Answer :
Sure, let's solve each problem step-by-step:
Problem 4: Solve for [tex]\( h \)[/tex] in the equation [tex]\( 4h + \frac{1}{3} = \frac{3}{4} \)[/tex].
1. Subtract [tex]\(\frac{1}{3}\)[/tex] from both sides:
[tex]\[ 4h = \frac{3}{4} - \frac{1}{3} \][/tex]
2. Find a common denominator for [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex], which is 12:
[tex]\[ \frac{3}{4} = \frac{9}{12}, \quad \frac{1}{3} = \frac{4}{12} \][/tex]
So,
[tex]\[ \frac{9}{12} - \frac{4}{12} = \frac{5}{12} \][/tex]
3. Divide both sides by 4:
[tex]\[ h = \frac{5}{12} \times \frac{1}{4} = \frac{5}{48} \][/tex]
Thus, [tex]\( h \approx 0.1042 \)[/tex].
Problem 5: Solve for [tex]\( f \)[/tex] in the equation [tex]\( \frac{1}{7}f - 5 \frac{1}{2} = \frac{9}{14} \)[/tex].
1. Convert [tex]\( 5 \frac{1}{2} \)[/tex] to an improper fraction:
[tex]\( 5 \frac{1}{2} = \frac{11}{2} \)[/tex]
2. Subtract [tex]\(\frac{11}{2}\)[/tex] from both sides:
[tex]\[ \frac{1}{7}f = \frac{9}{14} + \frac{11}{2} \][/tex]
3. Find a common denominator for [tex]\(\frac{9}{14}\)[/tex] and [tex]\(\frac{11}{2}\)[/tex], which is 14:
[tex]\(\frac{11}{2} \approx \frac{77}{14}\)[/tex] and then,
[tex]\[ \frac{9}{14} + \frac{77}{14} = \frac{86}{14} \approx \frac{43}{7} \][/tex]
4. Multiply both sides by 7:
[tex]\[ f = 43 \][/tex]
Problem 7: Determine how many skateboards were repaired.
1. Write the equation:
[tex]\( 15 \times n + 132.49 = 192.49 \)[/tex]
2. Subtract 132.49 from both sides:
[tex]\[ 15n = 60 \][/tex]
3. Divide by 15 to solve for [tex]\( n \)[/tex]:
[tex]\[ n = 4 \][/tex]
So, 4 skateboards were repaired.
Problem 8: Find the flat fee for the music download service.
1. Set up the equation:
[tex]\( \text{flat fee} + 0.99 \times 27 = 42.72 \)[/tex]
2. Calculate [tex]\( 0.99 \times 27 = 26.73 \)[/tex]
3. Subtract 26.73 from both sides:
[tex]\[ \text{flat fee} = 42.72 - 26.73 = 15.99 \][/tex]
Problem 9: Solve for [tex]\( m \)[/tex] in [tex]\(-5(m+4)=27\)[/tex].
1. Distribute [tex]\(-5\)[/tex]:
[tex]\(-5m - 20 = 27\)[/tex]
2. Add 20 to both sides:
[tex]\(-5m = 47\)[/tex]
3. Divide by [tex]\(-5\)[/tex]:
[tex]\[ m = -\frac{47}{5} \approx -9.4\][/tex]
Problem 10: Solve for [tex]\( a \)[/tex] in [tex]\(-12(a-2)=-50\)[/tex].
1. Distribute [tex]\(-12\)[/tex]:
[tex]\(-12a + 24 = -50\)[/tex]
2. Subtract 24 from both sides:
[tex]\(-12a = -74\)[/tex]
3. Divide by [tex]\(-12\)[/tex]:
[tex]\[ a = \frac{74}{12} \approx \frac{37}{6} \approx 6.1667 \][/tex]
Problem 11: Simplify [tex]\(-5x - 2x + 3\)[/tex].
Combine like terms:
[tex]\[ -7x + 3 \][/tex]
Problem 12: Find the lengths of the two unknown sides of the triangle.
1. Let the shorter unknown side be [tex]\( x \)[/tex], then the longer side is [tex]\( 2x \)[/tex].
2. Using the perimeter equation:
[tex]\[ 12 + x + 2x = 60 \][/tex]
3. Combine like terms:
[tex]\[ 12 + 3x = 60 \][/tex]
4. Subtract 12 from both sides:
[tex]\[ 3x = 48 \][/tex]
5. Divide by 3:
[tex]\[ x = 16 \][/tex]
So, the lengths of the unknown sides are 16 and 32 feet.
Problem 4: Solve for [tex]\( h \)[/tex] in the equation [tex]\( 4h + \frac{1}{3} = \frac{3}{4} \)[/tex].
1. Subtract [tex]\(\frac{1}{3}\)[/tex] from both sides:
[tex]\[ 4h = \frac{3}{4} - \frac{1}{3} \][/tex]
2. Find a common denominator for [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex], which is 12:
[tex]\[ \frac{3}{4} = \frac{9}{12}, \quad \frac{1}{3} = \frac{4}{12} \][/tex]
So,
[tex]\[ \frac{9}{12} - \frac{4}{12} = \frac{5}{12} \][/tex]
3. Divide both sides by 4:
[tex]\[ h = \frac{5}{12} \times \frac{1}{4} = \frac{5}{48} \][/tex]
Thus, [tex]\( h \approx 0.1042 \)[/tex].
Problem 5: Solve for [tex]\( f \)[/tex] in the equation [tex]\( \frac{1}{7}f - 5 \frac{1}{2} = \frac{9}{14} \)[/tex].
1. Convert [tex]\( 5 \frac{1}{2} \)[/tex] to an improper fraction:
[tex]\( 5 \frac{1}{2} = \frac{11}{2} \)[/tex]
2. Subtract [tex]\(\frac{11}{2}\)[/tex] from both sides:
[tex]\[ \frac{1}{7}f = \frac{9}{14} + \frac{11}{2} \][/tex]
3. Find a common denominator for [tex]\(\frac{9}{14}\)[/tex] and [tex]\(\frac{11}{2}\)[/tex], which is 14:
[tex]\(\frac{11}{2} \approx \frac{77}{14}\)[/tex] and then,
[tex]\[ \frac{9}{14} + \frac{77}{14} = \frac{86}{14} \approx \frac{43}{7} \][/tex]
4. Multiply both sides by 7:
[tex]\[ f = 43 \][/tex]
Problem 7: Determine how many skateboards were repaired.
1. Write the equation:
[tex]\( 15 \times n + 132.49 = 192.49 \)[/tex]
2. Subtract 132.49 from both sides:
[tex]\[ 15n = 60 \][/tex]
3. Divide by 15 to solve for [tex]\( n \)[/tex]:
[tex]\[ n = 4 \][/tex]
So, 4 skateboards were repaired.
Problem 8: Find the flat fee for the music download service.
1. Set up the equation:
[tex]\( \text{flat fee} + 0.99 \times 27 = 42.72 \)[/tex]
2. Calculate [tex]\( 0.99 \times 27 = 26.73 \)[/tex]
3. Subtract 26.73 from both sides:
[tex]\[ \text{flat fee} = 42.72 - 26.73 = 15.99 \][/tex]
Problem 9: Solve for [tex]\( m \)[/tex] in [tex]\(-5(m+4)=27\)[/tex].
1. Distribute [tex]\(-5\)[/tex]:
[tex]\(-5m - 20 = 27\)[/tex]
2. Add 20 to both sides:
[tex]\(-5m = 47\)[/tex]
3. Divide by [tex]\(-5\)[/tex]:
[tex]\[ m = -\frac{47}{5} \approx -9.4\][/tex]
Problem 10: Solve for [tex]\( a \)[/tex] in [tex]\(-12(a-2)=-50\)[/tex].
1. Distribute [tex]\(-12\)[/tex]:
[tex]\(-12a + 24 = -50\)[/tex]
2. Subtract 24 from both sides:
[tex]\(-12a = -74\)[/tex]
3. Divide by [tex]\(-12\)[/tex]:
[tex]\[ a = \frac{74}{12} \approx \frac{37}{6} \approx 6.1667 \][/tex]
Problem 11: Simplify [tex]\(-5x - 2x + 3\)[/tex].
Combine like terms:
[tex]\[ -7x + 3 \][/tex]
Problem 12: Find the lengths of the two unknown sides of the triangle.
1. Let the shorter unknown side be [tex]\( x \)[/tex], then the longer side is [tex]\( 2x \)[/tex].
2. Using the perimeter equation:
[tex]\[ 12 + x + 2x = 60 \][/tex]
3. Combine like terms:
[tex]\[ 12 + 3x = 60 \][/tex]
4. Subtract 12 from both sides:
[tex]\[ 3x = 48 \][/tex]
5. Divide by 3:
[tex]\[ x = 16 \][/tex]
So, the lengths of the unknown sides are 16 and 32 feet.