Magical Math Adventure

Problem 1 🌟

If the difference between a number and its reciprocal is 24/5, find the number.

Let the number be x

Its reciprocal is 1/x

Given: x - (1/x) = 24/5

Multiply both sides by x: x² - 1 = (24/5)x

Rearrange: 5x² - 24x - 5 = 0

Solve quadratic equation: x = [24 ± √(576 + 100)]/10

Calculate: x = [24 ± 26]/10

Solutions: x = 5 or x = -1/5

Answer: The number is 5 or -1/5 🎉

Problem 2 🌳

A garden measuring 12m by 16m is to have a pedestrian pathway that is 'w' meters wide installed all the way around so that it increases the total area to 285 m². What is the width of the pathway?

Original garden area: 12 × 16 = 192 m²

With pathway, dimensions become (12+2w) by (16+2w)

Total area with pathway: (12+2w)(16+2w) = 285

Expand: 192 + 56w + 4w² = 285

Simplify: 4w² + 56w - 93 = 0

Divide by 4: w² + 14w - 23.25 = 0

Solve quadratic: w = [-14 ± √(196 + 93)]/2

Calculate: w = [-14 ± 17]/2

Only positive solution: w = 1.5 m

Answer: The pathway is 1.5 meters wide 🌈

Problem 3 🚌

A bus covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hour more it would have taken 30 minutes less for the journey. Find the original speed of the bus.

Let original speed = x km/h

Original time = 90/x hours

New speed = (x + 15) km/h

New time = 90/(x + 15) hours

Time difference: 90/x - 90/(x+15) = 0.5 hours

Multiply by 2x(x+15): 180(x+15) - 180x = x(x+15)

Simplify: 2700 = x² + 15x

Rearrange: x² + 15x - 2700 = 0

Solve quadratic: x = [-15 ± √(225 + 10800)]/2

Calculate: x = [-15 ± 105]/2

Positive solution: x = 45 km/h

Answer: Original speed was 45 km/h 🚀

Problem 4 👧

A girl is twice as old as her sister. Five years hence, the product of their ages (in years) will be 375. Find their present ages.

Let sister's age = x years

Girl's age = 2x years

Five years hence: sister = x+5, girl = 2x+5

Product: (x+5)(2x+5) = 375

Expand: 2x² + 15x + 25 = 375

Simplify: 2x² + 15x - 350 = 0

Divide by 2: x² + 7.5x - 175 = 0

Solve quadratic: x = [-7.5 ± √(56.25 + 700)]/2

Calculate: x = [-7.5 ± 27.5]/2

Positive solution: x = 10 years

Girl's age: 2x = 20 years

Answer: Sister is 10, girl is 20 years old 🎂

Problem 5 ⚡

A pole has to be erected at a point on the boundary of a circular ground of diameter 20 m in such a way that the difference of its distances from two diametrically opposite fixed gates P and Q on the boundary is 4 m. Is it possible to do so? If answer is yes at what distance from the two gates should the pole be erected?

Diameter = 20 m ⇒ radius = 10 m

Let pole be at point R on circumference

Given: |PR - QR| = 4 m

Since P and Q are diametrically opposite, PQ = 20 m

Triangle PQR is right-angled at R (Thales' theorem)

Let PR = x ⇒ QR = √(400 - x²)

Given: |x - √(400 - x²)| = 4

Square both sides: x² + (400 - x²) - 2x√(400 - x²) = 16

Simplify: 400 - 16 = 2x√(400 - x²)

Square again: (384)² = 4x²(400 - x²)

Solve to get: x² = 256 or x² = 144

Thus, x = 16 m or x = 12 m

Answer: Yes! Pole should be 16m from one gate and 12m from the other ✨

Problem 6 🐝

From a group of 2x² black bees, square root of half of the group went to a tree. Again eight-ninth of the bees went to the same tree. The remaining two got caught up in a fragrant lotus. How many bees were there in total?

Total bees = 2x²

First group: √(2x²/2) = √(x²) = x bees

Remaining bees: 2x² - x

Second group: (8/9)(2x² - x) bees

Remaining bees: (1/9)(2x² - x) = 2

Thus: 2x² - x = 18

Rearrange: 2x² - x - 18 = 0

Solve quadratic: x = [1 ± √(1 + 144)]/4

Positive solution: x = (1 + 12.2)/4 ≈ 3.3

Since x must be integer, try x=3: 2(9)-3=15≠18

Try x=3.3: 2(10.89)-3.3≈18.48 (close)

Exact solution: x = (1 + √145)/4 ≈ 3.26

Answer: There were approximately 21.3 bees (problem may have integer solution with exact calculation) 🍯