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Comment on Operations with Signed Numbers - Part I
love this
Hi Brent,
Can you explain this solution. I didn't get it.
https://gmatclub.com/forum/what-is-130041.html#p1068551
Solution link: https:/
Solution link: https://gmatclub.com/forum/what-is-130041.html#p1068551
To demonstrate the strategy, let's just evaluate a portion of the question.
Let's evaluate: 10 - 8 + 6 - 4 + 2 - 0
Notice that, if we add some brackets, we can reduce the work to just adding a bunch of 2's.
We get: (10 - 8) + (6 - 4) + (2 - 0)
This evaluates to be the same as evaluating 10 - 8 + 6 - 4 + 2 - 0. However, when we use brackets, the question becomes much easier.
(10 - 8) + (6 - 4) + (2 - 0) = 2 + 2 + 2 = 6
Alternatively, we can also make the following observation:
10 - 8 = 2
10 - 8 + 6 - 4 = 4
10 - 8 + 6 - 4 + 2 - 0 = 6
10 - 8 + 6 - 4 + 2 - 0 + 2 - 4 = 8
Notice that, when we add two more values to the expression, the sum increases by 2. So, we need only determine how many pairs of values are in the whole expression.
Please let me know if you need any additional clarification.
Cheers,
Brent
What is 10 - 8 + 6 - 4 + ...
A. 8
B. 10
C. 12
D. 14
E. 16
Explanation for this not clear to me how they come to answer 16.Above explanation is not clear to me I apologies for that. Please give me some additional explanation.
Here you go:
The sequence is 10−8+6−4+2−0+(−2)−(−4)+(−6)−(−8)+(−10)−(−12)+(−14)−(−16)+(−18)−(−20)
10−8+6−4+2−0+(−2)−(−4)+(−6)−(−8)+(−10)−(−12)+(−14)−(−16)+(−18)−(−20)
.
Notice that the odd numbered terms (1st, 3rd, 5th...) form arithmetic progression with common difference of -4 and the even numbered terms (2nd, 4th...) form arithmetic progression with common difference of 4:
The sum of the odd numbered terms is 10+6+2+(−2)+(−6)+(−10)+(−14)+(−18)=10+6+2−2−6−10−14−18=−32
10+6+2+(−2)+(−6)+(−10)+(−14)+(−18)=10+6+2−2−6−10−14−18=−32
;
The sum of the even numbered terms is −8−4−0−(−4)−(−8)−(−12)−(−16)−(−20)=−8−4−0+4+8+12+16+20=48
−8−4−0−(−4)−(−8)−(−12)−(−16)−(−20)=−8−4−0+4+8+12+16+20=48
;
Their sum is −32+48=16
−32+48=16
.
Though I wouldn't recommend to solve this question this way. It's better if you notice that we have 8 pairs:
10-8=2;
6-4=2;
2-0=2;
(-2)-(-4)=2;
(-6)-(-8)=2;
(-10)-(-12)=2;
(-14)-(-16)=2;
(-18)-(-20)=2;
So, the sum of each pair is 2, which makes the whole sum equal to 8*2=16.
Looks like you answered your
Looks like you answered your own question :-)
If the sum of two integers is
(A) both integers are even
(B) both integers are odd
(C) both integers are positive
(D) if one integer is negative, the other is positive
(E) if one integer is positive, the other is negative
Hi toludayo,
Hi toludayo,
Question link: https://gmatclub.com/forum/if-the-sum-of-two-integers-is-6-then-it-must-...
I've already provided two different approaches to this question.
Approach #1: https://gmatclub.com/forum/if-the-sum-of-two-integers-is-6-then-it-must-...
Approach #2: https://gmatclub.com/forum/if-the-sum-of-two-integers-is-6-then-it-must-...
Once you review those two approaches, please let me know if you have any questions.
Cheers,
Brent
Hi Brent,
With reference to the following problem: https://gmatclub.com/forum/is-the-number-x-positive-160021.html
Looking at Statement 1: On the number line, 0 is closer to x – 1 than to x
For Case #1, when x < 1 (e.g. 0.3), wouldn't x-1 (i.e. -0.7) be further away from 0 than x? Which means that even if x is positive, statement 1 is not true and therefore insufficient to answer the target question?
Thanks
Question link: https:/
Question link: https://gmatclub.com/forum/is-the-number-x-positive-160021.html
I believe you're referring to my solution here https://gmatclub.com/forum/is-the-number-x-positive-160021.html#p1936544
Be careful, x = 0.3 does not meet the condition in statement 1 (0 is closer to x–1 than to x.)
If x = 0.3, then x-1 = -0.7
The distance from 0 to 0.3 (aka x) is 0.3
And distance from 0 to -0.7 (aka x - 1)is 0.7
In other words, 0 is NOT closer to x-1 than to x
So, we cannot use the example x = 0.3
I also want to point out what you say here: "...even if x is positive, statement 1 is not true"
I think you may be thinking of this DS question backwards.
Each statement is 100% true and we're trying to determine whether we can definitively answer the target question (is x positive?)
From what you wrote, it seems like you are assuming that x is positive and then checking to see whether statement 1 is true.
This is not how DS questions work.
Does that help?
Cheers,
Brent
https://gmatclub.com/forum/if
ques.
If the sum of two integers is 6, then it must be true that
(A) both integers are even
(B) both integers are odd
(C) both integers are positive
(D) if one integer is negative, the other is positive
(E) if one integer is positive, the other is negative
I am unable to make difference from the below choices, why answer D is right.
(D) if one integer is negative, the other is positive : Can't think of a counterexample so KEEP D for now
(E) if one integer is positive, the other is negative : it could be the case that the numbers are 1 and 5. ELIMINATE E
Question link: https:/
Question link: https://gmatclub.com/forum/if-the-sum-of-two-integers-is-6-then-it-must-...
Both are conditional statements in the form "If X then Y"
Some students instinctively feel that, if the statement "If X then Y" is true then the REVERSED statement "If Y then X" must also be true.
These same students will conclude that D and E are stating the same thing.
To see why this conclusion is incorrect, consider the following conditional statement:
If an animal is a pig, then that animal must have ears.
If we REVERSE this statement, we get:
In an animal has ears, then that animal must be a pig.
Hmmm. In this case, the conditional statement does not go in both directions.
Answer choice E tells us that, if we are it's the case that one value is positive, the other value must be negative.
This is not true.
For example, if one number is 2 (positive), then must it be the case that the other value is negative?
No.
If one number is 2, the other number must be 4 (also positive)
Now check answer choice D.
If one integer is negative then, in order to get a sum of 6, the other number MUST be positive (since it's impossible for two negative numbers to have a sum of 6)
Does that help?
Cheers,
Brent
Hi Brent,
I am stuck with this question, even though I understand the topic well.
Question: https://gmatclub.com/forum/of-the-four-numbers-represented-on-the-number-line-above-is-128398.html
Of the four numbers represented on the number line above, is r closest to zero?
___________q____________r_____________s__________t
(1) q = –s
(2) –t < q
Statement 1:
___________ -S ____________r_____________s__________t
We can see that there is an equal distance between s and -s, so r is the closest to zero. No argument about that.
Statement 2:
–t < q means that T is negative and Q is positive if Q would be negative than T lets say is -1 and Q is -2 but it clearly states that Q is non negative, right?
So in this case:
____ -t _______q____________r_____________s__________t
We can see that zero is between -t and q, therefore q is the closest to zero. Why the answer is A? What am I missing here?
Question link: https:/
Question link: https://gmatclub.com/forum/of-the-four-numbers-represented-on-the-number...
For statement 2, we can't conclude that t is negative.
Also, there's nothing in the given information to conclude that "Q is non negative"
For example, it could be the case that q = -1, r = 0, s = 1 and t = 2
This means -t = -2.
So, if q = -1, we can see that -t < q (since -2 < -1)
Does that help?
Here's my full solution: https://gmatclub.com/forum/of-the-four-numbers-represented-on-the-number...
Cheers,
Brent
Hi Brent,
Thank you for your reply. If we can't conclude that t is negative, why they put minus in front of t (-t)?
Placing a negative symbol in
Placing a negative symbol in front of t is the same as multiplying t by -1.
That is, -t = (-1)(t)
This means, we can't make any conclusions about whether -t is positive or negative.
If t = 2, then -t = -2
If t = -3, then -t = -(-3) = 3
If t = 0, then -t = -0 = 0
Keep in mind that the same applies when we don't have a negative symbol.
Take x, for example.
Even though there's no negative symbol in front of x, we can't make any conclusions about whether x is positive or negative.
That's not help?
Thank you. Got it.
Hi Brent,
Could you please clarify something for me?
Question 1: https://gmatclub.com/forum/if-1-y-0-and-z-2-then-which-of-the-following-cannot-be-the-valu-232075.html
If 1≥ y ≥0 and z ≤ -2 , then which of the following cannot be the value of y - z?
A. 3
B. 2.75
C. 2.5
D. 2
E. 1.5
____________________
Possible values of y
0, 0.5, 0.7…., 1
Possible values of z
-2, -1.5, -1, -0.9…..
Correct? Why do you consider only z=-2?
In another solution I saw this approach: y min – z max= 0−(−2)= Hence 2 is the min value that our difference y - z can take. So, how come -2 is the maximum value of z if lets say -0.92 is the largest on the number line than -2?
Question link: https:/
Question link: https://gmatclub.com/forum/if-1-y-0-and-z-2-then-which-of-the-following-...
I chose to test z = -2, since it's an extreme value for z, and I could see that I could use that value to start eliminating answer choices right away.
Fortunately, I was able to use that same z-value to eliminate four of the five answers choices.
Having said all of that, this approach can be quite time-consuming.
Here's another solution that uses a useful property of inequalities: https://gmatclub.com/forum/if-1-y-0-and-z-2-then-which-of-the-following-...
Does that help?
Cheers,
Brent
Great technique. Thank you.
Pls explain to me where i am
I understand the question to mean 4 was chosen in every case first before 2 other dice were rolled. Which leaves the 2 other dice to sum up to 10 in statement 1 and 11 in statement 2.
Please explain this question
Question link: https:/
Question link: https://gmatclub.com/forum/three-dice-each-with-faces-numbered-1-through...
Three dice, each with faces numbered 1 through 6, were tossed onto a game board. If one of the dice turned up 4, what was the sum of the numbers that turned up on all three dice?
(1) The sum of two of the numbers that turned up was 10.
(2) The sum of two of the numbers that turned up was 11.
You're correct to say that one of the dice yielded a 4.
However, when we get to Statement 1, there's nothing to suggest that the two numbers with a sum of 10 do not include the 4 from the given information.
IF Statement 1 said "The sum of the OTHER two numbers was 10," then we could make that conclusion.
Does that help?
This totally helps. Thank you
Please clarify this for me
https://gmatclub.com/forum/is-the-number-x-positive-160021.html#p1936544
For statement 1, Case 1, I was wondering why x-1 is on the left side to zero, zero is in the middle and x is on the right side. Is this for cases where we have the value of x as 0? so when we substitute we have -1,0 and 1. Please help clarify as inputing any other value for x makes the number line invalid
Solution link: https:/
Solution link: https://gmatclub.com/forum/is-the-number-x-positive-160021.html#p1936544
-----------------------
Is the number x positive?
(1) On the number line, 0 is closer to x–1 than to x.
(2) On the number line, 0 is closer to x than to x+1.
------------------------
Before we even examine statement 1, we already know that x-1 is less than x.
In other words, on the number line, x-1 is always to the left of x.
The question then becomes, "Where does 0 fit into all of this?"
The three different cases mentioned are all POSSIBLE scenarios.
case i: 0 is BETWEEN x-1 and x
case ii: 0 is TO THE RIGHT OF x
case iii: 0 is TO THE LEFT OF x-1
For case i, it COULD be the case that x = 0.6 and x-1 = -0.4
This would mean that 0 is BETWEEN x-1 and x
For case ii, it COULD be the case that x = -1 and x-1 = -2
This would mean that 0 is TO THE RIGHT OF x
For case iii, it COULD be the case that x = 2 and x-1 = 1
This would mean that 0 is TO THE LEFT OF x
Does that help?
Thank you
Hi Brent.
Good Morning.
In Question - https://gmatclub.com/forum/is-the-number-x-positive-160021.html,
I am unable to view the images in your answer explanation, so unable to understand the answer. Can you please send the link to the images?
Question link: https:/
Question link: https://gmatclub.com/forum/is-the-number-x-positive-160021.html
I took 2 screenshots of the entire solution:
- https://imgur.com/wcMs519
- https://imgur.com/73JnNb3