Math Problem, help me understand

Factual Questions
1

The problem is:

A number line has points A and B with coordinates -4 and 6 respectively. Find both possible coordinates for point P if PB = 5 AP.

This is what I think I am being asked: I am given two points on a number line. I am supposed to find two additional points. I can get these points by solving:

§(6) = (5)(-4)§.

I end up with §(6) =§(-20)
6P=-20P
26P = 0

So that wasn’t right.

Then I tried §(6) =(-20)§ and divided both sides by P giving me
1 = -20/6, which it doesn’t.

What am I not understanding here?

2

You had it right the first time. If you just substitute in the values of A and B, you’ll get 6P = -20P, which lets you conclude that 26P = 0. Therefore, P must be 0. Plug that back into the original equation and you’ll see it works. In fact, P = 0 is the only solution.

ETA: The reason that dividing both sides by P doesn’t work is that P = 0, and you can’t divide by 0.

3

One solution will be a point P which is located between A and B. In this case, PA + PB = AB

The other solution will be a point P which is located on the other side of A from B. In that case, PA + AB = PB

Solve each for the distance PA, and that will tell you where P must \be in each case.

Please tell me this isn’t homework? With your join date, I assume not…

4

I think this is incorrect. The question asks for two coordinates. This can’t be given if there is only one correct answer.

5

The question’s incorrect. P = 0 is the only possible solution.

PA + PB = AB is equivalent to -4P + 6P = -24, which gives P = -12. Not only is that not between A and B, but PB = -72 and 5AP = -240.

ETA: The reason that I know there’s only one solution is that (B- 5A)P is a first degree polynomial in P, and will only have one root.

6

The answers are

P = -6.5
P = -2.3333 (recurring)

7

If the length of PB is 5 times greater than the length of AP, then there are a couple ways this can happen:

One is for point P to be 1/6 of the way between point A and B. The length of AP will be 1/6 of the length of AB, and the length of PB will be 5/6 of the length of AB.

The second way is for point P to lie somewhere to the left of point A (i.e. more negative). If the length of AP in this case is 1/4 of the length of AB, then the length of PB should be 5 times greater.

But IMHO, this is easier to figure out using simple equations. For the solution between A and B:

B - P = 5 * (P - A)
6 - P = 5 * (P + 4)
6 - P = 5P + 20
-6P = 14
P = -7/3

Length of AP = -7/3 + 4 = 5/3
Length of PB = 6 + 7/3 = 25/3

For the solution left of point A:

B - P = 5 * (A - P)
6 - P = 5 * (-4 - P)
6 - P = -20 - 5P
4P = -26
P = -13/2 = -6.5

Length of AP = 2.5
Length of PB = 12.5

8 

-----+-------+-----------------+-------
     A       P                 B

For the distance between P and B to be 5 times the distance between A and P,
| P – 6 | = 5 * | P – (–4) |

For the absolute values to be equal, the expressions inside must be either equal to each other, or opposites of each other:
P – 6 = 5(P + 4) or P – 6 = –5(P + 4)

When I solve for P, I get P = –7/3 or P = –13/2.

(The first of these points is between A and B; the second is to the left of A.)

9

A coworker brought this into work. I am fairly sure it is his child’s homework. 3 of us played around with for much too long and came up blank. It is driving me crazy.

Following your advice I come up with -10 and -2.

10

I’d take ‘AB’ to mean ‘distance from A to B’, not multiplication of the respective coordinates, in this case; however, the notation is somewhat ambiguous.

11

You seem to be taking AP and BP as multiplications? Presumably, they are meant to indicate distances instead; thus, there’s an absolute value involved here, as Thudlow Boink pointed out.

12

As for why I (and presumably Half Man Half Wit) am going with the distance interpretation, rather than the multiplication interpretation, I don’t think anyone would bother phrasing things as “coordinates of points on a number line” if all they meant to talk about was numbers in themselves and their multiplication, with no geometric interpretation.

13

My working is:

First, imagine that P is before A. Call the distance PA; X
<----------5X------------>

P<—X—>A<-----10—>B
In the crude diagram above we see that A is 10 away from B (6 - -4), and X away from P. Also P is 5X away from B.

So

5X = X + 10
4X = 10
X = 2.5

So P is 2.5 to the left of A,
therefore P = -6.5

Also we can check our answer, -6.5 must be 5 times further from B than A and indeed (6 - -6.5) = 12.5 = 5 * 2.5
Now what about if P is to the right of A?

A<----X---->P<-----------5X---------->B
<----------------------10---------------->

This time X + 5X = 10

So 6X = 10
X = 10/6
X = 1.666 (recurring)

So X is 1.666 to the right of A
P = -2.333 (recurring)

Again, we can check our answer:

6 - -2.333
= 8.33333

8.333 / 1.666 = approx 5

On edit, I see lots of people posted their working before me, and used fractions to avoid the recurring decimal.
Oh well.

…I posted the first correct solution :slight_smile:

14

Ah. In my defense, the OP was treating the concatenation as a product as well and I assumed that was the sort of problem involved.

15

Eh, ignore this post

16

How?
Mijin and others have given the correct solutions.

17

Your posts are very dynamic things in their first few minutes of existence… :wink:

18

I know. I was treating AB and PB as multiplication, not distances. I understand the answer now.

19

Yeah, if I recall correctly, though I’d been lurking for years, I didn’t end up joining the SDMB until right after the edit ability was introduced. Clearly, I couldn’t have managed to post otherwise… :slight_smile: