other approach?

[nimshad20]

for the question on guide 4, pg 123, question 10.

in the solution given, it says "the easiest approach is to pick numbers". What is the other approach possible?

[tommywallach]

Hey Nimshad,

Apologies for this, but we use a digital version of the book that doesn't paginate, so I can't find questions by page number, only by book, chapter, and section of chapter. Or you could simply post the question itself. Also, in future, please title your posts on the forum with all this information, so that people can search by chapter/question (there would be no way to know what you were asking about with the title "other approach"). Thanks!

-t

[nimshad20]

guide 4,chapter 8,pg 135,problem set (the problem set just before the start of the 9th chapter), question 10

the question

the question has a line drawn with 3 points (q,s,r) marked. point s is in the middle, point q to the left and point r to the right
given information: s is the midpoint of qr
r = -2q
quantity A quantity B
s 0

[n00bpron00bpron00b]

Given:

# q,s,r lie on a line | s midpoint | q to the left of it | r to the right

# since "s" is the midpoint ; the value of "s" can be derived by midpoint formula i.e (q+r)/2

# for all possible range of values ; q,s,r will follow the condition
q<s<r (as values on a number line increase from left to right and decrease from right to left)

# Also given => r=-2q

Lets consider few values to see how the values interact -

Using the given condition r=-2q

a) If q=0 ; r=0 ; and s=0 (using midpoint formula) =>this case is invalid

b) If q=1 ; r=-2 => again this case is invalid

So clearly, q has to be negative

c) If q = -1 r=2 and s = 0.5

Okay, now we clearly know q will be negative, r will be positive. Now let's try to prove "D" (Q.C)

If q = -0.1 r = 0.2 s = 0.05
Minimum value for s is 0.05

q = -0.2 r = 0.4 s = 0.1

q = -1 r = 2 s = 0.5

q = -2 r = 4 s = 1

So the value of s just keeps increasing to higher positive values
and satisfies the condition S>0

S can never be 0 or any value less than 0 but all values greater than 0

Ans: A

P.S - I believe for most number line based questions testing values/moving values around is a great way to get started.

[tommywallach]

Great post from Noob, as usual.

We could also use the algebra to substitute:

s = (q + r)/2

r = -2q

Substitute:

s = (q + -2q)/2

s = -q/2

We should know pretty quickly from the original equation that q and r must have opposite signs, and because of the picture, q must be negative, so this equation now tells us that s will always be positive.

-t

[kpkanupriyakhmi]

why should signs of q and r be opposite?

[tommywallach]

Hey KP,

Because it says:

r=-2q

This means that if one is negative, the other one won't be, and vice-versa. You can prove it with numbers if it isn't immediately apparent.

If q = 2
Then r = -4

If q = -3
Then r = 6

Of course, only the second example is possible, because q is shown as being to the LEFT of r on the number line.