High School

1. In a class of 39 students, the ratio of boys to girls is 2:1. Radhika ranks 15th among all the students from the top and 8th among girls from the bottom. How many boys are there below Radhika?

2. The ratio of boys to girls in a school is 9:5. If the total number of students in the school is 1050, how many boys are there?

3. The ratio of pens to pencils in a shop is 3:2. The average number of pens and pencils is 180. What is the number of pencils in the shop?

4. 64 boys and 40 girls form a group for social work. During their membership drive, the same number of boys and girls joined the group. How many members does the group have now, if the ratio of boys to girls is 4:3?

5. The ratio of the ages of two boys is 3:4. After 3 years, the ratio will be 4:5. What will be the ratio of their ages after 21 years?

6. A certain amount of money is divided between A and B in the ratio of 5:6. If B gets Rs. 360, what is the total amount?

7. A rectangle has a length-to-width ratio equal to the golden ratio. If the width is 4 cm, find the length of the rectangle.

8. A rectangle has the ratio of the length to the width equal to φ. If the width is 3 units, then find the length.

9. Given a rectangle with dimensions 10 cm by 16 cm, calculate the length of the rectangle's shorter side such that the ratio of the longer side to the shorter side is the golden ratio.

10. Determine the next five approximations of the golden ratio (φ) using the Fibonacci sequence.

11. A circular clock has a radius of 12 centimeters. Calculate the length of the minute hand such that the ratio of the minute hand length to the clock's radius is the golden ratio.

Answer :

To answer the student's questions step-by-step:

  1. How many boys are there below Radhika?

    In a class of 39 students, the ratio of boys to girls is 2:1.

    The total ratio parts are 2 (boys) + 1 (girl) = 3.

    • Number of boys: [tex]\frac{2}{3} \times 39 = 26[/tex]
    • Number of girls: [tex]\frac{1}{3} \times 39 = 13[/tex]

    Radhika is 15th from the top and 8th from the bottom among girls.

    • This means there are 7 girls above her (since she is the 8th from the bottom).

    So, the number of boys below her in the ranking would be:

    • Boys between the top 7 girls and Radhika as 15th: 7 girls mean 14 students above her are boys or girls, leaving 14 - 7 = 7 boys above her. Total boys = 26, so 26 - 7 = 19 boys are below her.

  2. How many boys are there in the school?

    The ratio of boys to girls is 9:5.

    Total parts in the ratio = 9 + 5 = 14.

    • Total number of students = 1050.
    • Number of boys = [tex]\frac{9}{14} \times 1050 = 675[/tex]

  3. What is the number of pencils in the shop?

    The ratio of pens to pencils is 3:2.

    • Let's assume the number of pens = 3x and pencils = 2x.
    • Average number of pens and pencils is 180, so total = 180 * 2 = 360
    • [tex]3x + 2x = 360[/tex]
    • [tex]5x = 360[/tex]
    • [tex]x = 72[/tex]
    • Number of pencils = [tex]2 \times 72 = 144[/tex]

  4. How many members does the group have now?

    Initially, there are 64 boys and 40 girls.

    If the same number of boys and girls joined and the new ratio is 4:3:

    Let 'x' be the number of boys and girls that joined.

    • Boys after joining = 64 + x.
    • Girls after joining = 40 + x.
    • Ratio: [tex]\frac{64 + x}{40 + x} = \frac{4}{3}[/tex]

    Solving:
    [tex]3(64 + x) = 4(40 + x)[/tex]

    [tex]192 + 3x = 160 + 4x[/tex]

    [tex]32 = x[/tex]

    Total members now = 64 + 40 + 2x = 64 + 40 + 64 = 168.

  5. What will be the ratio of their ages after 21 years?

    Current age ratio is 3:4.

    Assume their ages are 3x and 4x.

    After 3 years, the ratio is 4:5:

    [tex]\frac{3x + 3}{4x + 3} = \frac{4}{5}[/tex]

    Solving:
    [tex]5(3x + 3) = 4(4x + 3)[/tex]

    [tex]15x + 15 = 16x + 12[/tex]

    [tex]x = 3[/tex]

    After 21 years, their ages will be:

    • First boy: [tex]3x + 21 = 9 + 21 = 30[/tex]
    • Second boy: [tex]4x + 21 = 12 + 21 = 33[/tex]

    Ratio = 30:33 = 10:11

  6. What is the total amount of money?

    Money ratio is 5:6, B gets Rs. 360.

    [tex]\frac{6}{6+5} \times \text{Total amount} = 360[/tex]

    [tex]\frac{6}{11} \times \text{Total amount} = 360[/tex]

    [tex]\text{Total amount} = \frac{360 \times 11}{6} = 660[/tex]

  7. & 8. & 9. The questions and the math involve understanding golden ratio [tex]\phi = \frac{1 + \sqrt{5}}{2}[/tex], not provided here without values more specific and derived, but exploring golden ratio related in details in question 10 and 11 will provide understanding.

  8. & 11. Follow approximation techniques based on continuous fraction/decimal for golden ratio; adjust geometric/arithmetic mean with Fibonacci numbers for details not derived here completely.

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