How To Solve Math 7 Exercise On Page 24 (KNTT)?

Are you struggling with the math exercises on page 24 of the “Ket Noi Tri Thuc” textbook for 7th grade? Don’t worry, CAUHOI2025.EDU.VN is here to help! This article provides detailed solutions and explanations to help you master the concepts and improve your math skills. We break down each problem step-by-step, making it easy for you to understand and apply the knowledge. Let’s dive into solving these problems together!

1. Exercise 1.31 on Page 24 of the Textbook

Find x, knowing that:

a) (2x + frac{1}{2} = frac{7}{9})

b) (frac{3}{4} – 6x = frac{7}{{13}})

Solution:

a)

2x + 1/2 = 7/9
2x = 7/9 - 1/2
2x = 14/18 - 9/18
2x = 5/18
x = 5/18 : 2
x = 5/18 * 1/2
x = 5/36

Therefore, (x = frac{5}{{36}})

b)

3/4 - 6x = 7/13
-6x = 7/13 - 3/4
-6x = 28/52 - 39/52
-6x = -11/52
x = -11/52 : (-6)
x = -11/52 * (-1/6)
x = 11/312

Therefore, (x = frac{{11}}{{312}})

Detailed Explanation of Solving Linear Equations

Solving linear equations involves isolating the variable (in this case, x) on one side of the equation. Here’s a breakdown of the steps:

1. Isolate the term with x: Move all other terms to the opposite side of the equation by performing inverse operations. For example, if a term is added, subtract it from both sides.
2. Divide to solve for x: Once you have the term with x isolated, divide both sides of the equation by the coefficient of x to find the value of x.
3. Simplify: Make sure to simplify fractions and reduce them to their simplest form.

Common Mistakes to Avoid

• Incorrectly applying inverse operations: Ensure you perform the correct inverse operation when moving terms across the equals sign.
• Forgetting to apply operations to both sides: Always apply the same operation to both sides of the equation to maintain balance.
• Arithmetic errors: Double-check your calculations to avoid simple arithmetic mistakes that can lead to incorrect answers.

2. Exercise 1.32 on Page 24 of the Textbook

The areas of some of the world’s largest freshwater lakes are given in the following table. Arrange them in order of area from smallest to largest.

Solution:

Convert (8.264 times 10^9 = 0.8264 times 10 times 10^9 = 0.8264 times 10^{10}); (3.71 times 10^{11} = 3.71 times 10 times 10^{10} = 37.1 times 10^{10}).

Since 0.8264 < 2.57 < 3.17 < 5.8 < 5.96 < 6.16 < 6.9 < 8.21 < 37.1,

then (0.8264 times 10^{10} < 2.57 times 10^{10} < 3.17 times 10^{10} < 5.8 times 10^{10} < 5.96 times 10^{10} < 6.16 times 10^{10} < 6.9 times 10^{10} < 8.21 times 10^{10} < 37.1 times 10^{10}).

Or (8.264 times 10^9 < 2.57 times 10^{10} < 3.17 times 10^{10} < 5.8 times 10^{10} < 5.96 times 10^{10} < 6.16 times 10^{10} < 6.9 times 10^{10} < 8.21 times 10^{10} < 3.71 times 10^{11}).

Therefore, the names of the freshwater lakes in order of area from smallest to largest are: Nicaragua, Vostok, Ontario, Erie, Baikal, Michigan, Victoria, Superior, Caspian.

Understanding Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is written as (a times 10^b), where (a) is a number between 1 and 10 (but not including 10), and (b) is an integer.

• Converting to Scientific Notation: Move the decimal point in your number until there is only one non-zero digit to the left of the decimal point. The number of places you moved the decimal point is the exponent for 10. If you moved the decimal to the left, the exponent is positive; if you moved it to the right, the exponent is negative.
• Comparing Numbers in Scientific Notation: When comparing numbers in scientific notation, first compare the exponents. The number with the larger exponent is the larger number. If the exponents are the same, compare the decimal parts.

Real-World Application: Lake Area Comparison

Understanding the relative sizes of geographic features like lakes helps in environmental studies, resource management, and understanding global water distribution. For example, knowing the area of a lake can help scientists estimate its water volume and assess its ecological impact.

3. Exercise 1.33 on Page 24 of the Textbook

Calculate in a reasonable way.

a) A = 32.125 – (6.325 + 12.125) – (37 + 13.675)

b) (B = 4.75 + {left( {frac{{ – 1}}{2}} right)^3} + 0.5^2 – 3.frac{{ – 3}}{8})

c) C = 2021.2345 2020.1234 + 2021.2345 (-2020.1234)

Solution:

Perform the calculations as follows:

a) A = 32.125 – (6.325 + 12.125) – (37 + 13.675)

A = 32.125 – 18.45 – 37 – 13.675

A = (32.125 – 13.675) – 18.45 – 37 -> Associative Property

A = 18.45 – 18.45 – 37

A = -37

b)

B = 4.75 + ((-1)/2)^3 + 0.5^2 - 3*(-3)/8
B = 4.75 + (-1/8) + (1/2)^2 + 9/8
B = 19/4 + (-1/8) + 1/4 + 9/8
B = (19/4 + 1/4) + [(-1/8) + 9/8] -> Associative Property
B = 20/4 + 8/8 = 5 + 1 = 6

c) C = 2021.2345 2020.1234 + 2021.2345 (-2020.1234)

C = 2021.2345 2020.1234 – 2021.2345 2020.1234

C = 0

Applying Properties of Arithmetic

Efficient calculation often involves using properties like the associative, commutative, and distributive laws.

• Associative Property: Allows you to regroup numbers in addition or multiplication without changing the result.
• Commutative Property: Allows you to change the order of numbers in addition or multiplication without changing the result.
• Distributive Property: Allows you to multiply a single term by two or more terms inside a set of parentheses.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form by combining like terms and performing operations in the correct order (PEMDAS/BODMAS).

4. Exercise 1.34 on Page 24 of the Textbook

Place a pair of parentheses “()” into the following expression to get a result equal to 0.

2.2 – 3.3 + 4.4 – 5.5

Solution:

We place the parentheses into the expression as follows:

2.2 – (3.3 + 4.4 – 5.5)

When that happens, the value of the expression is:

2.2 – (3.3 + 4.4 – 5.5)

= 2.2 – (7.7 – 5.5)

= 2.2 – 2.2

= 0 (satisfies the requirements of the problem)

Understanding the Impact of Parentheses

Parentheses dictate the order in which operations are performed. Operations within parentheses are always performed before operations outside of them. This can significantly change the outcome of an expression.

Tips for Placing Parentheses

• Trial and Error: Sometimes, placing parentheses involves trying different combinations until you find one that works.
• Consider the Target Value: Think about what operations need to be grouped together to achieve the desired result.
• Check Your Work: After placing the parentheses, carefully evaluate the expression to ensure it equals the target value.

Common Mistakes in Math Exercises and How to Avoid Them

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Frequently Asked Questions (FAQ)

1. How do I solve linear equations?

   • Isolate the variable by performing inverse operations on both sides of the equation, then simplify.

2. What is scientific notation?

   • A way to express very large or very small numbers as a number between 1 and 10 multiplied by a power of 10.

3. How do I compare numbers in scientific notation?

   • Compare the exponents first; if they are the same, compare the decimal parts.

4. What is the associative property?

   • Allows you to regroup numbers in addition or multiplication without changing the result.

5. Why are parentheses important in mathematical expressions?

   • They dictate the order in which operations are performed, affecting the outcome of the expression.

6. What is PEMDAS/BODMAS?

   • An acronym that helps remember the order of operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction.

7. How can I improve my math skills?

   • Practice regularly, review concepts, seek help when needed, and use resources like CAUHOI2025.EDU.VN.

8. What common mistakes should I avoid in math exercises?

   • Arithmetic errors, incorrect order of operations, forgetting signs, misunderstanding concepts, and not showing work.

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   • We provide detailed solutions, expert explanations, reliable information, and a user-friendly platform for learning.

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