Problems and Solutions

Jai Ganesh · 2006-01-15 17:02:39

irspow's solution to Problem # k + 59 is correct. I shall wait for some more time before posting the solutions to unanswered problems.

Jai Ganesh · 2006-01-15 17:11:31

Problem # k + 86

Out of the total 390 students studying in a college of Arts and Science, boys and girls are in the ratio of 7 : 6 respectively and the number of students studying Arts and Science are in the ratio of 3 : 7 respectively. The boys and girls studying Arts are in the ratio of 4 : 5 respectively. How many boys are studying Science?

Jai Ganesh · 2006-01-15 17:42:03

Excellent, irspow's solution to problem # k + 48 is correct too!

irspow · 2006-01-16 10:19:05

Jai Ganesh · 2006-01-16 19:36:28

Well done, irspow

Jai Ganesh · 2006-02-14 02:08:05

Problem # k + 87

Prove that there exists atleast one multiple of 5 between 10^k and 10^k+1 where k is a Natural number.

Ricky · 2006-02-14 02:42:22

Jai Ganesh · 2006-02-14 19:18:31

I made a mistake while posting problem # k + 87.
The problem should have read 'Prove that there exists atleast one power of 5 between 10^k and 10^k+1 where k is a Natural number.

Problem # k + 88

Show that the sum of any number of terms of the series
1/1*2, 1/2*3, 1/3*4, 1/4*5, 1/5*6,................ is less than 1.

Jai Ganesh · 2006-02-15 03:11:01

Problem # k + 89

In a bolt factory, machines A, B, and C manufacture respectively 25%, 35% and 40% of the total bolts. Of their output, 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random. If the bolt drawn is found to be defective, what is the probability that it is manufactured by machine B?

John E. Franklin · 2006-02-15 06:12:39

mathsyperson · 2006-02-15 09:10:32

ganesh wrote:

I made a mistake while posting problem # k + 87.
The problem should have read 'Prove that there exists atleast one power of 5 between 10^k and 10^k+1 where k is a Natural number.

Ah. That makes more sense. I thought there was a mistake somewhere because as it was, the proof was really obvious, so it didn't seem sensible.

I've got a vague idea for k+88, but I'll wait for it to mature a bit more before posting it.

irspow · 2006-02-15 11:45:49

Last edited by irspow (2006-02-15 11:49:14)

mathsyperson · 2006-02-15 12:53:51

I think my and irspow's methods are pretty much the same, just worded differently. We get the same answer, anyway.

irspow · 2006-02-15 13:04:42

Sorry, I didn't mean to jump in your spot. The wording of this problem is what had me scratching though. I wasn't sure if ganesh wanted the over-all probability of drawing a defective bolt from B or the probability that I (or should I say we?) calculated.

mathsyperson · 2006-02-15 13:54:23

Don't worry, you're in your own spot. Our spots just happen to look similar.
Anyway, I think we're right. Ganesh told us that the bolt was defective, and wanted to know tha probability of it being from B.

Jai Ganesh · 2006-02-16 00:21:55

Both mathsyperson and irspow are correct. Well done.

Problem # k + 90

Find the formula for sum of n terms of
(1*2*3), (2*3*4), (3*4*5), (4*5*6),................. Prove that the formula is correct by Mathematical Induction.

Ricky · 2006-02-16 05:03:04

k+90's formula is:

Anybody want to try their hand at an induction proof, or should I?

Jai Ganesh · 2006-02-16 16:32:01

Ricky is correct. I shall wait for someone to post a proof.

Jai Ganesh · 2006-02-16 16:51:11

Problem # k + 91

Find three consecutive terms in Geometric Progression such that their sum is 21 and the sum of their squares is 189.

mathsyperson · 2006-02-17 00:11:28

Jai Ganesh · 2006-02-17 00:36:45

mathsyperson's solution to problem # k + 91 is correct. I shall post the method if none of the others are able to give a proper solution with the steps.

Problem # k + 92

If x=9 + 4√5, find the value of √x - 1/√x.

Ricky · 2006-02-17 08:21:57

I have the induction proof written up on paper, but I'm a bit too lazy to type it all up now. Maybe later tonight.

irspow · 2006-02-17 11:17:34

Jai Ganesh · 2006-02-17 20:47:57

Ricky and irspow are correct!
Very%20Well%20Done.jpg

Jai Ganesh · 2006-02-17 21:35:43

Problem # k + 93

A right ciruclar cone is cut by two planes parallel to the base, trisecting the altitude. What is the ratio of the volumes of the three parts, the top, middle and the bottom respectively?

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