Maths Puzzles

 The puzzle is A car goes uphill a certain distance at a speed of A km per hour. Then it comes back the same distance at a speed of B km per hour. A and B are not equal.  Which of the averages are greater - Average of A and B or average speed of the total travel?  The answer is average of A and B. Solution :    Let us say the distance the car travels in one direction is D. Let the time taken for uphill travel is t1 and time taken for downhill travel is t2. Total distance traveled = 2D Total time = t1+t2 = D/A + D/B = DB+DA / AB = D (A+B)/AB Average speed of the travel = distance / time = 2D /(D (A+B) / AB)                     = 2AB/(A+B)    Average of A and B  = (A+B)/2 Which of these two is greater? 2AB/(A+B)                     A+B / 2 2AB                                 (A+B)*(A+B)/2      --multiplying both the sides by A+B 4AB                                  (A+B)*(A+B)        ---Multiplying both sides by 2 4AB                                   A 2 +B 2 +2AB         --Expanding

A car goes uphill a certain distance at a speed of A km per hour. Then it comes back the same distance at a speed of B km per hour. A and B are not equal.  Which of the averages are greater - Average of A and B or average speed of the total travel?

 The question was  What is wrong with this proof ? a=b multiply both the sides by a     --1 a2 = ab  add -b2 to both the side           --2 a2-b2 =ab-b2  factorizing                                --3 (a+b)(a-b) = b(a-b)  divide both the sides by (a-b)    ---4 a+b=b Substituting a with b,              ---5 2b = b Dividing both the sides by b    ---6 2 = 1   The problem is with step 4. When a is equal to b, a-b is zero. You can not divide numbers by zero because division is only defined for non-zero numbers.  

 The puzzle was   The sum of two numbers is 10. The product of these numbers is 20. What is the sum of their reciprocals? Instead of writing complicated equations, we can use a simple tweak.  If the numbers are x and y, then sum of their reciprocal is 1/x+1/y = (x+y)/xy Since we have both sum - x+y and product - xy of the numbers, the answer is 10/20 = 1/2

The question was In the following subtraction problem, each letter uniquely repre- sents one digit from 0 to 9. At least one digit is not 0. Find the values of A, B, and C. ABA – CA= AB  Now if A, B and C represent digits from 0 to 9, the following equation will be 100A+10B+A - (10C+A) = 10xA+B  90A+9B = 10C  9(10A+B) = 10C 10A+B = 10/9 C  To have integer values for A and B, C has to be 9. By replacing C with 9 we get, 10A+B = 10 If A and B have to between 0 to 9, A has to be 1 and B has to be 0. So A =1, B = 0 and C=9  

In the following subtraction problem, each letter uniquely represents one digit from 0 to 9. At least one digit is not 0. Find the values of A, B, and C.   ABA  –      CA -------------      AB

 The sum of two numbers is 10. The product of these numbers is 20. What is the sum of their reciprocals?