Given a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5, then what is

Question

Given a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5, then what is the value of a + b + c + d?

A) -1
B) −(10/3)
C) 12
D) 6
E) -2

anshu2602 · Accepted Answer

a+1=a+b+c+d+5
b+2=a+b+c+d+5
c+3=a+b+c+d+5
d+4=a+b+c+d+5

Adding all the equations we get:
a+b+c+d+10=4(a+b+c+d)+20

Solving for (a+b+c+d) we get:

a+b+c+d=(-10/3)

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kysb · Answer

chetan2u ,  Bunuel ,  VeritasKarishma ,  Gladiator59 ,  generis

Please Help needed why its should be B not D

Regards

K.y.Shrenik Bimbsar

Gladiator59 · Answer

Hi  ShrenikBimbsar ,

When so many variables are present the first order of business should be to convert everything into a single variable. This is easily possible in this case as multiple equations are given.

It comes out that,
b = a-1 , c= a-2 & d= a-3

On substituting the above and converting the everything into a it comes out that a = 2/3

And a+b+c+d = 4a - 6

Hence the answer is -10/3.

Hope it helps.

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kysb · Answer

Thanks

Regards 
K.y.Shrenik Bimbsar

kysb · Answer

Thanks

Regards 
K.y.Shrenik Bimbsar

generis · Answer

You do not have to solve this problem all the way.

(1)  Express three of the variables in terms of one other. 
Use  a  or  d  for the sake of visual simplicity; both are on one end of the first string of equalitites.

d+4=c+3 
 c=(d+1)

d+4=b+2 
 b=(d+2)

d+4=a+1 
 a=(d+3)

(2)  Solve for  d  using the final equality.
Substitute for  a, b,  and  c 
 (d+4)=a+b+c+d+5 
 d+4=d+3+d+2+d+1+d+5 
 d+4=4d+11 
 -7=3d 
 3d=-7 
 d=-\frac{7}{3}

Stop . The answer must be a fraction with a denominator of 3. 
If we add an integer to a fraction, the answer is a fraction with the same denominator.

Only one option has a denominator of 3.

The answer is (B).

Why  stop?

a, b,  and  c  all have values that equal 
 d (a fraction) + an integer
 
Test: If we add  1  to  -\frac{7}{3} , then  1  must be changed to  \frac{3}{3}

c=(d+1) 
 c=(-\frac{7}{3})+(\frac{3}{3}) 
 c=-\frac{4}{3} 
 c  = another fraction with a denominator of 3.
 a  and  b  will also be fractions with a denominator of 3. 
-- The sum of all four variables will be a fraction with a denominator of 3.
-- Only one option has a denominator of 3.
 The answer is (B).

(3)  if you are not sure, do all of the arithmetic

Find each value

a=(d+3) 
 a=(-\frac{7}{3})+(\frac{9}{3}) 
 a=\frac{2}{3}

b=(d+2) 
 b=(-\frac{7}{3})+(\frac{6}{3}) 
 b=-\frac{1}{3}

From above,  c=-\frac{4}{3}

a+b+c+d=

\frac{2}{3}+((-\frac{1}{3})+(-\frac{4}{3})+(-\frac{7}{3}))=

\frac{2}{3}+(-\frac{12}{3})=(\frac{2}{3}-\frac{12}{3})=-\frac{10}{3}

Answer B

KarishmaB · Answer

So this is what I observed:

a + 1 = b + 2 = c + 3 = d + 4
This means the values of the variables are reducing by 1 as we move to the right. So b is 1 less than a. c is 1 less than b and so on. 
So a, b, c and d could be something like 4, 3, 2, 1 respectively.

d + 4 = a + b + c + d + 5
a + b + c + d = d - 1

So (a + b + c + d) is further 1 less than d so taking the above example, it will go down one step further and be 0. 
Hence, a, b, c, d, a+b+c+d are consecutive numbers in decreasing order.

Now this is not possible if a, b, c and d are all positive numbers since their sum will be greater than individual numbers. 
So a+b+c+d cannot be 6 or 12.

If a+b+c+d = -1, then d = 0, c = 1, b = 2 and a = 3 
Does not satisfy.

If a+b+c+d = -2, then d = -1, c = 0, b = 1 and a = 2 
Does not satisfy.

If a+b+c+d = -(10/3), then d = -7/3, c = -4/3, b = -1/3 and a = 2/3
Satisfies.

Answer (B)

zEnkIz · Answer

a+1 = b+2 
=> b = a-1 ——— (1)
a+1 = c+3 
=> c = a-2 ——— (2)
a+1 = d+4 
=> d = a-3 ——— (3)

a+1 = a+b+c+d+5 
=> b+c+d = -4 ——— (4)

(1)+(2)+(3) 
=> b+c+d = 3a-6 ——— (5)

From (4) and (5)
3a-6 = -4 
=> a = 2/3 ——— (6)

From (4) and (6)
a+b+c+d = 2/3 - 4
=> -10/3

Answer (B)

m1033512 · Answer

I have attached the solution.

All are equal, let us assume that all are equal to K (any constant)

we can find the value of K by solving equation in the picture

amd then find the value of required expression

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TRINA6 · Answer

Hi "VeritasKarishma"

I have taken another approach, what will be your suggestion?

Thanks in advance.

hiranmay · Answer

Given a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5, then what is the value of a + b + c + d?

A) -1
B) −(10/3) -->  correct 
C) 12
D) 6
E) -2

Solution :
a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5 = k i.e let say x = a + b + c + d,   x+5 =k ---(i)   
=>k + k + k + k + k = 5k
(a + 1)+(b + 2)+(c + 3)+(d + 4)+(a + b + c + d + 5) = 5k = 5*(x+5) from (i)
=> x + 1+2+3+4+ x + 5 = 5x+25
=> 2x+15 = 5x+25
=> 3x = -10
=> x = -10/3

CrackverbalGMAT · Answer

Given that a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5

Adding the first 4 terms = 4 times the last term

(a + 1) + (b + 2) + (c + 3) + (d + 4) = 4 * (a + b + c + d + 5)

(a + b + c + d) + 10 = 4 * (a + b + c + d) + 20

3 * (a + b + c + d) = -10

(a + b + c + d) = -10/3

Option B

Arun Kumar

MathRevolution · Answer

a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5

=> a + 1 = a + b + c + d + 5
=> b + 2 = a + b + c + d + 5
=> c + 3 = a + b + c + d + 5
=> d + 4 = a + b + c + d + 5

Adding all, we get

a + b + c + d + 10 = 4 (a + b + c + d + 5)

=> a + b + c + d = 4((a + b + c + d ) + 20 - 10

=>a + b + c + d = 4((a + b + c + d ) + 20 - 10

=>a + b + c + d = 4((a + b + c + d ) + 10

=> -10 = 4 (a + b+ c+ d) - (a + b+ c+ d)

=> -10 = 3 (a + b+ c+ d)

=> a + b + c + d =  \frac{-10}{3}

Answer B

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