If a, b, c, and d are numbers on the number line shown and if

Question

https://gmatclub.com/forum/download/file.php?id=30570 If a, b, c, and d are numbers on the number line shown and if the tick marks are equally spaced, what is the value of a +c ? (1) a + b = -8 (2) a + d = 0 Source: GMAT OG 2016 Quant Review teste.jpg

FacelessMan · Accepted Answer

Imagine these points represent elements on an arithmetic progression, and the equal spacing between them is k. Then, b = a+k, c = a+2k and d = a+3k. We need to calculate a+c = a+a+2k = 2a + 2k. So we need to know the values of a and k.
S1 : a+b = a+ a+k => 2a+k = -8. Not sufficient.
S2 : a+d = a + a + 3k => 2a +3k = 0. Not sufficient.

Combining both statements, we have 2 equations and 2 variables. Thus we can solve for the values of a and k. Thus C.

ScottTargetTestPrep · Answer

We need to determine the value of a + c. Notice that b is the midpoint of a and c, thus b = 
(a + c)/2. So if we know the value of b, then we can determine a + c.

Statement One Alone:

a + b = -8

We see that we don’t have sufficient information to answer the question. For example, if a = -5 and b = -3, then c = -1 and a + c = -6. However, if a = -6 and b = -2, then c = 2 and a + c = -4. Statement one alone is not sufficient.

Statement Two Alone:

a + d = 0

Since the numbers are evenly spaced, we see that if a + d = 0, then b + c = 0. However, we still can’t determine the value of a + c. For example, if a = -3 and d = 3, then b = -1 and c = 1 and a + c = -2. However, if a = -6 and d = 6, then b = -2 and c = 2 and a + c = -4. Statement two alone is not sufficient.

Statements One and Two Together:

Recall that b = (a + c)/2. Therefore, from statement one, if a + b = -8, then

a + (a + c)/2 = -8

a + a/2 + c/2 = -8

2a + a + c = -16

3a + c = -16
 
Similarly, from statement two, if b + c = 0, then

(a + c)/2 + c = 0

a/2 + c/2 + c = 0

a + c + 2c = 0

a + 3c = 0

Adding 3a + c = -16 and a + 3c = 0, we have

4a + 4c = -16

a + c = -4

Answer: C

IanStewart · Answer

Generating any scenarios at all will establish that neither statement is sufficient alone. Using both Statements, if we just add the two equations, we get

2a + b + d = -8
a + (b+d)/2 = -4

and if the tick marks are equally spaced, (b+d)/2, the midpoint of b and d, must equal c. So a + c = -4.

helloksin · Answer

I approached by testing values-

Question- a+c=? This means, we need to know the value of a and the value of the equi-distant space between each of the points, or we need to know the value of a and c separately.

S1: It is only given that the sum of the  values  of a and b result to -8. So,
 Case1 : a  -ve and  b  -ve
If a=-7 and b=-1, then a+b = -8 ,  distance = 6 and c= 5

Case2 : a  -ve and  b  +ve
If a=-10 and b=2, then a+b = -8 ,  distance = 12 and c= 14

Two different values for c (and hence a+c) , thus Statement 1 INSUFFICIENT.

S2: a+d = 0,
Just tells us that d=-a, or that 0 is midpoint between b and c because of the equi-distances between each pair of points. 
 |a+b+ b/2| = |c/2 + c+d|  ]
So a and b are negative, while c and d are positive.
However, this doesn't provide any information about actual value of  c  or the value of equal distance. Hence, INSUFFICIENT.

Combining (1) and (2), 
From statement 2, we know that a and b MUST be to the left of zero (negative) and 0 should be midpoint between b and c. From Statement 1, the sum of a and b should be -8.
Only a=-6 and b=-2 (  distance = 4, c=2) satisfy these conditions. (ie. if a=-7 and b=-1 then c=5, but we know from S2, that 0 has to be mid-point of b and c, so these are not correct values of a and b).
Since the values of a and b are locked, so is the value of c locked.

Hence, we can answer the question with certainty, and the statements together are SUFFICIENT.
Hence, answer= C.

GMATSkilled · Answer

This is how it's spaced

a b c d

Let a = x and the spacing between a & b = y. Since, they are equally spaced, the spacing between b & c and c & d is also y.

So lets convert this in terms of x & y

a = x

b = x + y

c = x + 2y

d = x + 3y

We want a + c => x + (x + 2y) = 2x + 2y {we basically need the values for x & y}

Now, lets looks at each statement:

A -> a + b = -8
By using the above expression we get
x + (x + y) = -8 ........... (1)

This is clearly insufficient as we have both x & y in the expression

b -> a + d = 0
By using the above expression we get
x + (x + 3y) = 0 ............ (2)

This is clearly insufficient as we have x in terms of y.

On combining (1) & (2)
we get the values of x & y.

Hence  C  is the answer

nick1816 · Answer

b=a+x
c=a+2x
d=a+3x

We need to find a+a+2x=2(a+x)

Statement1 2a+x=-8, insufficient
Statement2 a+3x=-a
2a+3x=0
Insufficient

Combining 2 statements 
Add both equations
4a+4x=-8
2(a+x)=-4
Sufficient

Posted from my mobile device

Vegita · Answer

Hey  ScottTargetTestPrep

Since we have two variables and two equations after combining both the statements 1 and 2, should we even bother to go this route?

ScottTargetTestPrep · Answer

Solution:

We have to be very careful about the solutions of linear equation systems, especially when we are solving data sufficiency questions. Here, we actually start with three variables and three equations; but we obtain two variables and two equations after making the substitution b = (a + c)/2. A system with two variables and two equations can have infinitely many solutions (for instance, the system x + y = 1 and 2x + 2y = 2) or have no solutions (for instance, the system x + y = 1 and 2x + 2y = 3). That is why for certain DS questions, it is a good idea to actually make sure that the information given to us is indeed sufficient.

ScottTargetTestPrep · Answer

https://gmatclub.com/forum/download/file.php?id=30570 
If a, b, c, and d are numbers on the number line shown and if the tick marks are equally spaced, what is the value of a +c ?

(1) a + b = -8
(2) a + d = 0

Solution:

We have to be very careful about the solutions of linear equation systems, especially when we are solving data sufficiency questions. Here, we actually start with three variables and three equations; but we obtain two variables and two equations after making the substitution b = (a + c)/2. A system with two variables and two equations can have infinitely many solutions (for instance, the system x + y = 1 and 2x + 2y = 2) or have no solutions (for instance, the system x + y = 1 and 2x + 2y = 3). That is why for certain DS questions, it is a good idea to actually make sure that the information given to us is indeed sufficient.

KarishmaB · Answer

The numbers are evenly spaced. We need the value of (a + c). The first thing that crosses my mind is that if b = 0 on the number line, then a + c = 0 because they will be equidistant from b on opposite sides. In any case, we move on to the stmnts. (1) a + b = -8 We don't know where 0 is on the number line. Both a and b could be negative such as a = -5, b = -3 etc or a could be negative while b positive such as a = -10 and b = 2 etc. In any case, we have no idea about what c is. (2) a + d = 0 Our previous idea comes into play here. If a + d = 0, it means 0 is at their mid point which will also be the mid point of b and c. So b and c are equidistant from 0. Then, if c = x, b = -x, d = 3x and a = -3x. So a + c = - 3x + x = -2x But we don't know x. Using both statements together, a + b = -3x - x = -8 So x = 2 Then a + c = -2*2 = -4 Similar questions to practice: https://gmatclub.com/forum/of-the-four- ... l#p1052116 https://gmatclub.com/forum/the-number-l ... ml#p915379 https://gmatclub.com/forum/on-the-numbe ... ml#p812434

rickygo01 · Answer

Let's find the values of a,b,c,d in relation to a ; as we would normally do when we look at sequences evenly spaced.

(x is the space between them)
a=a
b=a+x 
c=a+2x 
d=a+3x

We are looking for a+c which is = to a+a+2x= 2a + 2x. So if we manage to find what's the value for 2a+2x we reached a conclusion

Statement 1 says a+b=-8 which is equal to say a+a+x=-8 ----> 2a+x= -8. not sufficient because we want to find 2a+2x

Statement 2 says a+d=0 which is equal to say a+a+3x=0-----> 2a+3x=0 not sufficient because we want to find 2a+2x

If we put them together we have that 2x=-8 and as a consequence by substitution we are able to find a.

So we will be able to obtain the value of 2x+2a which is what the exercise was asking for

answer C

AnumFatima · Answer

Hi  KarishmaB ,
Can we approach it the following way as well?

After we have established that (1) is insufficient alone and (2) is also insufficient alone, we combine (1) and (2) as below:
Subtract the equations given in (2) and (1)

a+d = 0 
a+b= -8
----------
d-b=8

So the distance between d and b on the number line is 8. Since all the points are equidistant, c is midway between b and d. And since bd = 8, cd = 8/2=4. Also all the other segments are equal to 4 (ab=bc=cd=4).

From (2) we know that the midpoint of bc is 0. And c is to the right of 0. So c=0+2 = 2 
and a is to the left of 0. So a= 0-2-4 = -6 
a+c=-6+2=-4

KarishmaB · Answer

Sure though you are essentially using the number line concept only. If you try to visualise it on the line, you will arrive at all these conclusions without the need for explicit calculations.

Check out this video that discusses how to use number line on GMAT: youtu.be/3gxVx3Y9xJA[/quote]

Thank you KarishmaB! The clarification helps a lot

AnumFatima · Answer

Sure though you are essentially using the number line concept only. If you try to visualise it on the line, you will arrive at all these conclusions without the need for explicit calculations.

Check out this video that discusses how to use number line on GMAT: youtu.be/3gxVx3Y9xJA

Thank you  KarishmaB ! The clarification helps a lot

Bunuel · Answer

Problem Solving:

https://gmatclub.com/forum/on-the-numbe ... 16629.html GMAT Prep (Focus)
https://gmatclub.com/forum/on-a-laborat ... 24877.html GMAT Prep (Focus)
https://gmatclub.com/forum/the-tick-mar ... 25298.html GMAT Prep (Focus)
https://gmatclub.com/forum/if-the-tick- ... 08185.html GMAT Prep
https://gmatclub.com/forum/on-the-numbe ... 08085.html GMAT Prep
https://gmatclub.com/forum/on-the-numbe ... 04204.html Official Guide
https://gmatclub.com/forum/a-straight-p ... 45031.html Official Guide
https://gmatclub.com/forum/on-the-numbe ... 98106.html GMAT Club Tests
https://gmatclub.com/forum/if-the-tick- ... 63234.html
https://gmatclub.com/forum/in-the-numbe ... 54299.html
https://gmatclub.com/forum/on-the-numbe ... 91631.html
https://gmatclub.com/forum/marks-are-eq ... 44534.html
https://gmatclub.com/forum/kim-finds-a- ... 40038.html
https://gmatclub.com/forum/on-the-numbe ... 26136.html

Data Sufficiency:
https://gmatclub.com/forum/if-the-succe ... 44053.html Official Guide
https://gmatclub.com/forum/if-a-b-c-and ... 20097.html Official Guide
https://gmatclub.com/forum/if-the-tick- ... 04050.html
https://gmatclub.com/forum/if-the-tick- ... 49455.html
https://gmatclub.com/forum/on-the-numbe ... 72416.html

Hope it helps.

SMQazi · Answer

not certain my method was correct, but I simply chose statement 2, which says, a+d=0, and since they're equally spaced, a+b+c+d=0, now plug a+b = -8, this gives us c+d=8, so combining (1) and (2) gives us a solution.

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