Math worksheet from Tutorified focusing on solving numerical expressions, including simplification, algebraic evaluation, and real-world application problems.

Problem: Solving Numerical Expressions

We will solve each part of the problem step by step.

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#### Part 1: Simplify the following expressions.

a. \( 10 + (25 - 10) \div 5 \)

1. Solve the expression inside the parentheses:
\[
25 - 10 = 15
\]
2. Perform the division:
\[
15 \div 5 = 3
\]
3. Add the result to 10:
\[
10 + 3 = 13
\]

Solution:
\[
\boxed{13}
\]

---

b. \( 4 \div [25 - (5 - 2)] \)

1. Solve the innermost parentheses:
\[
5 - 2 = 3
\]
2. Subtract the result from 25:
\[
25 - 3 = 22
\]
3. Perform the division:
\[
4 \div 22 = \frac{4}{22} = \frac{2}{11}
\]

Solution:
\[
\boxed{\frac{2}{11}}
\]

---

c. \( \left(4 \frac{1}{2} + 3 \frac{3}{2}\right) \div \left(\frac{7}{12} - \frac{1}{2}\right) \)

1. Convert mixed numbers to improper fractions:
\[
4 \frac{1}{2} = \frac{9}{2}, \quad 3 \frac{3}{2} = \frac{9}{2}
\]
2. Add the fractions:
\[
\frac{9}{2} + \frac{9}{2} = \frac{18}{2} = 9
\]
3. Subtract the fractions in the denominator:
\[
\frac{7}{12} - \frac{1}{2} = \frac{7}{12} - \frac{6}{12} = \frac{1}{12}
\]
4. Divide the results:
\[
9 \div \frac{1}{12} = 9 \times 12 = 108
\]

Solution:
\[
\boxed{108}
\]

---

d. \( 4.5 \div 3 \times 5 \times (3.2 + 1.1) \)

1. Solve the expression inside the parentheses:
\[
3.2 + 1.1 = 4.3
\]
2. Perform the division and multiplication from left to right:
\[
4.5 \div 3 = 1.5
\]
\[
1.5 \times 5 = 7.5
\]
\[
7.5 \times 4.3 = 32.25
\]

Solution:
\[
\boxed{32.25}
\]

---

e. \( \frac{2}{5} + 7 - \frac{1}{2} \times 8 \)

1. Perform the multiplication:
\[
\frac{1}{2} \times 8 = 4
\]
2. Subtract the result from 7:
\[
7 - 4 = 3
\]
3. Add the fraction:
\[
\frac{2}{5} + 3 = \frac{2}{5} + \frac{15}{5} = \frac{17}{5}
\]

Solution:
\[
\boxed{\frac{17}{5}}
\]

---

#### Part 2: Solve the following algebraic expressions using the given value/s of the variable.

a. \( 2(4p + 5y) - (3p + 7y) \) if \( p = 1 \) and \( y = 2 \)

1. Substitute \( p = 1 \) and \( y = 2 \):
\[
2(4(1) + 5(2)) - (3(1) + 7(2))
\]
2. Simplify inside the parentheses:
\[
2(4 + 10) - (3 + 14) = 2(14) - 17
\]
3. Perform the multiplication and subtraction:
\[
28 - 17 = 11
\]

Solution:
\[
\boxed{11}
\]

---

b. \( 3n \times \left(\frac{5}{n} - \frac{1}{3}\right) \) if \( n = 2 \)

1. Substitute \( n = 2 \):
\[
3(2) \times \left(\frac{5}{2} - \frac{1}{3}\right)
\]
2. Simplify the expression inside the parentheses:
\[
\frac{5}{2} - \frac{1}{3} = \frac{15}{6} - \frac{2}{6} = \frac{13}{6}
\]
3. Multiply:
\[
6 \times \frac{13}{6} = 13
\]

Solution:
\[
\boxed{13}
\]

---

c. \( \frac{1}{2} \times (10m - 2) + 3 \times 2 \) if \( m = \frac{2}{3} \)

1. Substitute \( m = \frac{2}{3} \):
\[
\frac{1}{2} \times \left(10 \left(\frac{2}{3}\right) - 2\right) + 3 \times 2
\]
2. Simplify inside the parentheses:
\[
10 \left(\frac{2}{3}\right) = \frac{20}{3}
\]
\[
\frac{20}{3} - 2 = \frac{20}{3} - \frac{6}{3} = \frac{14}{3}
\]
3. Perform the multiplication:
\[
\frac{1}{2} \times \frac{14}{3} = \frac{14}{6} = \frac{7}{3}
\]
4. Add the remaining term:
\[
\frac{7}{3} + 6 = \frac{7}{3} + \frac{18}{3} = \frac{25}{3}
\]

Solution:
\[
\boxed{\frac{25}{3}}
\]

---

d. \( \frac{1}{8} \times (2t + 1) + 5 \) if \( t = \frac{1}{2} \)

1. Substitute \( t = \frac{1}{2} \):
\[
\frac{1}{8} \times \left(2 \left(\frac{1}{2}\right) + 1\right) + 5
\]
2. Simplify inside the parentheses:
\[
2 \left(\frac{1}{2}\right) = 1
\]
\[
1 + 1 = 2
\]
3. Perform the multiplication:
\[
\frac{1}{8} \times 2 = \frac{2}{8} = \frac{1}{4}
\]
4. Add the remaining term:
\[
\frac{1}{4} + 5 = \frac{1}{4} + \frac{20}{4} = \frac{21}{4}
\]

Solution:
\[
\boxed{\frac{21}{4}}
\]

---

e. \( m + \frac{(n-1)}{4} \div 5 \) if \( m = \frac{3}{4} \) and \( n = 2 \)

1. Substitute \( m = \frac{3}{4} \) and \( n = 2 \):
\[
\frac{3}{4} + \frac{(2-1)}{4} \div 5
\]
2. Simplify inside the parentheses:
\[
2 - 1 = 1
\]
\[
\frac{1}{4} \div 5 = \frac{1}{4} \times \frac{1}{5} = \frac{1}{20}
\]
3. Add the terms:
\[
\frac{3}{4} + \frac{1}{20} = \frac{15}{20} + \frac{1}{20} = \frac{16}{20} = \frac{4}{5}
\]

Solution:
\[
\boxed{\frac{4}{5}}
\]

---

#### Part 3: Use the expression \( 5m + 4 \times 2(m - 4) \) to compute for the entrance fee for 3, 5, and 7 people. (In dollars)

The expression is:
\[
5m + 4 \times 2(m - 4)
\]

1. Simplify the expression:
\[
5m + 8(m - 4) = 5m + 8m - 32 = 13m - 32
\]

2. Compute for \( m = 3 \):
\[
13(3) - 32 = 39 - 32 = 7
\]

3. Compute for \( m = 5 \):
\[
13(5) - 32 = 65 - 32 = 33
\]

4. Compute for \( m = 7 \):
\[
13(7) - 32 = 91 - 32 = 59
\]

Solution:
\[
\boxed{7, 33, 59}
\]

---

#### Part 4: You bought a magazine for \$5 and four erasers for \$b each. How much did you spend on magazine and erasers? Evaluate the expression if \( b = 0.75 \)?

The total cost is:
\[
5 + 4b
\]

1. Substitute \( b = 0.75 \):
\[
5 + 4(0.75) = 5 + 3 = 8
\]

Solution:
\[
\boxed{8}
\]

---

#### Part 5: Evaluate the expression \( \frac{3p}{4} - \frac{p+2}{3} + \frac{p+2}{12} \) if \( p = 3 \). Choose the letter of the correct answer.

The expression is:
\[
\frac{3p}{4} - \frac{p+2}{3} + \frac{p+2}{12}
\]

1. Substitute \( p = 3 \):
\[
\frac{3(3)}{4} - \frac{3+2}{3} + \frac{3+2}{12} = \frac{9}{4} - \frac{5}{3} + \frac{5}{12}
\]

2. Find a common denominator (12):
\[
\frac{9}{4} = \frac{27}{12}, \quad \frac{5}{3} = \frac{20}{12}, \quad \frac{5}{12} = \frac{5}{12}
\]

3. Combine the fractions:
\[
\frac{27}{12} - \frac{20}{12} + \frac{5}{12} = \frac{27 - 20 + 5}{12} = \frac{12}{12} = 1
\]

Solution:
\[
\boxed{1}
\]

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Final Answers:

1. a. \( \boxed{13} \)
b. \( \boxed{\frac{2}{11}} \)
c. \( \boxed{108} \)
d. \( \boxed{32.25} \)
e. \( \boxed{\frac{17}{5}} \)

2. a. \( \boxed{11} \)
b. \( \boxed{13} \)
c. \( \boxed{\frac{25}{3}} \)
d. \( \boxed{\frac{21}{4}} \)
e. \( \boxed{\frac{4}{5}} \)

3. \( \boxed{7, 33, 59} \)

4. \( \boxed{8} \)

5. \( \boxed{1} \)