Let's go through each part of the question step-by-step to solve it clearly and accurately.
Part b: Simplify [tex]\(\frac{5}{8}\left(1 \frac{1}{2} + \frac{3}{4} \div \frac{1}{4}\right)\)[/tex]
First, we convert the mixed number [tex]\(1 \frac{1}{2}\)[/tex] to an improper fraction:
[tex]\[ 1 \frac{1}{2} = 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2} \][/tex]
Next, we handle the division within the parentheses:
[tex]\[ \frac{3}{4} \div \frac{1}{4} = \frac{3}{4} \times \frac{4}{1} = 3 \][/tex]
Now we add the results inside the parentheses:
[tex]\[ 1 \frac{1}{2} + 3 = \frac{3}{2} + 3 = \frac{3}{2} + \frac{6}{2} = \frac{9}{2} \][/tex]
Finally, we multiply by [tex]\(\frac{5}{8}\)[/tex]:
[tex]\[ \frac{5}{8} \times \frac{9}{2} = \frac{5 \times 9}{8 \times 2} = \frac{45}{16} \][/tex]
Thus,
[tex]\[ \frac{45}{16} = 2.8125 \][/tex]
Part c: Evaluate [tex]\(\frac{4.56 \times 3.6}{0.12}\)[/tex] leaving your answer in standard form
First, perform the multiplication in the numerator:
[tex]\[ 4.56 \times 3.6 = 16.416 \][/tex]
Next, divide by the denominator:
[tex]\[ \frac{16.416}{0.12} = 136.8 \][/tex]
Convert 136.8 to standard form (scientific notation):
[tex]\[ 136.8 = 1.368 \times 10^2 \][/tex]
Part 2E: Solve [tex]\( \frac{\frac{112}{50}}{1000} \times \frac{112}{10} \)[/tex]
Simplify the fraction within the numerator:
[tex]\[ \frac{112}{50} = 2.24 \][/tex]
Convert the divisor [tex]\(1000\)[/tex] to a fraction:
[tex]\[ \frac{2.24}{1000} = 0.00224 \][/tex]
Now multiply by [tex]\( \frac{112}{10} \)[/tex]:
[tex]\[ 0.00224 \times 11.2 = 0.025088 \][/tex]
Part b (inequality): Solve [tex]\(3x - 9 \geq 12(0 - 3)\)[/tex] and illustrate the answer on the number line
First, simplify the right-hand side:
[tex]\[ 12(0 - 3) = 12 \times (-3) = -36 \][/tex]
Now we write the inequality:
[tex]\[ 3x - 9 \geq -36 \][/tex]
Add 9 to both sides:
[tex]\[ 3x - 9 + 9 \geq -36 + 9 \][/tex]
[tex]\[ 3x \geq -27 \][/tex]
Divide by 3:
[tex]\[ x \geq -9 \][/tex]
Part c: Evaluate [tex]\(\frac{20}{a} - b\)[/tex] if [tex]\(a = 30\)[/tex] and [tex]\(b = 1\)[/tex]
Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the expression:
[tex]\[ \frac{20}{30} - 1 \][/tex]
[tex]\[ \frac{2}{3} - 1 = \frac{2}{3} - \frac{3}{3} = -\frac{1}{3} \][/tex]
Thus,
[tex]\[ -\frac{1}{3} = -0.3333 \][/tex]
Part 4a: Find the area of a semi-circle whose radius is 7 cm (using [tex]\(\pi = \frac{22}{7}\)[/tex])
First, find the area of a full circle:
[tex]\[ \text{Area} = \pi r^2 = \frac{22}{7} \times 7^2 = \frac{22}{7} \times 49 = 154 \][/tex]
The area of a semi-circle is half of the full circle:
[tex]\[ \text{Area of semi-circle} = \frac{154}{2} = 77 \text{ sq cm} \][/tex]
To summarize, here are the results:
- Part b: [tex]\(2.8125\)[/tex]
- Part c: [tex]\(1.37 \times 10^2\)[/tex]
- Part 2E: [tex]\(0.025088\)[/tex]
- Part b (inequality): [tex]\(x \geq -9\)[/tex]
- Part c (expression evaluation): [tex]\(-0.3333\)[/tex]
- Part 4a: [tex]\(77 \text{ sq cm}\)[/tex]
