Advanced grammar practice worksheet with exercises on sentence structure and usage.

Problem Analysis:

The task involves solving a series of problems related to fractions and their applications. Let's break down each part of the problem and solve it step by step.

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Part 1: Calculating Fractions

#### Question 1: Calculate:
1. \( \frac{3}{4} + \frac{5}{6} \)
2. \( \frac{7}{8} - \frac{3}{4} \)
3. \( \frac{2}{3} \times \frac{9}{10} \)
4. \( \frac{5}{6} \div \frac{10}{12} \)

##### Solution:
1. Addition: \( \frac{3}{4} + \frac{5}{6} \)
- Find the least common denominator (LCD) of 4 and 6, which is 12.
- Convert fractions:
\[
\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}, \quad \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}
\]
- Add the fractions:
\[
\frac{9}{12} + \frac{10}{12} = \frac{19}{12}
\]

2. Subtraction: \( \frac{7}{8} - \frac{3}{4} \)
- Find the LCD of 8 and 4, which is 8.
- Convert fractions:
\[
\frac{7}{8} = \frac{7}{8}, \quad \frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}
\]
- Subtract the fractions:
\[
\frac{7}{8} - \frac{6}{8} = \frac{1}{8}
\]

3. Multiplication: \( \frac{2}{3} \times \frac{9}{10} \)
- Multiply the numerators and denominators:
\[
\frac{2}{3} \times \frac{9}{10} = \frac{2 \times 9}{3 \times 10} = \frac{18}{30}
\]
- Simplify the fraction:
\[
\frac{18}{30} = \frac{3}{5} \quad (\text{divide numerator and denominator by 6})
\]

4. Division: \( \frac{5}{6} \div \frac{10}{12} \)
- Rewrite division as multiplication by the reciprocal:
\[
\frac{5}{6} \div \frac{10}{12} = \frac{5}{6} \times \frac{12}{10}
\]
- Multiply the fractions:
\[
\frac{5}{6} \times \frac{12}{10} = \frac{5 \times 12}{6 \times 10} = \frac{60}{60}
\]
- Simplify the fraction:
\[
\frac{60}{60} = 1
\]

#### Final Answers for Part 1:
1. \( \frac{19}{12} \)
2. \( \frac{1}{8} \)
3. \( \frac{3}{5} \)
4. \( 1 \)

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Part 2: Solving Word Problems

#### Question 2: A farmer has a field that is divided into 8 equal parts. He plants corn in 3 parts and wheat in 2 parts. What fraction of the field is planted with either corn or wheat?

##### Solution:
- Total parts of the field: 8
- Parts planted with corn: 3
- Parts planted with wheat: 2
- Total parts planted with either corn or wheat:
\[
3 + 2 = 5
\]
- Fraction of the field planted with either corn or wheat:
\[
\frac{5}{8}
\]

#### Final Answer for Question 2:
\[
\boxed{\frac{5}{8}}
\]

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#### Question 3: In a class of 30 students, \(\frac{2}{5}\) of the students are girls. How many boys are there in the class?

##### Solution:
- Total number of students: 30
- Fraction of students who are girls: \( \frac{2}{5} \)
- Number of girls:
\[
\frac{2}{5} \times 30 = \frac{2 \times 30}{5} = \frac{60}{5} = 12
\]
- Number of boys:
\[
\text{Total students} - \text{Number of girls} = 30 - 12 = 18
\]

#### Final Answer for Question 3:
\[
\boxed{18}
\]

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Part 3: Applying Fractions to Real-Life Situations

#### Question 4: A recipe requires \(\frac{3}{4}\) cup of sugar. If you want to make half of the recipe, how much sugar do you need?

##### Solution:
- Original amount of sugar required: \( \frac{3}{4} \) cup
- To make half of the recipe, multiply the original amount by \( \frac{1}{2} \):
\[
\frac{3}{4} \times \frac{1}{2} = \frac{3 \times 1}{4 \times 2} = \frac{3}{8}
\]

#### Final Answer for Question 4:
\[
\boxed{\frac{3}{8}}
\]

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Part 4: Explaining Concepts

#### Question 5: Explain why multiplying two fractions results in a smaller fraction than either of the original fractions.

##### Solution:
When multiplying two fractions, both the numerator and the denominator are multiplied together. Since both fractions are less than 1 (as they are proper fractions), their product will be even smaller. This is because multiplying numbers less than 1 always results in a smaller number. For example:
- \( \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \)
Here, \( \frac{1}{6} \) is smaller than both \( \frac{1}{2} \) and \( \frac{1}{3} \).

#### Final Answer for Question 5:
\[
\boxed{\text{Multiplying two fractions less than 1 results in a smaller fraction because the product of two numbers less than 1 is always smaller than either of the original numbers.}}
\]

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Final Summary of Answers:

1. \( \frac{19}{12}, \frac{1}{8}, \frac{3}{5}, 1 \)
2. \( \boxed{\frac{5}{8}} \)
3. \( \boxed{18} \)
4. \( \boxed{\frac{3}{8}} \)
5. Explanation provided above.