a. Golden Ratio:
1. The length of a rectangle is 1.618 times its width. If the width is 5 cm, what is the length of the rectangle?
Solution: Length of the rectangle = 1.618 x 5 = 8.09 cm
2. The ratio of the distance between the navel and the feet to the distance between the navel and the top of the head in a human body is approximately equal to the Golden Ratio. If the distance from the navel to the top of the head is 60 cm, what is the distance from the navel to the feet?
Solution: Distance from the navel to the feet = 60 / 1.618 = 37.16 cm
3. A spiral galaxy has a diameter of 100,000 light years. If the ratio of the diameter of the galaxy to the distance between the center and the edge is equal to the Golden Ratio, what is the distance between the center and the edge of the galaxy?
Solution: Distance between the center and the edge of the galaxy = 100,000 / 1.618 = 61,856 light years
4. The length of a line segment is divided in such a way that the ratio of the longer segment to the shorter segment is equal to the Golden Ratio. If the length of the line segment is 10 cm, what is the length of the longer segment?
Solution: Length of the longer segment = 10 / 1.618 = 6.18 cm
5. A regular pentagon has a side length of 10 cm. If the ratio of the diagonal to the side length is equal to the Golden Ratio, what is the length of the diagonal?
Solution: Length of the diagonal = 10 x 1.618 = 16.18 cm
b. Translating English sentences into Mathematical sentences:
1. Five times a number plus 8 is equal to 23.
Solution: 5x + 8 = 23
2. The sum of two consecutive integers is 27.
Solution: x + (x+1) = 27
3. The product of a number and its reciprocal is equal to 1.
Solution: x * (1/x) = 1
4. One-sixth of a number is equal to 4.
Solution: x/6 = 4
5. The difference between twice a number and 5 is equal to 13.
Solution: 2x - 5 = 13
c. Translating Mathematical sentences into English sentences:
1. 9x - 2y = 16
Solution: Nine times x minus two times y equals sixteen.
2. 3(x + 5) = 21
Solution: Three times the sum of x and five is equal to twenty-one.
3. x² + 4x - 12 = 0
Solution: The quadratic equation x squared plus four times x minus twelve equals zero.
4. ∠ABC + ∠CBD = ∠ABD
Solution: The sum of angle ABC and angle CBD is equal to angle ABD.
5. If x is greater than y and y is greater than z, then x is greater than z.
Solution: If x is greater than y and y is greater than z, then x is greater than z.
d. Compound statements into Symbolic form:
1. If it rains, then the ground will be wet.
Solution: R → W
2. Either it is sunny or it is not raining.
Solution: S ∨ ¬R
3. If the temperature is below freezing and there is precipitation, then there will be snow.
Solution: (T < 0) ∧ P → S
4. If a number is divisible by 3 and 5, then it is divisible by 15.
Solution: (n % 3 = 0) ∧ (n % 5 = 0) → (n % 15 = 0)
5. For all real numbers x, if x is positive, then x squared is positive.
Solution: ∀x ∈ ℝ, x > 0 → x² > 0
e. Symbolic statements into Compound statements:
1. R ∨ (P ∧ Q)
Solution: Either it rains or it is both cloudy and windy.
2. (A → B) ∧ (C → D)
Solution: If A is true, then B is true, and if C is true, then D is true.
3. ¬(P ∧ Q) ∨ (R ∧ ¬S)
Solution: Either it is not both hot and humid or it is both rainy and not sunny.
4. (¬P ∧ ¬Q) → ¬R
Solution: If it is neither hot nor cold, then it is not rainy.
5. (A ∨ B) ∧ (C ∨ D)
Solution: Either A is true or B is true, and either C is true or D is true.
f. Inverse, Converse and Contrapositive:
1. If p is a prime number, then p is odd.
Inverse: If p is not a prime number, then p is not odd.
Converse: If p is odd, then p is a prime number.
Contrapositive: If p is not odd, then p is not a prime number.
2. If a triangle is equilateral, then it is also equiangular.
Inverse: If a triangle is not equilateral, then it is not equiangular.
Converse: If a triangle is equiangular, then it is also equilateral.
Contrapositive: If a triangle is not equiangular, then it is not equilateral.
3. If the dog barks, then someone is at the door.
Inverse: If the dog does not bark, then no one is at the door.
Converse: If someone is at the door, then the dog barks.
Contrapositive: If no one is at the door, then the dog does not bark.
4. If a number is divisible by 6, then it is divisible by 2.
Inverse: If a number is not divisible by 6, then it is not divisible by 2.
Converse: If a number is divisible by 2, then it is divisible by 6.
Contrapositive: If a number is not divisible by 2, then it is not divisible by 6.
5. If it is raining, then the grass is wet.
Inverse: If it is not raining, then the grass is not wet.
Converse: If the grass is wet, then it is raining.
Contrapositive: If the grass is not wet, then it is not raining.
