Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin

Question

Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Claudia have?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

KarishmaB · Accepted Answer

She has 12 coins.
We can obtain 17 distinct values with the 12 coins i.e. 17 distinct sums of money with the 12 coins.

If all 12 coins were 5 cent coins, we could have obtained a total of 60 cents in multiples of 5 i.e. 5 cents, 10 cents, 15 cents, 20 cents, ... 55 cents, 60 cents. These are just 12 distinct sums of money.
If all 12 coins were 10 cent coins, we could have obtained a total of 120 cents in multiples of 10 i.e. 10 cents, 20 cents, 30 cents, 40 cents, ... 110 cents, 120 cents. These are just 12 distinct sums of money.

To get 17 combinations, we must have a mix of 5 cent and 10 cent coins.

Now note that you can make each multiple of 5 till the max multiple of 5 as per the number of coins. 
Say of the 12 coins, 10 are 5 cent coins and 2 are 10 cent coins.
Max amount = 10*5 + 2*10 = 70 cents
5c, 10c, 15c (a 10c and a 5c), 20c (2 10c coins), ... for all other sums, just keep adding 5.

Similarly, if we have 17 distinct sums, it means the maximum amount is 85 cents (so 17 distinct sums will be 5c, 10c, 15c ... 85c)
So, if x is the number of 5 cent coins and y is the number of 10 cent coins, 
x + y = 12
5x + 10y = 85

Solve to get x = 7 and y = 5

Answer (C)

Method 2:

Use the options. Since number of 10 cent coins must be between 3 to 7, there are a mix of both type of coins. So this means we can make every multiple of 5 with the coins we have till the sum of all the coins together. 
Say even if we had only one coin of 5 cent, we would be able to make 5, 10, 15, 20, 25, 30, 35.... etc all.

So let's look at the options to see what gives us 17 combinations. I tried (C) first because it is right in the middle. It will tell me whether to go up or down.

5 cent coins - 7
10 cent coins - 5
Total value = 5*7 + 10*5 = 85 = 5 * 17
So here we have 17 distinct values: 5, 10, 15, 20, 25, ... 85

Answer (C)

Had we got more than 17 values here, we would have tried 3 or 4 and would have got our answer either way. 
Had we got fewer than 17 values here, we would have tried 6 or 7 and would have got our answer either way.

Shobhit7 · Answer

17 different combinations means: 5, 10, 15... up to 85.

Now, with 12 coins, we can create 85c as : 10c*5 + 5c*7

So, Total 10c coins: 5

Ans C

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agar123 · Answer

Hi,

Can you please explain in detail how you solved this?

ManjariMishra · Answer

chetan2u   Bunuel   VeritasKarishma  Can you please explain this .What 17 different combinations infer here ?

nick1816 · Answer

We can make 17(>12) different values. She must have atleast 1 5-cent coin; hence she can obtain all the multiples of 5 less than or equal to 5x+10y, where x is number of 5-cent coins and y is number of 10-cent coins.

x= \frac{120-17*5}{5}= 7

y=12-7=5

VIVA1060 · Answer

Hi  nick1816 , How do we get 120? What is that?

Fdambro294 · Answer

She has 12 coins: some are 5 cents and some are 10 cents

Any particular combination of a 10 cent coin or a 5 cent coin will always give us a multiple of 5.

Because we are allowed to make every unique combination of coins that is possible, it should be the case that we can make every multiple of 5 within the following range ————->

From 1 coin of 5 cents

To choosing all 12 coins to get the maximum SUM

We are told there are 17 unique totals that can be made.

The first 17 multiples of 5 are from:

The correct amount of 5 cent coins and 10 cent coins should give us a MAX SUM Total of 85 cents when we Choose All 12 coins

every other multiple of 5 prior to 85 will be able to be made as a SUM through choosing some combination of 5 cent and 10 cent coins

Thus, we are looking for which combination of 12 coins will give us a MAX total of 85 cents when we choose all 12 of them and add them up?

(C) Ten cent coins = 5

Which means Five cent coins = 7

If we choose all 12 coins, the MAX SUM total will be:

(5 coins) (10 cents) + (7 coins) (5 cents) =

50 + 35 = 85 cents

This is the amount we would get on our 17th final unique combination when we choose all the coins. we can not create any higher SUM and we should be able to create every other Multiple of 5 prior to 85 by choosing some combination of Five Cent and Ten Cent coins.

This will result in 17 possible Unique Sums: 5, 10, 15......... all the way to MAX 85

Thus,

(C) 5

Posted from my mobile device

ParitoshRawat99 · Answer

Answer = 5 (OPTION C)

First we need to find the 17 combinations which would be there:
They would be 5,10,15,20,25,30,35...........till n=17
N(17)= 5 + (17-1)5
N(17)= 5 + 80 = 85

now we know that 85 is the 17th unique combination made of 5's and 10's
Thus 85=5x+10y ---------------1
we also know x+y=12 ------------2

By solving the two equations we get
y=5
x=7

Rohan271 · Answer

Just one doubt in this: Why only multiples of 5 is being considered?

HarshavardhanR · Answer

https://gmatclub.com/forum/download/file.php?id=83583 Hi Rohan271 , Given that the coins are either 5 cent or 10 cent, the possible values obtained by addition can only be multiples of 5. For instance, can any combination of 5-cent and/or 10 cent coins give us a value of 17? Mathematically not possible. Above is a simpler example to understand what is happening. For this question Let the number of 10 cent coins be x. Then, the number of 5 cent coins = 12 - x. Now, we are given that there are 17 unique values that can be reached by taking various combinations of one or more coins. These values are: 5, 10, 15, 20,...............(max possible value i.e., counting all coins) In other words: 5, 10, 15, 20,....... 10x + 5 (12-x) 5, 10, 15,..............60 + 5x We are given that the number of terms of the above = 17. 60 + 5x = 5 + (17-1)5 => x = 5. Thus, the number of 10-cent coins = 5. Hope this helps! Harsha GMAT-Club-Forum-7i36g5qq.png

Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin

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