Solve The Following Mathematical Expressions: 39. (4/7) - (3/2) + 3 * (4/7) 40. (3/5) * (2/5) - 1 + (8/5) ÷ (5/2)

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Navigating the world of fractions can often feel like traversing a complex maze. However, with a systematic approach and a clear understanding of the order of operations, even the most intricate fractional expressions can be解开. In this comprehensive guide, we will delve into the step-by-step solutions of two challenging fraction problems: 47−32+3×47\frac{4}{7} - \frac{3}{2} + 3 \times \frac{4}{7}} and 35×25−1+85÷52{\frac{3{5} \times \frac{2}{5} - 1 + \frac{8}{5} \div \frac{5}{2}}. By meticulously dissecting each problem, we aim to equip you with the knowledge and skills necessary to confidently tackle any fraction-related challenge.

Problem 39: 解开 47−32+3×47{\frac{4}{7} - \frac{3}{2} + 3 \times \frac{4}{7}}

Let's embark on our journey by tackling the first expression: 47−32+3×47{\frac{4}{7} - \frac{3}{2} + 3 \times \frac{4}{7}}. This problem masterfully blends subtraction, addition, and multiplication, requiring a keen eye for the order of operations – often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Step 1: Multiplication First

According to PEMDAS, multiplication takes precedence over addition and subtraction. Therefore, our initial focus will be on the term 3×47{3 \times \frac{4}{7}}. To multiply a whole number by a fraction, we can simply treat the whole number as a fraction with a denominator of 1. Thus, we have:

3×47=31×47=3×41×7=127{3 \times \frac{4}{7} = \frac{3}{1} \times \frac{4}{7} = \frac{3 \times 4}{1 \times 7} = \frac{12}{7}}

Now, our expression transforms into:

47−32+127{\frac{4}{7} - \frac{3}{2} + \frac{12}{7}}

Step 2: Finding a Common Denominator

Before we can proceed with addition and subtraction, we must ensure that all fractions share a common denominator. The least common multiple (LCM) of 7 and 2 is 14. Therefore, we will convert each fraction to an equivalent fraction with a denominator of 14.

  • For 47{\frac{4}{7}}, we multiply both the numerator and denominator by 2:

    47×22=814{\frac{4}{7} \times \frac{2}{2} = \frac{8}{14}}

  • For 32{\frac{3}{2}}, we multiply both the numerator and denominator by 7:

    32×77=2114{\frac{3}{2} \times \frac{7}{7} = \frac{21}{14}}

  • For 127{\frac{12}{7}}, we multiply both the numerator and denominator by 2:

    127×22=2414{\frac{12}{7} \times \frac{2}{2} = \frac{24}{14}}

Our expression now reads:

814−2114+2414{\frac{8}{14} - \frac{21}{14} + \frac{24}{14}}

Step 3: Addition and Subtraction

With a common denominator in place, we can now perform the addition and subtraction from left to right:

814−2114=8−2114=−1314{\frac{8}{14} - \frac{21}{14} = \frac{8 - 21}{14} = \frac{-13}{14}}

Next, we add 2414{\frac{24}{14}}:

−1314+2414=−13+2414=1114{\frac{-13}{14} + \frac{24}{14} = \frac{-13 + 24}{14} = \frac{11}{14}}

Therefore, the final answer for the first expression is 1114{\frac{11}{14}}, which corresponds to option D.

Problem 40:解开 35×25−1+85÷52{\frac{3}{5} \times \frac{2}{5} - 1 + \frac{8}{5} \div \frac{5}{2}}

Let's shift our focus to the second expression: 35×25−1+85÷52{\frac{3}{5} \times \frac{2}{5} - 1 + \frac{8}{5} \div \frac{5}{2}}. This problem introduces division into the mix, further emphasizing the importance of adhering to the order of operations.

Step 1: Multiplication and Division

According to PEMDAS, multiplication and division are performed before addition and subtraction. Let's tackle the multiplication and division operations from left to right.

First, we multiply 35{\frac{3}{5}} by 25{\frac{2}{5}}:

35×25=3×25×5=625{\frac{3}{5} \times \frac{2}{5} = \frac{3 \times 2}{5 \times 5} = \frac{6}{25}}

Next, we handle the division. Dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of 52{\frac{5}{2}} is 25{\frac{2}{5}}. Therefore, we have:

85÷52=85×25=8×25×5=1625{\frac{8}{5} \div \frac{5}{2} = \frac{8}{5} \times \frac{2}{5} = \frac{8 \times 2}{5 \times 5} = \frac{16}{25}}

Our expression now becomes:

625−1+1625{\frac{6}{25} - 1 + \frac{16}{25}}

Step 2: Finding a Common Denominator (Again)

Before we can perform the subtraction and addition, we need a common denominator. We can rewrite 1 as a fraction with a denominator of 25:

1=2525{1 = \frac{25}{25}}

Our expression now reads:

625−2525+1625{\frac{6}{25} - \frac{25}{25} + \frac{16}{25}}

Step 3: Addition and Subtraction (One More Time)

Now, we can perform the addition and subtraction from left to right:

625−2525=6−2525=−1925{\frac{6}{25} - \frac{25}{25} = \frac{6 - 25}{25} = \frac{-19}{25}}

Next, we add 1625{\frac{16}{25}}:

−1925+1625=−19+1625=−325{\frac{-19}{25} + \frac{16}{25} = \frac{-19 + 16}{25} = \frac{-3}{25}}

Therefore, the final answer for the second expression is −325{\frac{-3}{25}}.

Key Takeaways for Mastering Fraction Operations

  • Embrace PEMDAS: The order of operations is your guiding star. Always prioritize Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This fundamental rule ensures consistent and accurate results.
  • Common Denominators are Crucial: Before adding or subtracting fractions, always ensure they share a common denominator. The least common multiple (LCM) is your best friend in this endeavor. Finding the LCM simplifies the process and prevents errors.
  • Division is Multiplication in Disguise: Dividing by a fraction is the same as multiplying by its reciprocal. This clever trick transforms division problems into multiplication problems, making them easier to solve.
  • Practice Makes Perfect: Like any mathematical skill, proficiency in fraction operations requires practice. The more you practice, the more comfortable and confident you will become.

Conclusion: Conquering Fractions with Confidence

Through this detailed exploration of two challenging fraction problems, we've not only arrived at the solutions but also uncovered the underlying principles that govern fraction operations. By diligently applying the order of operations, mastering the art of finding common denominators, and understanding the relationship between division and multiplication, you can confidently conquer any fraction-related challenge that comes your way.

Remember, the journey to mathematical mastery is paved with practice and perseverance. Embrace the challenges, celebrate your successes, and never stop exploring the fascinating world of numbers and equations. With dedication and the right tools, you can unlock the full potential of your mathematical abilities.