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# If 3a – 2b – 2c = 32 and 3a - 2b + 2c = 4, what is the value of a + b

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Math Expert
Joined: 02 Sep 2009
Posts: 43828
If 3a – 2b – 2c = 32 and 3a - 2b + 2c = 4, what is the value of a + b [#permalink]

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31 Dec 2015, 04:52
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If 3a – 2b – 2c = 32 and $$\sqrt{3a}- \sqrt{2b + 2c} = 4$$, what is the value of a + b + c ?

A. 3
B. 9
C. 10
D. 12
E. 14
[Reveal] Spoiler: OA

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Math Expert
Joined: 02 Aug 2009
Posts: 5654
Re: If 3a – 2b – 2c = 32 and 3a - 2b + 2c = 4, what is the value of a + b [#permalink]

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31 Dec 2015, 06:43
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duahsolo wrote:
If 3a – 2b – 2c = 32 and √3a-√(2b+2c)=4, what is the value of a + b + c?

A) 3
B) 9
C) 10
D) 12
E) 14

Hi,
when we look at the two equations, we can relize some similarity, so lets work on it..
3a – 2b – 2c = 32 can be written as √3a^2-√(2b+2c)^2=32
{√3a-√(2b+2c)}{√3a+√(2b+2c)}=32..
or 4*√3a+√(2b+2c)=32..
or √3a+√(2b+2c)=8..

now lets work on these two equations
1)√3a-√(2b+2c)=4..
2)√3a+√(2b+2c)=8..

√3a+√(2b+2c)+√3a-√(2b+2c)=12..
2√3a=12..
or √3a=6..
3a=36..
a=12.

B) subtract 1 from 2..
√3a+√(2b+2c)-√3a+√(2b+2c)=4..
2√(2b+2c)=4..
√(2b+2c)=2..
2b+2c=4..
or b+c=2..

from A and B a+b+c=12+2=14..
E
Hope it helped
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Math Expert
Joined: 02 Sep 2009
Posts: 43828
If 3a – 2b – 2c = 32 and 3a - 2b + 2c = 4, what is the value of a + b [#permalink]

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02 Oct 2017, 03:48
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If 3a – 2b – 2c = 32 and 3a - 2b + 2c = 4, what is the value of a + b [#permalink]

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02 Oct 2017, 04:19
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Bunuel wrote:
If 3a – 2b – 2c = 32 and $$\sqrt{3a}- \sqrt{2b + 2c} = 4$$, what is the value of a + b + c ?

A. 3
B. 9
C. 10
D. 12
E. 14

$$3a-2b-2c=32 => 5a-2a-2b-2c=32 =>5a-2(a+b+c)=32$$

therefore $$a+b+c=\frac{5a}{2}-16$$ ----------------------------(1)

Also $$2b+2c=3a-32$$

Now, $$\sqrt{3a}- \sqrt{2b + 2c} = 4$$ substitute the value of $$2b+2c$$

$$\sqrt{3a}- \sqrt{3a-32} = 4$$. square both sides to get

$$3a+3a-32-2\sqrt{3a(3a-32)}=16$$. this can be written as

$$6a-48=2\sqrt{3a(3a-32)}$$ $$=>$$ $$3a-24=\sqrt{3a(3a-32)}$$. Again square both sides to get

$$9a^2+24^2-2*3a*24=9a^2-96a$$. solve this to get $$a=12$$

Substitute the value of $$a$$ in equation (1) or $$a+b+c=\frac{5*12}{2}-16=14$$

Option E

Last edited by niks18 on 02 Oct 2017, 04:50, edited 1 time in total.
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Re: If 3a – 2b – 2c = 32 and 3a - 2b + 2c = 4, what is the value of a + b [#permalink]

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02 Oct 2017, 04:49
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niks18 wrote:
Bunuel wrote:
If 3a – 2b – 2c = 32 and $$\sqrt{3a}- \sqrt{2b + 2c} = 4$$, what is the value of a + b + c ?

A. 3
B. 9
C. 10
D. 12
E. 14

$$3a-2b-2c=32 => 5a-2a-2b-2c=32 =>5a-2(a+b+c)=32$$

therefore $$a+b+c=\frac{5a}{2}-16$$ ----------------------------(1)

Also $$2b+2c=3a-32$$

Now, $$\sqrt{3a}- \sqrt{2b + 2c} = 4$$ substitute the value of $$2b+2c$$

$$\sqrt{3a}- \sqrt{3a-32} = 4$$. square both sides to get

$$3a+3a-32-2\sqrt{3a(3a-32)}=16$$. this can be written as

$$6a-48=2\sqrt{3a(3a-32)}$$ $$=>$$ $$3a-24=\sqrt{3a(3a-32)}$$. Again square both sides to get

$$9a^2+24^2-2*3a*24=9a^2-96$$. solve this to get $$a=12$$

Substitute the value of $$a$$ in equation (1) or $$a+b+c=\frac{5*12}{2}-16=14$$

Option E

Same answer. The highlighted portion would be 9a^2 - 96a?
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Re: If 3a – 2b – 2c = 32 and 3a - 2b + 2c = 4, what is the value of a + b [#permalink]

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02 Oct 2017, 04:52
Quote:
Same answer. The highlighted portion would be 9a^2 - 96a?

Hi Richak91

Thanks for highlighting the typo error
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Re: If 3a – 2b – 2c = 32 and 3a - 2b + 2c = 4, what is the value of a + b [#permalink]

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02 Oct 2017, 05:22
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Bunuel wrote:
If 3a – 2b – 2c = 32 and $$\sqrt{3a}- \sqrt{2b + 2c} = 4$$, what is the value of a + b + c ?

A. 3
B. 9
C. 10
D. 12
E. 14

Lets start solving this question.
Formula required : (x−y)(x+y)=(x2−y2)

$$\sqrt{3a}- \sqrt{2b + 2c} = 4$$ ..............(i)

3a – 2b – 2c = 32
($$\sqrt{3a}- \sqrt{2b + 2c}$$)($$\sqrt{3a}+ \sqrt{2b + 2c}$$) = 32
(4)($$\sqrt{3a}+ \sqrt{2b + 2c}$$) = 32
($$\sqrt{3a}+ \sqrt{2b + 2c}$$) = 8 ........................ (ii)

Adding (i) & (ii) , we get ($$2 \sqrt{3a}$$) = 12
a = 12

Subtracting (i) from (ii) , we get ($$2 \sqrt{2b + 2c}$$) = 4
b+c = 2

So, a+b+c = 12+2 = 14

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Re: If 3a – 2b – 2c = 32 and 3a - 2b + 2c = 4, what is the value of a + b [#permalink]

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02 Oct 2017, 10:44
Bunuel wrote:
If 3a – 2b – 2c = 32 and $$\sqrt{3a}- \sqrt{2b + 2c} = 4$$, what is the value of a + b + c ?

A. 3
B. 9
C. 10
D. 12
E. 14

I will use plugging in . Given the equations and the options, a is likely 12 and a+b+c is 14. For further checking, 2b+2c = 4 and \sqrt{2b + 2c} = 2, making a=b=1. Hence, a+b+c = 12+2(1) = 14.
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Re: If 3a – 2b – 2c = 32 and 3a - 2b + 2c = 4, what is the value of a + b [#permalink]

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02 Oct 2017, 11:07
rulingbear wrote:
Bunuel wrote:
If 3a – 2b – 2c = 32 and $$\sqrt{3a}- \sqrt{2b + 2c} = 4$$, what is the value of a + b + c ?

A. 3
B. 9
C. 10
D. 12
E. 14

I will use plugging in . Given the equations and the options, a is likely 12 and a+b+c is 14. For further checking, 2b+2c = 4 and \sqrt{2b + 2c} = 2, making a=b=1. Hence, a+b+c = 12+2(1) = 14.

I don't think plugging in is a good idea for this question.... How did u directly arrived at a = 12 ??
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If 3a – 2b – 2c = 32 and 3a - 2b + 2c = 4, what is the value of a + b [#permalink]

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02 Oct 2017, 11:19
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shashankism wrote:
rulingbear wrote:
Bunuel wrote:
If 3a – 2b – 2c = 32 and $$\sqrt{3a}- \sqrt{2b + 2c} = 4$$, what is the value of a + b + c ?

A. 3
B. 9
C. 10
D. 12
E. 14

I will use plugging in . Given the equations and the options, a is likely 12 and a+b+c is 14. For further checking, 2b+2c = 4 and \sqrt{2b + 2c} = 2, making a=b=1. Hence, a+b+c = 12+2(1) = 14.

I don't think plugging in is a good idea for this question.... How did u directly arrived at a = 12 ??

Good question. It needs keen observation

1. From the first equation, 3a is likely to be greater than 32
2. From the second equation, [m]\sqrt{3a} is likely to be greater than 4

Put this inference together and a could be 12, i.e 3*12= 36>32, and \sqrt{3*12} = 6>4

The same could be used to know that a= b= 1, as explained in my earlier post. Even without doing this you should know by now that the answer is E.

This is a good must be true type of question, it could also make a decent data sufficiency question. Just observing keenly will obviously save a lot of time!

Another way to look at it is the form of the equation, take the first equation, 3a - 2a- 2b= 32. This can be rearranged as 3a - 2(a+b)= 32, therefore a must be an even number and a+b will also be even. Given this, a+b+c would be an even number of the x+2y. Only 10,12 and 14 are viable. Looking at the second equation will tell you that x/a would be 12 as 3a would form the only proper square.

Last edited by rulingbear on 02 Oct 2017, 11:50, edited 1 time in total.
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If 3a – 2b – 2c = 32 and 3a - 2b + 2c = 4, what is the value of a + b [#permalink]

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02 Oct 2017, 11:30
rulingbear wrote:

Good question. It needs keen observation

1. From the first equation, 3a is likely to be greater than 32
2. From the second equation, [m]\sqrt{3a} is likely to be greater than 4

Put this inference together and a could be 12, i.e 3*12= 36>32, and \sqrt{3*12} = 6>4

The same could be used to know that a= b= 1, as explained in my earlier post. Even without doing this you should know by now that the answer is E.

This is a good must be true type of question, it could also make a decent data sufficiency question. Just observing keenly will obviously save a lot of time!

3a>32 already includes [m]\sqrt{3a} is be greater than 4.
So we should further work on 3a>32..

But your observation is appreciable and surely can be used here to solve the problem using plugging method..
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Re: If 3a – 2b – 2c = 32 and 3a - 2b + 2c = 4, what is the value of a + b [#permalink]

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02 Oct 2017, 11:55
shashankism wrote:
rulingbear wrote:

Good question. It needs keen observation

1. From the first equation, 3a is likely to be greater than 32
2. From the second equation, [m]\sqrt{3a} is likely to be greater than 4

Put this inference together and a could be 12, i.e 3*12= 36>32, and \sqrt{3*12} = 6>4

The same could be used to know that a= b= 1, as explained in my earlier post. Even without doing this you should know by now that the answer is E.

This is a good must be true type of question, it could also make a decent data sufficiency question. Just observing keenly will obviously save a lot of time!

3a>32 already includes [m]\sqrt{3a} is be greater than 4.
So we should further work on 3a>32..

But your observation is appreciable and surely can be used here to solve the problem using plugging method..

There is no need to actually work further once you know 3a>32, a is most likely 12 and the answer can only be 14, given the options.
Re: If 3a – 2b – 2c = 32 and 3a - 2b + 2c = 4, what is the value of a + b   [#permalink] 02 Oct 2017, 11:55
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