:arrow: Sounds like a good fool proof approach but surely a lengthy one.Rahul@gurome wrote:We need to find whether linear combination of two equations can result in a third equation or not. By linear combination I mean simple arithmetic operations that can done on two equations, like addition/subtraction of two equations, multiplication/division of an equation by a constant.goyalsau wrote:The cost of 3 chocolates, 5 biscuits, and 5 ice creams is 195. What is the cost of 7 chocolates, 11 biscuits and 9 ice creams?
(A) The cost of 5 chocolates, 7 biscuits and 3 ice creams is 217.
(B) The cost of 4 chocolates, 1 biscuit and 3 ice creams is 141.
Please note that this the methodical solution which uses some higher level mathematical concept (namely linear independence). Solving proper GMAT question does not need such concepts.
Given: Say cost of one chocolate = a, cost of one biscuit = b and cost of one ice-cream = c. Then (3a + 5b + 5c) = 195 => (3a + 5b + 5c - 195) = 0. We have to find (7a + 11b + 9c) = ?.
Say, (7a + 11b + 9c) = x => (7a + 11b + 9c - x) = 0
Statement 1: (5a + 7b + 3c) = 217 => (5a + 7b + 3c - 217) = 0
Now the question is whether we can complete the required equation by linearly combining (3a + 5b + 5c) = 195 and (5a + 7b + 3c) = 217. If we can, then the following relation must hold for some constant m and n,As a cannot contribute in b, b in c and so on, m and n must follow the following relations, (the relations are obtained by equating the coefficients of a, b and c)
- m*(3a + 5b + 5c - 195) + n*(5a + 7b + 3c - 217) = (7a + 11b + 9c -x)
If there exists a set of value for m and n for which all the three relations are satisfied, then we can easily find x. In fact x will be equal to (195m + 217m).
- 1. 3m + 5n = 7
2. 5m + 7n = 11
3. 5m + 3n = 9
For this case we can find such a set of value for m and n: m = 3/2 and n = 1/2
Sufficient.
Statement 1: (4a + b + 3c) = 141 => (4a + b + 3c - 141) = 0
Applying same procedure as above, m and n must satisfy the following relations,Try to solve these three relations, you'll find there is no such (m, n) for which all the three relations are satisfied. Thus we cannot complete the required equation.
- 1. 3m + 4n = 7
2. 5m + n = 11
3. 5m + 3n = 9
Not sufficient.
Correct answer is A.
Chocolates.
-
shubhamkumar
- Senior | Next Rank: 100 Posts
- Posts: 72
- Joined: Mon Apr 11, 2011 6:45 pm
- Thanked: 2 times
-
Ganesh hatwar
- Senior | Next Rank: 100 Posts
- Posts: 97
- Joined: Sun Jun 24, 2012 11:23 pm
GMATGuruNY wrote:Given equation: 3c + 5b + 5i = 195.goyalsau wrote:The cost of 3 chocolates, 5 biscuits, and 5 ice creams is 195. What is the cost of 7 chocolates, 11 biscuits and 9 ice creams?
(A) The cost of 5 chocolates, 7 biscuits and 3 ice creams is 217.
(B) The cost of 4 chocolates, 1 biscuit and 3 ice creams is 141.
Question: What is the value of 7c + 11b + 9i (or a multiple of this expression)?
Statement 1: 5c + 7b + 3i = 217
Try to combine this equation with the given equation so that the coefficient in front of c is a multiple of 7.
Multiplied by 3, the given equation 3c + 5b + 5i = 195 becomes 9c + 15b + 15i = 585.
Stacking this equation with the equation in statement 1, we get:
9c + 15b + 15i = 585
5c + 7b + 3i = 217
Adding the two equations, we get:
14c + 22b + 18i = 802.
Dividing the equation above by 2, we get:
7c + 11b + 9i = 401.
Sufficient.
Statement 2: 4c + 1b + 3i = 141.
Try to combine this equation with the given equation so that the coefficient in front of c is a multiple of 7.
Stacking the given equation with the equation in statement 2, we get:
3c + 5b + 3i = 195
4c + 1b + 3i = 141
Adding the two equations, we get:
7c + 6b + 6i = 336.
No way to determine the value of 7c + 11b + 9i.
Insufficient.
The correct answer is A.
Without going into complexities of linear equations, your solution offer much simpler approach that is easily comprehensible.
Thankyou !!
-
hemant_rajput
- Master | Next Rank: 500 Posts
- Posts: 447
- Joined: Sun Apr 22, 2012 7:13 am
- Thanked: 46 times
- Followed by:13 members
- GMAT Score:700
I'm not sure if this method will work all the time.GMATGuruNY wrote:Given equation: 3c + 5b + 5i = 195.goyalsau wrote:The cost of 3 chocolates, 5 biscuits, and 5 ice creams is 195. What is the cost of 7 chocolates, 11 biscuits and 9 ice creams?
(A) The cost of 5 chocolates, 7 biscuits and 3 ice creams is 217.
(B) The cost of 4 chocolates, 1 biscuit and 3 ice creams is 141.
Question: What is the value of 7c + 11b + 9i (or a multiple of this expression)?
Statement 1: 5c + 7b + 3i = 217
Try to combine this equation with the given equation so that the coefficient in front of c is a multiple of 7.
Multiplied by 3, the given equation 3c + 5b + 5i = 195 becomes 9c + 15b + 15i = 585.
Stacking this equation with the equation in statement 1, we get:
9c + 15b + 15i = 585
5c + 7b + 3i = 217
Adding the two equations, we get:
14c + 22b + 18i = 802.
Dividing the equation above by 2, we get:
7c + 11b + 9i = 401.
Sufficient.
Statement 2: 4c + 1b + 3i = 141.
Try to combine this equation with the given equation so that the coefficient in front of c is a multiple of 7.
Stacking the given equation with the equation in statement 2, we get:
3c + 5b + 3i = 195
4c + 1b + 3i = 141
Adding the two equations, we get:
7c + 6b + 6i = 336.
No way to determine the value of 7c + 11b + 9i.
Insufficient.
The correct answer is A.
Please correct me if I wrong, I guess that you are saying that only by making the coefficient of
one of a/b/c you'll be able to derive the solution of required equation?
I also tried doing it your way but I got stuck. To try and get the coefficient of C to be 7, I took the given equation and multiplied my 10 then added the eqn from statement 1:GMATGuruNY wrote:Given equation: 3c + 5b + 5i = 195.goyalsau wrote:The cost of 3 chocolates, 5 biscuits, and 5 ice creams is 195. What is the cost of 7 chocolates, 11 biscuits and 9 ice creams?
(A) The cost of 5 chocolates, 7 biscuits and 3 ice creams is 217.
(B) The cost of 4 chocolates, 1 biscuit and 3 ice creams is 141.
Question: What is the value of 7c + 11b + 9i (or a multiple of this expression)?
Statement 1: 5c + 7b + 3i = 217
Try to combine this equation with the given equation so that the coefficient in front of c is a multiple of 7.
Multiplied by 3, the given equation 3c + 5b + 5i = 195 becomes 9c + 15b + 15i = 585.
Stacking this equation with the equation in statement 1, we get:
9c + 15b + 15i = 585
5c + 7b + 3i = 217
Adding the two equations, we get:
14c + 22b + 18i = 802.
Dividing the equation above by 2, we get:
7c + 11b + 9i = 401.
Sufficient.
Statement 2: 4c + 1b + 3i = 141.
Try to combine this equation with the given equation so that the coefficient in front of c is a multiple of 7.
Stacking the given equation with the equation in statement 2, we get:
3c + 5b + 3i = 195
4c + 1b + 3i = 141
Adding the two equations, we get:
7c + 6b + 6i = 336.
No way to determine the value of 7c + 11b + 9i.
Insufficient.
The correct answer is A.
30C+50B+50i=1950
5c+7b+3i=217
Adding the two eqn
35C+57B+53i=2167
The plan was to divide by 5 but as you can see the other coefficients have a plan of their own. I can't understand why your way works and not mine.
-
The Iceman
- Master | Next Rank: 500 Posts
- Posts: 194
- Joined: Mon Oct 15, 2012 7:14 pm
- Location: India
- Thanked: 47 times
- Followed by:6 members
This question can be solved by two approaches:goyalsau wrote:The cost of 3 chocolates, 5 biscuits, and 5 ice creams is 195. What is the cost of 7 chocolates, 11 biscuits and 9 ice creams?
(A) The cost of 5 chocolates, 7 biscuits and 3 ice creams is 217.
(B) The cost of 4 chocolates, 1 biscuit and 3 ice creams is 141.
1>hit and trial (can give quick results but may backfire if the coefficients do not strike)
2>traditional method (Appropriate way to do; basically a superset of approach 1; will take less time once you realise that you just need to establish whether a rational solution exists for a pair of variables)
GMAT should give numbers that would strike instantly. In case they don't, fall back to approach 2.
Statement A:
(3 5 5)*x
(5 7 3)*y
------
7 11 9
We need an unique solution in (x,y)such that 3x+5y=7; 5x+7y=11; 5x+3y=9 => (x,y) does have a unique solution
Statement B:
(3 5 5)*x
(4 1 3)*y
------
7 11 9
We need an unique solution in (x,y)such that 3x+4y=7; 5x+y=11; 5x+3y=9 => (x,y) does not have a unique solution
Hence A.
-
rajeshsinghgmat
- Master | Next Rank: 500 Posts
- Posts: 171
- Joined: Tue Jan 08, 2013 7:24 am
- Thanked: 1 times
Hi,GMATGuruNY wrote:Given equation: 3c + 5b + 5i = 195.goyalsau wrote:The cost of 3 chocolates, 5 biscuits, and 5 ice creams is 195. What is the cost of 7 chocolates, 11 biscuits and 9 ice creams?
(A) The cost of 5 chocolates, 7 biscuits and 3 ice creams is 217.
(B) The cost of 4 chocolates, 1 biscuit and 3 ice creams is 141.
Question: What is the value of 7c + 11b + 9i (or a multiple of this expression)?
Statement 1: 5c + 7b + 3i = 217
Try to combine this equation with the given equation so that the coefficient in front of c is a multiple of 7.
Multiplied by 3, the given equation 3c + 5b + 5i = 195 becomes 9c + 15b + 15i = 585.
Stacking this equation with the equation in statement 1, we get:
9c + 15b + 15i = 585
5c + 7b + 3i = 217
Adding the two equations, we get:
14c + 22b + 18i = 802.
Dividing the equation above by 2, we get:
7c + 11b + 9i = 401.
Sufficient.
Statement 2: 4c + 1b + 3i = 141.
Try to combine this equation with the given equation so that the coefficient in front of c is a multiple of 7.
Stacking the given equation with the equation in statement 2, we get:
3c + 5b + 3i = 195
4c + 1b + 3i = 141
Adding the two equations, we get:
7c + 6b + 6i = 336.
No way to determine the value of 7c + 11b + 9i.
Insufficient.
The correct answer is A.
I really would like to know how you came up understanding that equations had to be added together in order to have a miltiple of 7.
Thank you and regards
-
sarathought777
- Newbie | Next Rank: 10 Posts
- Posts: 7
- Joined: Thu Dec 05, 2013 8:07 am
- Thanked: 1 times
The Iceman wrote:This question can be solved by two approaches:goyalsau wrote:The cost of 3 chocolates, 5 biscuits, and 5 ice creams is 195. What is the cost of 7 chocolates, 11 biscuits and 9 ice creams?
(A) The cost of 5 chocolates, 7 biscuits and 3 ice creams is 217.
(B) The cost of 4 chocolates, 1 biscuit and 3 ice creams is 141.
1>hit and trial (can give quick results but may backfire if the coefficients do not strike)
2>traditional method (Appropriate way to do; basically a superset of approach 1; will take less time once you realise that you just need to establish whether a rational solution exists for a pair of variables)
GMAT should give numbers that would strike instantly. In case they don't, fall back to approach 2.
Statement A:
(3 5 5)*x
(5 7 3)*y
------
7 11 9
We need an unique solution in (x,y)such that 3x+5y=7; 5x+7y=11; 5x+3y=9 => (x,y) does have a unique solution
Statement B:
(3 5 5)*x
(4 1 3)*y
------
7 11 9
We need an unique solution in (x,y)such that 3x+4y=7; 5x+y=11; 5x+3y=9 => (x,y) does not have a unique solution
Hence A.
Can u please explain what do u mean by unique solution?
-
sarathought777
- Newbie | Next Rank: 10 Posts
- Posts: 7
- Joined: Thu Dec 05, 2013 8:07 am
- Thanked: 1 times
Please explain, what do u mean by unique solution?The Iceman wrote:This question can be solved by two approaches:goyalsau wrote:The cost of 3 chocolates, 5 biscuits, and 5 ice creams is 195. What is the cost of 7 chocolates, 11 biscuits and 9 ice creams?
(A) The cost of 5 chocolates, 7 biscuits and 3 ice creams is 217.
(B) The cost of 4 chocolates, 1 biscuit and 3 ice creams is 141.
1>hit and trial (can give quick results but may backfire if the coefficients do not strike)
2>traditional method (Appropriate way to do; basically a superset of approach 1; will take less time once you realise that you just need to establish whether a rational solution exists for a pair of variables)
GMAT should give numbers that would strike instantly. In case they don't, fall back to approach 2.
Statement A:
(3 5 5)*x
(5 7 3)*y
------
7 11 9
We need an unique solution in (x,y)such that 3x+5y=7; 5x+7y=11; 5x+3y=9 => (x,y) does have a unique solution
Statement B:
(3 5 5)*x
(4 1 3)*y
------
7 11 9
We need an unique solution in (x,y)such that 3x+4y=7; 5x+y=11; 5x+3y=9 => (x,y) does not have a unique solution
Hence A.
-
Bill@VeritasPrep
- GMAT Instructor
- Posts: 1248
- Joined: Thu Mar 29, 2012 2:57 pm
- Location: Everywhere
- Thanked: 503 times
- Followed by:192 members
- GMAT Score:780
I think The Iceman means that you need exactly one value for x and y. It's not sufficient if you have multiple values.
Join Veritas Prep's 2010 Instructor of the Year, Matt Douglas for GMATT Mondays
Visit the Veritas Prep Blog
Try the FREE Veritas Prep Practice Test
Visit the Veritas Prep Blog
Try the FREE Veritas Prep Practice Test
-
nikhilgmat31
- Legendary Member
- Posts: 518
- Joined: Tue May 12, 2015 8:25 pm
- Thanked: 10 times
-
Max@Math Revolution
- Elite Legendary Member
- Posts: 3991
- Joined: Fri Jul 24, 2015 2:28 am
- Location: Las Vegas, USA
- Thanked: 19 times
- Followed by:37 members
In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem.
Remember equal number of variables and equations ensures a solution.
In original condition, 3c+5b+5i=195 then question is 7c+11b+9i=?
In this case, we have 3 variables (c,b,i) and 1 equation (3c+5b+5i=195)
We need 2 more equations. In this case, (1) is 1 equation, (2) is 1 equation so we have 2 equations. So C is the answer.
Why C? If you know our own innovative logics to find the answer, you don't need to solve the problem.
www.mathrevolution.com
Our world's first Variable Approach (DS) and IVY Approach (PS) help students dramatically reduce their time spent per question and improve accuracy. You will have 10 min. to spare before the test ends.
Remember equal number of variables and equations ensures a solution.
In original condition, 3c+5b+5i=195 then question is 7c+11b+9i=?
In this case, we have 3 variables (c,b,i) and 1 equation (3c+5b+5i=195)
We need 2 more equations. In this case, (1) is 1 equation, (2) is 1 equation so we have 2 equations. So C is the answer.
Why C? If you know our own innovative logics to find the answer, you don't need to solve the problem.
www.mathrevolution.com
Our world's first Variable Approach (DS) and IVY Approach (PS) help students dramatically reduce their time spent per question and improve accuracy. You will have 10 min. to spare before the test ends.
