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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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31 Dec 2015, 04:52
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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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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.. A) add the two eq.. √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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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 \(3a2b2c=32 => 5a2a2b2c=32 =>5a2(a+b+c)=32\) therefore \(a+b+c=\frac{5a}{2}16\) (1) Also \(2b+2c=3a32\) Now, \(\sqrt{3a} \sqrt{2b + 2c} = 4\) substitute the value of \(2b+2c\) \(\sqrt{3a} \sqrt{3a32} = 4\). square both sides to get \(3a+3a322\sqrt{3a(3a32)}=16\). this can be written as \(6a48=2\sqrt{3a(3a32)}\) \(=>\) \(3a24=\sqrt{3a(3a32)}\). Again square both sides to get \(9a^2+24^22*3a*24=9a^296a\). 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 \(3a2b2c=32 => 5a2a2b2c=32 =>5a2(a+b+c)=32\) therefore \(a+b+c=\frac{5a}{2}16\) (1) Also \(2b+2c=3a32\) Now, \(\sqrt{3a} \sqrt{2b + 2c} = 4\) substitute the value of \(2b+2c\) \(\sqrt{3a} \sqrt{3a32} = 4\). square both sides to get \(3a+3a322\sqrt{3a(3a32)}=16\). this can be written as \(6a48=2\sqrt{3a(3a32)}\) \(=>\) \(3a24=\sqrt{3a(3a32)}\). Again square both sides to get \(9a^2+24^22*3a*24= 9a^296\). 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 ESame answer. The highlighted portion would be 9a^2  96 a?



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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 Richak91Thanks 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 Answer E
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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
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