Math Is Fun Forum

mathster

Member

Registered: 2015-10-09

Posts: 15

Algebra problems.

Two trains travel directly toward each other. One of the trains travels at a rate of 12 km/h while the other travels at a rate of 20 km/h. When the trains are 72 km apart a conductor at the front of one of the trains releases the insane pigeon Hyde. Hyde flies first from the slower of the two trains to the faster train at which point Hyde doubles back toward the slower train. Hyde continues to fly back and forth between the trains as they approach, always at a constant speed of 48 km/h. Assuming the trains never change speed until they meet and magically stop, how many kilometers has Hyde flown when the trains meet?

The terms of a particular sequence are determined according to the following rules:
* If the value of a given term is an odd positive integer s, then the value of the following term is 3s - 9
* If the value of a given term is an even positive integer t, then the value of the following term is 2t - 7.
Suppose that the terms of the sequence alternate between two positive integers (a,b,a,b,...). What is the sum of the two positive integers?

Given positive integers

and

such that

and

, what is the smallest possible positive value for

?

Help would be very appreciated. Thanks!

Last edited by mathster (2015-10-18 08:04:47)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Algebra problems.

Hi;

Given positive integers

and

such that

and

, what is the smallest possible positive value for

?

I am getting 49 as the smallest.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Algebra problems.

hi mathster,

Q1.  The trains are approaching at a relative speed of (12 + 20) Kph and the distance travelled is 72Km.  So calculate the time until they meet and apply this to the pigeon's travel.

Q2.  Suppose that a is odd. Then

b = 3a-9 which will be even and so
a = 2b-7

You can solve for a and b from this.

I'll leave you to consider the case where a is even. 

Q3.  I did this:

A sketch http://www.mathsisfun.com/data/function … 12/(x-12)) shows that after x = 12 the graph has a minimum somewhere between x = 20 and 30.  You can use calculus to get the exact point.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathster

Member

Registered: 2015-10-09

Posts: 15

Re: Algebra problems.

Thanks so much guys! Question 3 was quite confusing, although I don't know calculus, apparently you can solve this with SFFT (Simon's Favorite Factoring Trick). Quite clever.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Algebra problems.

hi mathster,

"Simon's Favorite Factoring Trick".  I've not met this before.  Looked up on AoPS but still none the wizer.

Please would you post how you used it.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathster

Member

Registered: 2015-10-09

Posts: 15

Re: Algebra problems.

hi mathster,

"Simon's Favorite Factoring Trick".  I've not met this before.  Looked up on AoPS but still none the wizer.

Please would you post how you used it.

Bob

Multiplying

by

gives

.

Using the trick, it's factorable to

.

We would probably want to set

and

to

, but we remember that

.

The next closest factors are

and

.

This gives

and

. Therefore

and

. Finally we come to

.

Last edited by mathster (2015-10-19 04:14:11)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Algebra problems.

This is an Egyptian fraction so even Simon is not needed here.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Wangotango

Member

Registered: 2016-07-29

Posts: 6

Re: Algebra problems.

How do you finish question number 1?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Algebra problems.

hi Wangotango,

Two trains travel directly toward each other. One of the trains travels at a rate of 12 km/h while the other travels at a rate of 20 km/h. When the trains are 72 km apart a conductor at the front of one of the trains releases the insane pigeon Hyde. Hyde flies first from the slower of the two trains to the faster train at which point Hyde doubles back toward the slower train. Hyde continues to fly back and forth between the trains as they approach, always at a constant speed of 48 km/h. Assuming the trains never change speed until they meet and magically stop, how many kilometers has Hyde flown when the trains meet?

From the viewpoint of one of the trains, the other is 72Km away and approaching at (12+22) Km/h.  So using time = distance/speed you can calculate the time until the trains meet.  That is also the time that Hyde flies, so use distance = speed x time to calculate the distance Hyde has travelled.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Algebra problems.

(2) a+b=11

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}