The cost of a certain school trip was $16 for each child and $24 for

Question

The cost of a certain school trip was $16 for each child and $24 for each adult. What was the ratio of the number of children to the number of adults on the trip?

(1) The ratio of the total cost of the trip for all children to the total cost of the trip for all adults was 8 to 3.
(2) The total cost of the trip for all children and adults together was $2,200.

gmatophobia · Accepted Answer

Number of adults =  a   
 Number of children =  c

Question  :  \frac{c}{a}

Statement 1

(1) The ratio of the total cost of the trip for all children to the total cost of the trip for all adults was 8 to 3.

\frac{16c }{ 24a} = \frac{8}{3}

We can find the ratio of  \frac{c}{a} .

Hence, this statement is sufficient. We can eliminate B, C, and E.

Statement 2

(2) The total cost of the trip for all children and adults together was $2,200

16c + 24a = 2200

2c + 3a = 275

We can have multiple values of c and a satisfying the equation.

c = 22, a = 77  ;  \frac{c }{ a} = \frac{2}{7}

c = 46, a = 71  ;  \frac{c }{ a} = \frac{46}{71}

The statement alone is not sufficient.

Option A

KarishmaB · Answer

We use the concept of integer solutions to equations in 2 variables here. The moment we get one solution, we know all other possible solutions, if there are any. It has been discussed here: https://anaprep.com/algebra-integer-sol ... variables/ Given equation: 2c+3a=275 We can see straightaway that a = 1 will satisfy because 275 - 3 = 272 (an even number) so we will get 2c = 272 which will give us c = 136 Hence a = 1, c = 136 is a possible solution. Next possible solution is a = 3, c = 133 Next possible solution is a = 5, c = 130 and so on... (The logic has been discussed in the blog post given above)

JeffTargetTestPrep · Answer

The prices were $16 per child and $24 per adult, and we need to answer the question:

# of children (c) : # of adults (a)= ?

Statement One Alone:

=> The ratio of the total cost of the trip for all children to the total cost of the trip for all adults was 8 to 3.

Total cost for children : Total cost for adults = 8 : 3

16c/24a = 8/3
2c/3a = 8/3
c/a = 4/1 => c : a = 4 : 1

Statement one is sufficient. Eliminate answer choices B, C, and E.

Statement Two Alone:

=> The total cost of the trip for all children and adults together was $2,200.

16c + 24a = 2,200
2c + 3a = 275

If there were 100 children and 25 adults, then c : a = 100 : 25 = 4 : 1.

Whereas, if there were 25 children and 75 adults, then c : a = 25 : 75 = 1 : 3.

Statement two is not sufficient.

Answer: A

KS1409 · Answer

I always have trouble during the exam to check such possibilities of different numbers that can satisfy an equation. I end up either picking the wrong combos, getting only 1 pair to work and calling such a statement sufficient OR end up wasting time either way. Any work arounds or suggestions would be very helpful  JeffTargetTestPrep  !

illegallyblonde · Answer

Second that.  KarishmaB   chetan2u   gmatophobia   Bunuel   GMATCoachBen  Experts, if you could please advise.

rishabhm99 · Answer

If you are facing issues with Diophantine equations, you can refer to the blog post in Karishma’s response above, or watch a video on the topic on YouTube by Anthony Ritz. The video is titled: Linear Diophantine Equations: Ax + By = C, and is hosted on the GMAT club channel.

Borrat · Answer

is there a way of quickly finding out if an expression such as 2a + 3b = 275 has multiple solutions? , not this one in particular but maybe something harder

FrozenDragon · Answer

You don't have to remember this every time. Just remember that a single linear equation with two different equation in addition typically have many solution.

MarkGMATMentor · Answer

On Data Sufficiency, often you don't need to solve everything to get the right answer.
  
The prompt tells you it's $16 per child, so let's call the total cost for the children 16c. It's $24 per adult, so the total cost for the adults is 24a.

The question asks:  "What is c/a?"

Statement 1: 
 The first statement gives you the ratio of the total costs:
16c/24a = 8/3

You can't solve for c or a individually. But you don't need to. You can solve for c/a, so Statement 1 is  sufficient .

Statement 2:

Statement 2 gives you the total cost of the trip, so you'll have:
16c + 24a = $2,200

Again you have one equation, two variables, so there's no way to solve for a or c individually. But notice: three children cost the same as two adults, $48.

So we could exchange three children for two adults, or two adults for three children, without changing the total cost at all.

The ratio of c/a can be many things, so Statement 2 is  insufficient .

The answer is A.

Notice that we never found out the actual number of children or adults.

On Data Sufficiency, stay focused on exactly what you're looking for. You don't need to solve, you only need to know if you    can    solve, and only for exactly what they're asking.

I go in more depth in the video solution:
 https://www.youtube.com/watch?v=wk2jXW_2w-k

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