Factorial Worksheet - Fill and Sign Printable Template Online - Free Printable

Problem Analysis:

The image contains a series of mathematical problems that need to be solved. The tasks involve basic arithmetic operations, simplifications, and conversions to decimal form. Let's solve each problem step by step.

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Problem 1:

$$
\frac{2}{3} \div \frac{4}{5}
$$

#### Solution:
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
$$
\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4}
$$
Now, multiply the numerators and denominators:
$$
\frac{2 \times 5}{3 \times 4} = \frac{10}{12}
$$
Simplify the fraction by dividing both numerator and denominator by their greatest common divisor (GCD), which is 2:
$$
\frac{10 \div 2}{12 \div 2} = \frac{5}{6}
$$

#### Final Answer:
$$
\boxed{\frac{5}{6}}
$$

---

Problem 2:

$$
\frac{3}{7} \div \frac{9}{14}
$$

#### Solution:
Similarly, divide the fractions by multiplying by the reciprocal:
$$
\frac{3}{7} \div \frac{9}{14} = \frac{3}{7} \times \frac{14}{9}
$$
Multiply the numerators and denominators:
$$
\frac{3 \times 14}{7 \times 9} = \frac{42}{63}
$$
Simplify the fraction by dividing both numerator and denominator by their GCD, which is 21:
$$
\frac{42 \div 21}{63 \div 21} = \frac{2}{3}
$$

#### Final Answer:
$$
\boxed{\frac{2}{3}}
$$

---

Problem 3:

$$
5 \div 2^2
$$

#### Solution:
First, calculate the exponent:
$$
2^2 = 4
$$
Now, perform the division:
$$
5 \div 4 = \frac{5}{4}
$$

#### Final Answer:
$$
\boxed{\frac{5}{4}}
$$

---

Problem 4:

$$
(2 \cdot 3)^2
$$

#### Solution:
First, calculate the product inside the parentheses:
$$
2 \cdot 3 = 6
$$
Now, square the result:
$$
6^2 = 36
$$

#### Final Answer:
$$
\boxed{36}
$$

---

Problem 5:

$$
\frac{n - 2p}{(n + 1)^2}
$$

#### Solution:
This expression cannot be simplified further without specific values for \( n \) and \( p \). It remains in its current form.

#### Final Answer:
$$
\boxed{\frac{n - 2p}{(n + 1)^2}}
$$

---

Problem 6:

$$
\frac{m^2}{(w + 1)^2}
$$

#### Solution:
This expression also cannot be simplified further without specific values for \( m \) and \( w \). It remains in its current form.

#### Final Answer:
$$
\boxed{\frac{m^2}{(w + 1)^2}}
$$

---

Problem 7:

$$
\frac{w(w - 3)}{(w + 1)(w - 2)}
$$

#### Solution:
This expression is already in factored form. Without additional information or specific values for \( w \), it cannot be simplified further.

#### Final Answer:
$$
\boxed{\frac{w(w - 3)}{(w + 1)(w - 2)}}
$$

---

Problem 8:

$$
\frac{(x + y)^2}{(w + 1)^2}
$$

#### Solution:
This expression is already in its simplest form. Without specific values for \( x \), \( y \), and \( w \), it cannot be simplified further.

#### Final Answer:
$$
\boxed{\frac{(x + y)^2}{(w + 1)^2}}
$$

---

Problem 9:

Convert the following fractions to decimal form:
1. \( \frac{10}{8} \)
2. \( \frac{19}{16} \)
3. \( \frac{20}{18} \)
4. \( \frac{50}{45} \)
5. \( \frac{100}{90} \)

#### Solution:
1. \( \frac{10}{8} \):
$$
\frac{10}{8} = 1.25
$$

2. \( \frac{19}{16} \):
$$
\frac{19}{16} = 1.1875
$$

3. \( \frac{20}{18} \):
Simplify the fraction:
$$
\frac{20}{18} = \frac{10}{9} \approx 1.1111\ldots = 1.\overline{1}
$$

4. \( \frac{50}{45} \):
Simplify the fraction:
$$
\frac{50}{45} = \frac{10}{9} \approx 1.1111\ldots = 1.\overline{1}
$$

5. \( \frac{100}{90} \):
Simplify the fraction:
$$
\frac{100}{90} = \frac{10}{9} \approx 1.1111\ldots = 1.\overline{1}
$$

#### Final Answers:
$$
\boxed{1.25, 1.1875, 1.\overline{1}, 1.\overline{1}, 1.\overline{1}}
$$

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Final Summary of All Answers:

1. \( \boxed{\frac{5}{6}} \)
2. \( \boxed{\frac{2}{3}} \)
3. \( \boxed{\frac{5}{4}} \)
4. \( \boxed{36} \)
5. \( \boxed{\frac{n - 2p}{(n + 1)^2}} \)
6. \( \boxed{\frac{m^2}{(w + 1)^2}} \)
7. \( \boxed{\frac{w(w - 3)}{(w + 1)(w - 2)}} \)
8. \( \boxed{\frac{(x + y)^2}{(w + 1)^2}} \)
9. \( \boxed{1.25, 1.1875, 1.\overline{1}, 1.\overline{1}, 1.\overline{1}} \)