in no time!

I hope this has helped you. If you have any questions feel free to ask. Keep up the learning, and I'll see you in the next math adventure!