Maths Puzzles

  At ten o'clock one morning Mr. Smith and his wife left their house in Connecticut to drive to the home of Mrs. Smith's parents in Penn- sylvania. They planned to stop once along the way for lunch at Patri- cia Murphy's Candlelight Restaurant in Westchester. The prospective visit with his in-laws, combined with business worries, put Mr. Smith in a sullen, uncommunicative mood. It was not until eleven o'clock that Mrs. Smith ventured to ask: "How far have we gone, dear?" Mr. Smith glanced at the mileage meter. "Half as far as the distance from here to Patricia Murphy's," he snapped. They arrived at the restaurant at noon, enjoyed a leisurely lunch, then continued on their way. Not until five o'clock, when they were 200 miles from the place where Mrs. Smith had asked her first question, did she ask a second one. "How much farther do we have to go, dear?" "Half as far," he grunted, "as the distance from here to Patricia

The question was  Two missiles speed directly toward each other, one at 9,000 miles per hour and the other at 21,000 miles per hour. They start 1,317 miles apart. Without using pencil and paper, calculate how far apart they are one minute before they collide.  If two objects are moving in opposite direction in speeds x and y , their relative speeds are x+y. (If they are moving in the same direction, their relative speeds are x-y). In this question, they are moving towards each other at a speed of 21000+9000 = 30000 miles/hour = 30000/60 = 500 miles/minute. Which means, just one minute before their collision, they were 500 miles apart.

 Two missiles speed directly toward each other, one at 9,000 miles per hour and the other at 21,000 miles per hour. They start 1,317 miles apart. Without using pencil and paper, calculate how far apart they are one minute before they collide. Find the solution here .

The question was  What is the pair of numbers whose product is equal to the product of its least common multiple (LCM) and greatest common divisor (GCD)? The solution is any pair of positive integers. Product of LCM and GCD is equal to product of the given numbers for any two positive integers.

Solution: Draw a diagonal AB from the same corner.   Area of triangle ABD is 1/2 of area of trisection. i.e. it is 1/6 area of rectangle. Area of triangle DBC is 1/3 area of rectangle. As these two triangles have a common altitude BC, their bases must be in the ratio 1:2 Ratio of AD:DC = 1:2 So point D trisects the line AC. Similar thing is true for  the the other line.  

 From one corner of a square, draw two straight lines such that the lines trisect area of the triangle exactly. In what ratio, do these line cut the two sides of the square?