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alex.gellatly: Master | Next Rank: 500 Posts; Posts: 435; Joined: Wed Nov 16, 2011 7:27 am; Thanked: 48 times; Followed by:16 members

Is there a faster way>

by alex.gellatly » Sat Jun 23, 2012 7:16 pm

Billy has an unlimited supply of the following coins: pennies (1Â¢), nickels (5Â¢), dimes (10Â¢), quarters (25Â¢), and half-dollars (50Â¢). On Monday, Billy bought one candy for less than a dollar and paid for it with exactly four coins (i.e., he received no change). On Tuesday, he bought two of the same candy and again paid with exactly four coins. On Wednesday, he bought three of the candies, on Thursday four of the candies, and on Friday five of the candies; each day he was able to pay with exactly four coins. Which of the following could be the price of one candy in cents?

8Â¢
13Â¢
40Â¢
53Â¢
66Â¢

I solved this the very slow way of checking each answer choice.... is there a faster way?
Thanks

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Source: Beat The GMAT — Problem Solving | Sat Jun 23, 2012 7:16 pm

GMAT/MBA Expert

Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Sat Jun 23, 2012 9:43 pm

alex.gellatly wrote:Billy has an unlimited supply of the following coins: pennies (1Â¢), nickels (5Â¢), dimes (10Â¢), quarters (25Â¢), and half-dollars (50Â¢). On Monday, Billy bought one candy for less than a dollar and paid for it with exactly four coins (i.e., he received no change). On Tuesday, he bought two of the same candy and again paid with exactly four coins. On Wednesday, he bought three of the candies, on Thursday four of the candies, and on Friday five of the candies; each day he was able to pay with exactly four coins. Which of the following could be the price of one candy in cents?

Note that values of the all of the coins are multiple of 5 except the pennies. Hence, if the total price of the candies is of the form (5n + 4), then we need 4 pennies alone to cover the remainder of 4. Hence, the price of one or two or three or four or five candies cannot be of the form (5n + 4)

Let us check the options accordingly,

A. 8 --> 3*8 = 24 = (4*5 + 4) --> NO
B. 13 --> 3*13 = 39 = (7*5 + 4) --> NO
A. 40 --> All multiples are divisible by 5 --> YES
A. 53 --> 3*53 = 159 = (31*5 + 4) --> NO
A. 66 --> 4*66 = (52*5 + 4) --> NO

The correct answer is C.

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passionatepursuer!!: Newbie | Next Rank: 10 Posts; Posts: 2; Joined: Sat Jul 07, 2012 8:58 am

by passionatepursuer!! » Sun Jul 08, 2012 5:27 am

Hi Anurag,

Thanks for your post.
I have been able to solve it by plugging in options on equation however need quick solution. I don't quite understand the solution below. Please elaborate how you derived equation '5n+4' and why you multiplied each option to get '5n+4'?

Anurag@Gurome wrote:
alex.gellatly wrote:Billy has an unlimited supply of the following coins: pennies (1Â¢), nickels (5Â¢), dimes (10Â¢), quarters (25Â¢), and half-dollars (50Â¢). On Monday, Billy bought one candy for less than a dollar and paid for it with exactly four coins (i.e., he received no change). On Tuesday, he bought two of the same candy and again paid with exactly four coins. On Wednesday, he bought three of the candies, on Thursday four of the candies, and on Friday five of the candies; each day he was able to pay with exactly four coins. Which of the following could be the price of one candy in cents?
Note that values of the all of the coins are multiple of 5 except the pennies. Hence, if the total price of the candies is of the form (5n + 4), then we need 4 pennies alone to cover the remainder of 4. Hence, the price of one or two or three or four or five candies cannot be of the form (5n + 4)

Let us check the options accordingly,
A. 8 --> 3*8 = 24 = (4*5 + 4) --> NO
B. 13 --> 3*13 = 39 = (7*5 + 4) --> NO
A. 40 --> All multiples are divisible by 5 --> YES
A. 53 --> 3*53 = 159 = (31*5 + 4) --> NO
A. 66 --> 4*66 = (52*5 + 4) --> NO
The correct answer is C.

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Sun Jul 08, 2012 6:18 am

passionatepursuer!! wrote:... I don't quite understand the solution below. Please elaborate how you derived equation '5n+4' and why you multiplied each option to get '5n+4'?

Values of the all of the coins are multiples of 5 except the pennies.
Hence, if the total price of the candies is 4 more than multiple of 5 (like 14, 29, 44 etc), i.e. of the form (5n + 4) for any non-negative integer n, then we need 4 pennies alone to cover the remainder of 4.

Hence, the price of one or two or three or four or five candies cannot be 4 more than a multiple of 5, i.e. of the form (5n + 4)

So our job is to check whether the given options satisfy this condition.
For example, for option A, price of one candy is 8 cents.
Hence, price of three candies is 3*8 = 24 = 4 more than 20 --> Not possible
This is because we cannot pay 24 cents with any three of the given coins.

Hope that helps.