Question

1. alan has 4 red balls and 8 blue balls in a hat. what is the probability of alan picking blue balls?

a. \\(\frac{1}{3}\\)
b. \\(\frac{1}{4}\\)
c. \\(\frac{1}{2}\\)
d. \\(\frac{2}{3}\\)
e. \\(\frac{1}{8}\\)

1. what is the result if you rationalise the following? \\(\frac{3}{\sqrt{5}}\\)

a. 3
b. \\(\frac{3\sqrt{5}}{5}\\)
c. \\(5\sqrt{5}\\)
d. \\(\frac{3}{5}\\)
e. 5

1. find the shaded area correct to two decimal places.

a. \\(8.44\text{ m}^2\\)
b. \\(9.52\text{ m}^2\\)
c. \\(10.86\text{ m}^2\\)
d. \\(14.14\text{ m}^2\\)
e. none of these

1. a town has 2,500 residents of whom 60% voted in an election to fill a council vacancy. the result was that of those who voted 38% voted for p, 32% for q and 30% for r. under the voting system, p was elected. the number of residents who voted for p was

a. 450
b. 570
c. 1250
d. 950
e. none of these

1. if a rectangle has area \\(24\text{ cm}^2\\) which one of the following is a possible perimeter of this rectangle?

a. \\(20\text{ cm}\\)
b. \\(25\text{ cm}\\)
c. \\(21\text{ cm}\\)
d. \\(44\text{ cm}\\)
e. \\(30\text{ cm}\\)

Explanation:

Calculate probability of picking blue balls

Find the total number of balls and the fraction of blue balls.
Total balls:

$$ 4 + 8 = 12 $$

Probability of picking a blue ball:

$$ P(\text{Blue}) = \frac{8}{12} = \frac{2}{3} $$

Rationalize the denominator

Multiply the numerator and the denominator by the radical conjugate.
Given expression:

$$ \frac{5}{\sqrt{5}} $$

Multiply numerator and denominator by \(\sqrt{5}\):

$$ \frac{5 \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{5\sqrt{5}}{5} = \sqrt{5} $$

Since \(\sqrt{5}\) is not explicitly listed as a single term, let's check the options:
Option B is \(\frac{5\sqrt{5}}{5}\), which simplifies to \(\sqrt{5}\).

Find the shaded area

Calculate the area of the square and subtract the unshaded regions.
The side length of the square is:

$$ 1\text{ m} + 2\text{ m} + 1\text{ m} = 4\text{ m} $$

Area of the square:

$$ A_{\text{square}} = 4 \times 4 = 16\text{ m}^2 $$

The unshaded regions consist of four semi-circular cutouts on each side.
The diameter of each semi-circle is \(2\text{ m}\), so the radius \(r = 1\text{ m}\).
Four semi-circles make up two full circles:

$$ A_{\text{unshaded}} = 2 \times \pi r^2 = 2 \times \pi \times 1^2 \approx 2 \times 3.14159 = 6.283\text{ m}^2 $$

Shaded area:

$$ A_{\text{shaded}} = 16 - 6.283 = 9.717\text{ m}^2 $$

Since \(9.72\text{ m}^2\) is not listed, the correct choice is "none of these".

Calculate the number of residents who voted for P

Find the total voters first, then the votes for P.
Total voters:

$$ 2500 \times 0.60 = 1500 $$

Votes for P:

$$ 1500 \times 0.38 = 570 $$

Determine a possible perimeter of the rectangle

Find integer dimensions that yield an area of \(24\text{ cm}^2\) and check their perimeters.
Let the side lengths be \(l\) and \(w\) such that \(l \times w = 24\).
Possible integer pairs \((l, w)\) and their perimeters \(2(l+w)\):

• \((24, 1) \implies P = 2(24+1) = 50\text{ cm}\)
• \((12, 2) \implies P = 2(12+2) = 28\text{ cm}\)
• \((8, 3) \implies P = 2(8+3) = 22\text{ cm}\)
• \((6, 4) \implies P = 2(6+4) = 20\text{ cm}\)

Comparing with the options, \(20\text{ cm}\) is a possible perimeter.

Answer:

Question 19

• A. \(\frac{1}{3}\)
• B. \(\frac{1}{4}\)
• C. \(\frac{1}{2}\)
• D. \(\frac{2}{3}\) (Correct answer)
• E. \(\frac{1}{8}\)

Question 20

• A. \(3\)
• B. \(\frac{5\sqrt{5}}{5}\) (Correct answer)
• C. \(5\sqrt{5}\)
• D. \(\frac{3}{5}\)
• E. \(5\)

Question 21

• A. \(8.44\text{m}^2\)
• B. \(9.52\text{m}^2\)
• C. \(10.86\text{m}^2\)
• D. \(14.14\text{m}^2\)
• E. none of these (Correct answer)

Question 22

• A. \(450\)
• B. \(570\) (Correct answer)
• C. \(1250\)
• D. \(950\)
• E. none of these

Question 23

• A. \(20\text{cm}\) (Correct answer)
• B. \(25\text{cm}\)
• C. \(21\text{cm}\)
• D. \(44\text{cm}\)
• E. \(30\text{cm}\)