This is a Data Sufficiency problem from Session 5, Extra Practice HW 3 (Quantitative Strategy). So far, 63.5% of Knewton students have missed it. How would you approach it?

Try it out, then share your answers, questions, and thought processes in the comments below. Remember, if you’re in our GMAT class now, add your teacher name and session to your comment (e.g., Zwelling, MW 1:30). We’ll post the official answer (in video form!) in a week or so. Good luck!

Update: Once you’ve tried the question, click the video below to see the answer and an explanation from Kyle, one of our GMAT team members.

Eunice sold several cakes. If each cake sold for either exactly 17 or exactly 19 dollars, how many 19 dollar cakes did Eunice sell?

1. Eunice sold a total of 8 cakes.
2. Eunice made 140 dollars in total revenue from her cakes.

[A] Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
[B] Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
[C] BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
[D] EACH statement ALONE is sufficient.
[E] Statements (1) and (2) TOGETHER are NOT sufficient.

• Vijay

Correct Answer is B….just check the multiples of 17 & 19….only one condition satisfies \$140 revenue..

• http://www.knewton.com Knewton Team

Hey all,

For the official answer, check out the video by Kyle from our GMAT team. We just posted his explanation below the question, so let us know what you think!

• Vijay

Correct Answer is B….just check the multiples of 17 & 19….only one condition satisfies \$140 revenue..

• http://www.knewton.com Knewton Team

Hey all,

For the official answer, check out the video by Kyle from our GMAT team. We just posted his explanation below the question, so let us know what you think!

• Bhupreet Kaur

c – equation 1 – x+y=8, equation 2 – 17x+19y = 140

• vishnu

First of all understanding of the equation,
17x+19y=140
so think addition of x+y=8; if we consider multiple of y(2) cakes of \$19 and remaining amount in multiple of \$102 worth of cakes are x(6) \$17 each. Answer is B.

• Bhupreet Kaur

c – equation 1 – x+y=8, equation 2 – 17x+19y = 140

• vishnu

First of all understanding of the equation,
17x+19y=140
so think addition of x+y=8; if we consider multiple of y(2) cakes of \$19 and remaining amount in multiple of \$102 worth of cakes are x(6) \$17 each. Answer is B.

• Johnraine18

you used part A in your description which means the answer is C. Both together are needed. you can’t solve the single equation with two variables. you need the system of equations.

• agshin

You are right. I can’t understand. They teach us not to use two variables in one equation. But here in video they demonstrate counter version. I choose C, because it is right way to solve this problem.

1:Insufficient, BCE may be the ans
2:17a+19b=140
Try plugging values… 17*9= 153; hence a<9
So check for values of a<9 such that 17a= 140-19b (in other words, 140-multiple of 19)
17*6= 102 = 140- 19*2, Hence a=6 and b=2
statement 2 alone sufficient, Hence Ans B
Cheers!

1:Insufficient, BCE may be the ans
2:17a+19b=140
Try plugging values… 17*9= 153; hence a<9
So check for values of a<9 such that 17a= 140-19b (in other words, 140-multiple of 19)
17*6= 102 = 140- 19*2, Hence a=6 and b=2
statement 2 alone sufficient, Hence Ans B
Cheers!

• Ajith

The answer is B. ie statement 2 alone is sufficient. 17x + 19y = 140. Solve for x and y which is 6 and 2

• Ajith

The answer is B. ie statement 2 alone is sufficient. 17x + 19y = 140. Solve for x and y which is 6 and 2

• Anonymous

Statement#1 alone gives us multiple answers

Statement#2 alone gives us unique answer. 102 + 38= 17*(6) + 19*(2) ……this is the only equation we can get.

• Anonymous

Statement#1 alone gives us multiple answers

Statement#2 alone gives us unique answer. 102 + 38= 17*(6) + 19*(2) ……this is the only equation we can get.

• Anonymous

C. Both Statements Together are sufficient, but neither statement alone is sufficient.
Zwelling, Session 1

• Anonymous

C. Both Statements Together are sufficient, but neither statement alone is sufficient.
Zwelling, Session 1

• Andre C

B is correct
17x+19y=140

only one correct solution when you check the multiples of 17 and 19

solve for x and y

Rich Zwelling- T/TH 7pm CST

• Andre C

B is correct
17x+19y=140

only one correct solution when you check the multiples of 17 and 19

solve for x and y

Rich Zwelling- T/TH 7pm CST

• Pawan Agarwal

Lets say # of cakes sold for \$17 = x
# of cakes sold for \$19 = y.

We need to find y.

As per 1. x+y = 8. This does not help us get value of y.
As per 2.
17x+19y = 140.
x = one of {0, 1,2,3,4,5,6,7,8} (note max = 8 because @ x=9 17x exceeds 140 which is not possible}
y = one of {0,1,2,3,4,5,6,7}

When x=8 17x = 136.==> 19y = 140-136 = 4 ==> No possible valid integer y.
when x=7 17x = 119. ==> 19y =140-119 = 21 ==> No possible valid integer y.
when x=6 17x = 102 ==> 19y = 140-102 = 38 ==> y=2.
when x=5 17x = 85 ==> 19y = 140-85 = 55 ==> No possible valid integer Y
when x=4 17x = 68 ==> 19y = 140-68 = 72 ==> No possible valid integer Y
when x=3 17x = 51 ==> 19y = 140-51 = 89 ==> No possible valid integer Y
when x=2 17x = 34 ==> 19y = 140-34 = 106 ==> No possible integer Y.
when x=1 17x = 17 ==> 19y = 140-17 = 133 ==> no possible integer y.

So only one combination of x,y exist which solves equation 17x+19y = 140.

As a result we know y = 2.

• Pawan Agarwal

Lets say # of cakes sold for \$17 = x
# of cakes sold for \$19 = y.

We need to find y.

As per 1. x+y = 8. This does not help us get value of y.
As per 2.
17x+19y = 140.
x = one of {0, 1,2,3,4,5,6,7,8} (note max = 8 because @ x=9 17x exceeds 140 which is not possible}
y = one of {0,1,2,3,4,5,6,7}

When x=8 17x = 136.==> 19y = 140-136 = 4 ==> No possible valid integer y.
when x=7 17x = 119. ==> 19y =140-119 = 21 ==> No possible valid integer y.
when x=6 17x = 102 ==> 19y = 140-102 = 38 ==> y=2.
when x=5 17x = 85 ==> 19y = 140-85 = 55 ==> No possible valid integer Y
when x=4 17x = 68 ==> 19y = 140-68 = 72 ==> No possible valid integer Y
when x=3 17x = 51 ==> 19y = 140-51 = 89 ==> No possible valid integer Y
when x=2 17x = 34 ==> 19y = 140-34 = 106 ==> No possible integer Y.
when x=1 17x = 17 ==> 19y = 140-17 = 133 ==> no possible integer y.

So only one combination of x,y exist which solves equation 17x+19y = 140.

As a result we know y = 2.

• Abhishek

• Abhishek

• Anshu Mishra

x-> number of 17\$ cakes, y-> no. of 19 \$ cakes,

y = ?

Statement 1: x+y = 8
y = 8-x , INSUFFICIENT

Statement 2 : 17x+ 19 y = 140
Also, x and y are non-negative integers , so x,y < 9 ( since, 140/17 < 9)

The only integral solution, is when :

x = 6, y = 2. Sufficient

• Anshu Mishra

x-> number of 17\$ cakes, y-> no. of 19 \$ cakes,

y = ?

Statement 1: x+y = 8
y = 8-x , INSUFFICIENT

Statement 2 : 17x+ 19 y = 140
Also, x and y are non-negative integers , so x,y < 9 ( since, 140/17 < 9)

The only integral solution, is when :

x = 6, y = 2. Sufficient

• annmary

x:19\$ (I show how many cake selling in price19\$) & y:17\$ (I show how many cake selling in price17\$)
1.x+y=8
2.19x+17y=140
now we can to solve these equations…
good luck!

• annmary

x:19\$ (I show how many cake selling in price19\$) & y:17\$ (I show how many cake selling in price17\$)
1.x+y=8
2.19x+17y=140
now we can to solve these equations…
good luck!

• AlessandroMastracco

• AlessandroMastracco

• HemaChandra

Answer B.. only one combo of 19 and 17 multiples satisfies it .. 2 \$19 buns and 6 \$17 buns

• HemaChandra

Answer B.. only one combo of 19 and 17 multiples satisfies it .. 2 \$19 buns and 6 \$17 buns

• Pierre

For this very exact scenario, B is correct because there is only one combination of multiples that results in 140. But don’t expect this to apply to any problem of the same type. for example, if he were selling cookies at exactly \$2 or \$3, and (1) he sold 30 cookies (2) he made \$72 in total revenue. In this case you have to know both statements in order to get the correct answer.

• Pierre

For this very exact scenario, B is correct because there is only one combination of multiples that results in 140. But don’t expect this to apply to any problem of the same type. for example, if he were selling cookies at exactly \$2 or \$3, and (1) he sold 30 cookies (2) he made \$72 in total revenue. In this case you have to know both statements in order to get the correct answer.

• ayman

that was less complicated than that u should have done a system!

• me

when looking at just statement 2 you only have one equation

• ayman

that was less complicated than that u should have done a system!

• me

when looking at just statement 2 you only have one equation

• Kenneth

Correct ans is C
17x + 19y = 140
x+y = 8
Therefore y=2

• Kenneth

Correct ans is C
17x + 19y = 140
x+y = 8
Therefore y=2

• Samira Greche

• Samira Greche

• Anatkramer

I got B as well but I went through all of the possibilities. I think there is an easier way.
If we were told that one cake was \$3 and the other was \$5 and the total revenue was \$75, we would have at least two possibilities for a and b => 3a+5b=75 => a=20,b=3 or a=10,b=9 in which case the answer would be insufficient.

In this question we have the primes 17 and 19. The only way for there to be multiple values for a and be would be if a were a multiple of 19 and b was a multiple of 17. If we even plug in one of those numbers we can see that revenues would be too much there for there is only one answer choice. I would stop there.

If I really wanted to know what a and be was I would use both equations to find out. I know we are not supposed to look at 1 while evaluating 2 but I know they will not contradict so its a fast way to find a and b once I know that there is only one possible answer.

• Anatkramer

I got B as well but I went through all of the possibilities. I think there is an easier way.
If we were told that one cake was \$3 and the other was \$5 and the total revenue was \$75, we would have at least two possibilities for a and b => 3a+5b=75 => a=20,b=3 or a=10,b=9 in which case the answer would be insufficient.

In this question we have the primes 17 and 19. The only way for there to be multiple values for a and be would be if a were a multiple of 19 and b was a multiple of 17. If we even plug in one of those numbers we can see that revenues would be too much there for there is only one answer choice. I would stop there.

If I really wanted to know what a and be was I would use both equations to find out. I know we are not supposed to look at 1 while evaluating 2 but I know they will not contradict so its a fast way to find a and b once I know that there is only one possible answer.

• Tebteb

This problem was relatively easy. They all are, but they take me too long. This took me 3:20.

• Tebteb

This problem was relatively easy. They all are, but they take me too long. This took me 3:20.

• BSchool hopeful

Why did this video not just set up a system of equations (the second being x + y = 8) and solve them simultaneously after realizing the number must be 8?

• Jarrod

Because that wouldn’t prove that the answer should be B instead of C.

• BSchool hopeful

Why did this video not just set up a system of equations (the second being x + y = 8) and solve them simultaneously after realizing the number must be 8?

• Anonymous

That’s a great way to approach the problem. You have a choice whether to tackle the problem conceptually or algebraically, and each approach takes about the same amount of time.

• Anonymous

That’s a great way to approach the problem. You have a choice whether to tackle the problem conceptually or algebraically, and each approach takes about the same amount of time.

• Enrique

Why B is correct if you are using the statement# 1 (8 cakes). Could be the answer letter “C” ? I am confused now.

• Enrique

Sorry this is a better question from me. Why cannot I use the data of the statement# 1 (8)? It would be easy to do the math, so the answer could be letter “C”.

• Anonymous

Good question, Enrique. The trick is, we do not actually need to use Statement 1 to know that Eunice sold 8 cakes.

Consider Statement 2 alone–remember, when considering the statements individually, you have to ignore all the information from the other statement. We know that Eunice made a total of 140 dollars from selling the cakes, and so we set up the equation 17x + 19y = 140. Then, because the cakes are discrete items, x and y must each be a positive integer, which is a major restriction. Start watching the video at 0:35 if you would like more of an explanation on that. The key is that it turns out the only number of cakes Eunice could have sold is 8, and you know that based solely on Statement 2.

• Chuck

Kyle: I beg to differ with your solution/approach.

17x + 19y = 140 (constitutes an equation with 2 unknowns).

Obviously trial by inserting numbers will get us to the answer, but what if you had landed with a solution with 2 or more choices then what?

mathematically, there should be a second equation that should make choice C more appropriate.

However, with that said, i do not work for the GMAT group so no way to confirm who is right here

-Chuck

• Enrique

Why B is correct if you are using the statement# 1 (8 cakes). Could be the answer letter “C” ? I am confused now.

• Enrique

Sorry this is a better question from me. Why cannot I use the data of the statement# 1 (8)? It would be easy to do the math, so the answer could be letter “C”.

• Anonymous

Good question, Enrique. The trick is, we do not actually need to use Statement 1 to know that Eunice sold 8 cakes.

Consider Statement 2 alone–remember, when considering the statements individually, you have to ignore all the information from the other statement. We know that Eunice made a total of 140 dollars from selling the cakes, and so we set up the equation 17x + 19y = 140. Then, because the cakes are discrete items, x and y must each be a positive integer, which is a major restriction. Start watching the video at 0:35 if you would like more of an explanation on that. The key is that it turns out the only number of cakes Eunice could have sold is 8, and you know that based solely on Statement 2.

• Chuck

Kyle: I beg to differ with your solution/approach.

17x + 19y = 140 (constitutes an equation with 2 unknowns).

Obviously trial by inserting numbers will get us to the answer, but what if you had landed with a solution with 2 or more choices then what?

mathematically, there should be a second equation that should make choice C more appropriate.

However, with that said, i do not work for the GMAT group so no way to confirm who is right here

-Chuck

[B] There is only one combination of 17 x + 19 y = 140

[B] There is only one combination of 17 x + 19 y = 140

• EvaJager

I would rewrite the equation (20-3)x+(20-1)y=140 or 20(x+y)-3x-y=140. Since x and y are positive integers, and y cannot be greater than 7 and x cannot be greater than 8, from the previous form of the equation, I can deduce that 3x+y=20 (it must be a multiple of 20) and so x+y=8. So x and y can be immediately found, no messy calculations.

• EvaJager

I would rewrite the equation (20-3)x+(20-1)y=140 or 20(x+y)-3x-y=140. Since x and y are positive integers, and y cannot be greater than 7 and x cannot be greater than 8, from the previous form of the equation, I can deduce that 3x+y=20 (it must be a multiple of 20) and so x+y=8. So x and y can be immediately found, no messy calculations.

• EvaJager

I would rewrite the equation (20-3)x+(20-1)y=140 or 20(x+y)-3x-y=140. Since x and y are positive integers (140 is not divisible neither by 17, nor by 19, so neither can be 0), and y cannot be greater than 7 and x cannot be greater than 8, from the previous form of the equation, I can deduce that 3x+y=20 (it must be a multiple of 20 and the maximum is 3×8+7=31, and therefore x+y=8. So x and y can be immediately found, no messy calculations.

• EvaJager

I would rewrite the equation (20-3)x+(20-1)y=140 or 20(x+y)-3x-y=140. Since x and y are positive integers (140 is not divisible neither by 17, nor by 19, so neither can be 0), and y cannot be greater than 7 and x cannot be greater than 8, from the previous form of the equation, I can deduce that 3x+y=20 (it must be a multiple of 20 and the maximum is 3×8+7=31, and therefore x+y=8. So x and y can be immediately found, no messy calculations.

• Qwert2

• Evajager

No, the answer is definitely B. When solving equations involving positive integers, many times there is a finite number of solutions, and sometimes, as in this case, there is a unique solution. See my comment below (EvaJager).

• Qwert2

• Evajager

No, the answer is definitely B. When solving equations involving positive integers, many times there is a finite number of solutions, and sometimes, as in this case, there is a unique solution. See my comment below (EvaJager).

• arryputtar

The answer is D – both statements taken toghether`

Is there a way to find out for sure that given any linear equation, how many integer solutions satisfy it? This question, it is not obvious that there is only a single solution at first glance, if given any other equation it is possible that more than one integral sols exist, in which case the answer would have been c. Any ideas?

• Almazd2003

i don’t agree with the answer, it should be “C”

• Evajager

Why do you think the answer should be C? Please, provide a proof.
Did you understand the many solutions suggested in this forum?
With which part of the solutions you don’t agree?

• Syed

B alone is sufficient.

• Taylorbyu

The answer is B, but the real test is how fast can you solve the problem. I think anyone can quickly calculate that 1 & 2 together are sufficient. Additionally, one can quickly infer from 2 that a total of 8 cakes were sold (all 19 cakes means more than 7 cakes were sold and all 17 cakes means less than 9 cakes were sold so 8 cakes had to be sold); therefore, 2 is sufficient alone. Further it should be painfully clear that C cannot be the correct answer since 1 is implicit in 2.

• Marevik2

• Zan

I obtained the right answer…  But, I did it differently.   I asked myself “what multiple of 7 added to 9 results in a zero” I would have done the test in vice versa.. but there was no need.

In example, 7×3=21, 21+9=30…   Therefore, (17×3)+(19×1)=70…   Obviously (2) \$19 cakes.

• kntu24

But 2 isn’t the only answer which is why I chose “C” 19*4= 72 and 17*4 is 68. 68+72 is 140…..? I’m confused.

• mb

19*4 = 76 not 72

• Moko

it’s a pretty straightforward algebra problem. 19x + 14y = 140. solve for y. then take the other equation where x + y = 8, solve for either x or y, and plug into the first equation and BAM!!! you’ve got your answer. 2 unknowns require 2 equations to solve. therefore both statements ARE needed. 2 unknowns and 1 equation = under-defined problem. 1 unknown and 2 equations = over-defined problem.

• Moko

19x +17y, typo on last post

• Moko

you can solve by deduction too, which is lame, considering using 17x + 19y = 140, and substituting x = 8 – y, you get y = 2 in under 30 seconds…. so B is a correct statement. lame.

• caly

each statement is sufficient since he can sell them either 17 dollars or 19 dollars .option C is the answer