6-5 Additional Practice Answer Key

6-5 Additional Practice: Answer Key and Comprehensive Explanations

This document provides a detailed answer key and comprehensive explanations for the "6-5 Additional Practice" exercises, commonly found in mathematics textbooks at the 6th-grade level or equivalent. Understanding the underlying concepts is crucial for success in future math studies. This guide will not only provide the answers but will also break down the problem-solving process, clarifying the logic and techniques involved. We'll explore various mathematical concepts, including fractions, decimals, percentages, geometry, and data analysis, ensuring a solid grasp of the fundamental principles. This comprehensive resource will be beneficial for students, parents, and educators seeking a deeper understanding of the material.

Section 1: Number Sense and Operations

This section focuses on problems involving whole numbers, fractions, decimals, and percentages. Mastering these concepts is foundational for more advanced mathematics.

1.1 Fractions:

Problem 1: Simplify the fraction 12/18.

Answer: 2/3

Explanation: To simplify a fraction, find the greatest common divisor (GCD) of the numerator and denominator. The GCD of 12 and 18 is 6. Divide both the numerator and denominator by 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3. Therefore, the simplified fraction is 2/3.

Problem 2: Add the fractions 1/4 + 2/3.

Answer: 11/12

Explanation: To add fractions with different denominators, find a common denominator. The least common multiple (LCM) of 4 and 3 is 12. Rewrite the fractions with a denominator of 12: 1/4 = 3/12 and 2/3 = 8/12. Then add the numerators: 3/12 + 8/12 = 11/12.

Problem 3: Subtract the fractions 5/6 - 1/3.

Answer: 3/6 = 1/2

Explanation: Find a common denominator (which is 6). Rewrite 1/3 as 2/6. Then subtract: 5/6 - 2/6 = 3/6. This fraction can be simplified to 1/2.

1.2 Decimals:

Problem 4: Add the decimals 3.25 + 4.7.

Answer: 7.95

Explanation: Align the decimal points and add the numbers column by column, carrying over when necessary.

Problem 5: Subtract the decimals 8.6 - 2.35.

Answer: 6.25

Explanation: Align the decimal points and subtract the numbers column by column, borrowing when necessary. Remember to add zeros as placeholders if needed.

1.3 Percentages:

Problem 6: What is 25% of 80?

Answer: 20

Explanation: To find a percentage of a number, convert the percentage to a decimal (25% = 0.25) and multiply by the number: 0.25 * 80 = 20.

Problem 7: What percentage of 50 is 10?

Answer: 20%

Explanation: Divide the part (10) by the whole (50): 10 ÷ 50 = 0.2. Convert the decimal to a percentage by multiplying by 100: 0.2 * 100 = 20%.

Section 2: Geometry

This section covers basic geometric concepts, including area, perimeter, and volume.

2.1 Area and Perimeter:

Problem 8: Find the area of a rectangle with length 8 cm and width 5 cm.

Answer: 40 cm²

Explanation: The area of a rectangle is calculated by multiplying its length and width: Area = Length × Width = 8 cm × 5 cm = 40 cm².

Problem 9: Find the perimeter of a square with sides of 6 inches.

Answer: 24 inches

Explanation: The perimeter of a square is the sum of its four sides: Perimeter = 4 × Side = 4 × 6 inches = 24 inches.

2.2 Volume:

Problem 10: Find the volume of a rectangular prism with length 4 cm, width 3 cm, and height 2 cm.

Answer: 24 cm³

Explanation: The volume of a rectangular prism is calculated by multiplying its length, width, and height: Volume = Length × Width × Height = 4 cm × 3 cm × 2 cm = 24 cm³.

Section 3: Data Analysis

This section covers interpreting and analyzing data presented in various formats, such as tables and graphs.

3.1 Interpreting Data:

Problem 11: A table shows the number of apples sold each day: Monday (10), Tuesday (15), Wednesday (12), Thursday (8), Friday (20). What is the total number of apples sold?

Answer: 65 apples

Explanation: Add the number of apples sold each day: 10 + 15 + 12 + 8 + 20 = 65 apples.

Problem 12: A bar graph shows the number of students who like different colors: Red (15), Blue (10), Green (8), Yellow (12). Which color is most popular?

Answer: Red

Explanation: The color with the tallest bar in the graph represents the most popular choice. In this case, it's Red.

Section 4: Algebraic Thinking (Introduction)

This section introduces basic algebraic concepts, laying the groundwork for future algebra studies. At this level, it often focuses on solving simple equations.

Problem 13: Solve for x: x + 5 = 12

Answer: x = 7

Explanation: Subtract 5 from both sides of the equation: x + 5 - 5 = 12 - 5. This simplifies to x = 7.

Problem 14: Solve for y: y - 3 = 9

Answer: y = 12

Explanation: Add 3 to both sides of the equation: y - 3 + 3 = 9 + 3. This simplifies to y = 12.

Problem 15: Solve for z: 2z = 10

Answer: z = 5

Explanation: Divide both sides of the equation by 2: 2z / 2 = 10 / 2. This simplifies to z = 5.

Section 5: Word Problems

Word problems require translating real-world scenarios into mathematical equations and solving them.

Problem 16: John has 25 apples. He gives 10 to his friend. How many apples does John have left?

Answer: 15 apples

Explanation: Subtract the number of apples given from the initial number of apples: 25 - 10 = 15 apples.

Problem 17: Maria buys 3 bags of oranges, with 6 oranges in each bag. How many oranges does Maria have in total?

Answer: 18 oranges

Explanation: Multiply the number of bags by the number of oranges per bag: 3 × 6 = 18 oranges.

Problem 18: A train travels at a speed of 60 miles per hour for 2 hours. How far does the train travel?

Answer: 120 miles

Explanation: Multiply the speed by the time: 60 miles/hour × 2 hours = 120 miles.

Section 6: Frequently Asked Questions (FAQ)

This section addresses common questions and misconceptions related to the 6-5 additional practice exercises.

Q1: What if I get a different answer than the key?

A1: Carefully review your steps. Check for calculation errors, especially in problems involving fractions and decimals. Double-check your understanding of the underlying concepts. If you're still stuck, try working through similar problems to reinforce your understanding. Consider seeking help from a teacher or tutor.

Q2: Are there other ways to solve these problems?

A2: Often, there are multiple approaches to solve a math problem. While the solutions provided here represent efficient methods, exploring alternative strategies can deepen your understanding and problem-solving skills. Discuss different approaches with your teacher or classmates.

Q3: What if I don't understand a specific concept?

A3: Focus on the underlying concept. Revisit the relevant sections of your textbook or seek additional resources online or from your teacher. Practice similar problems until you feel confident in your understanding.

Section 7: Conclusion

This comprehensive guide provides detailed answers and explanations for the 6-5 additional practice exercises. By understanding the step-by-step solutions and the underlying mathematical concepts, you can build a strong foundation for future math studies. Remember that consistent practice and a clear understanding of fundamental principles are key to success in mathematics. Don't hesitate to seek help when needed and continue to explore different approaches to problem-solving. Mathematics is a journey of continuous learning and discovery. Embrace the challenge, and enjoy the process of expanding your mathematical knowledge.