Yvonne purchased an item and the total cost, including tax, was less.

Question

Yvonne purchased an item and the total cost, including tax, was less than $1. She gave the clerk a $1 bill and received the correct change, which consisted of exactly 4 coins, each with a value of $0.01, $0.05, $0.10 or $0.25. What is the total value of the change that Yvonne received?

(1) One of the 4 coins that Yvonne received had a value of $0.01.
(2) The correct change could have consisted of exactly 2 coins, each with a value of $0.01, $0.05, $0.10 or $0.25.

Show Spoiler

Attachment:

GMAT-Club-Forum-vcef8ch5.png [ 198.74 KiB | Viewed 7380 times ]

chetan2u · Accepted Answer

I would say it is slightly tricky and more of logic than any calculation.

To ease the calculation, I would multiply each by 100 or I would deal in cents.
So, the amounts are 1, 5, 10 and 25, and the total cost is less than 100.

(1) One of the 4 coins that Yvonne received had a value of $0.01 or 1.
Many possibilities.
Insufficient

(2) The correct change could have consisted of exactly 2 coins, each with a value of 1, 5, 10 or 25($0.01, $0.05, $0.10 or $0.25).
let us analyze the statements
 The total cost could be paid in  TWO  coins or FOUR coins. 
Ways: 
(a) Possibility of each of  two  coins being divisible in two parts, for example 10+10=(5+5)+(5+5)
(b) Possibility of one of the  two  coins being divisible in three parts and the second remaining, for example 25+10=(10+10+5)+(10)
Insufficient

Combined
as 1 can not be divided into two parts, we have to have the second of  two  coins being divided in three parts.
Now, none of 1, 5 or 10 can be divided in three parts with the lower value coins. Thus, the other coin is 25.
Total is 26 or ($0.01 + $0.25) = $0.26 = $0.01 + $0.05 + $0.10 + $0.10
Sufficient

C

Melisma_Behera · Answer

I marked and approached this question incorrectly in the mock, and came for looking answers and realised how to approach it. Anyways, here is my take on it :

(1) Insufficient as the values of other three coins can range from anything to anything.

(2) Okay lets see, we have 4 values - 0.01, 0.05, 0.10, 0.25. So the possible sum of 2 values: 0.26 (0.25 + 0.01), 0.11, 0.15, 0.06, 0.30, 0.35. Now values which can be represented as a sum of 4 values received by Yvonne - 0.26 (0.01+0.10+0.10+0.5), 0.30(0.10+0.10+0.5+0.5) and 0.35 (0.10+0.10+0.10+0.5).

Three possible combinations, so insufficient.

(1)+(2)

One value has to be 0.01 and that is only found in the sum of 0.26, we cant have the value in 0.30 or 0.35 and make the sum as required.

Answer - C

Please let me know if I did any mistake or missed out on something because this is my first time replying to a question on the gmat forum. Thank You!

MrWhite · Answer

Update #2: Refer to Chetan's answer below. My approach is incorrect but decided to leave it here so that one can read this question thread in full and get the most out of this question

Update: after spending some time splitting hair over this I think I might have a solution: Clearly 1) & 2) alone are insufficient. Taking them together: Following on from 1) 0.01 must be there and 2) Sum of 4 coins is same as sum of 2 other coin combinations So our desired sum must be either 0.02, 0.06, 0.11 or 0.26 to satisfy both conditions. "Clearly" of the 4 given coin denominations, achieving a sum of 0.02, 0.06, 0.11 using 4 coins in any combination is not possible. 0.26 = 0.01 + 0.25 = 0.10 + 0.10 + 0.05 + 0.01 The only possible variation which satisfies both conditions. Is this question this tough or is there an infinitely better way to approach this problem 😅 I am still not fully sure whether this is a robust solution. Posted from my mobile device

MrWhite · Answer

Thanks Chetan!! Having you weigh in on this really provided the much needed reassurance.

What are the odds your answer came in while I was literally editing my reply to include my answer hahaha

Posted from my mobile device

chetan2u · Answer

I believe your editing finished 15 minutes later, so there is a probability of you editing while answer was given.

Just a point on your solution. You have taken that the value given in statement 1 as one of the TWO coins (given in statement 2) straight forward which may not be correct. Had the statement 1 said that one of the coin was $0.05, then you would have taken total as 0.06, 0.10, 0.15 and 0.30. With these four sums, your observations would be that only 0.30 is possible as 0.10+0.10+0.05+0.05 and write C as answer. But this misses out on 0.35=0.10+0.10+0.10+0.5, thereby making E as the answer.

pranjalshah · Answer

My approach was slightly different. For (1) and (2) together:

The correct change could have consisted of exactly 2 coins, each with a value of 1, 5, 10 or 25 . So out of all possible combinations we can make, they'll be only of 2 types: either the sum will be a multiple of 5 or not a multiple of 5. For example, they can be (25,25) or (10,5) or (10,10) (sum is a multiple of 5) or (1,5), (1,10) (non-multiple of 5). We leave this here for now.

We also know that out of the 4 actual coins, one of them is 1. So for the remaining 3, we can see that there is no possible combination such that the final sum would be a multiple of 5. For example, (1,5,5,10), (1,1,10,25), (1,1,1,10). There's no way we can have a multiple of 5 as the sum.

Back to the previous finding: Since the sum is not a multiple of 5, therefore one of the coins out of the two has to be 1 (or else the sum would be a multiple of 5).

After we find this, we can make all possible combinations: (1,1) = 2, (1,5) = 6, (1,10) = 11, (1,25) = 26
Out of these four, only one is possible, i.e. (1,25) -> (1,5,10,10)

Sorry for the poor formatting.

KarishmaB · Answer

Responding to a pm: 4 coins each of - 1, 5, 10 or 25 cents Question : What is the total sum of all 4 coins? (1) One of the 4 coins that Yvonne received had a value of $0.01. No info on other 3 coins. They could be anything. Not sufficient alone (2) The correct change could have consisted of exactly 2 coins, each with a value of $0.01, $0.05, $0.10 or $0.25. The sum could be 20 cents which can be made up by 2 coins (two 10 cent coins) or by 4 coins (4 coins of 5 cents each) The sum could be 50 cents which can be made up by 2 coins (two 25 cent coins) or by 4 coins (20 + 20 + 5 + 5 cents) Not sufficient alone Using both statements - we must have at least one 1 cent coin. It means the sum cannot be a multiple of 5. So the sum cannot be x5 or x0. With the 2 coins, one of the coins must be 1 cent coin too since the sum cannot be a multiple of 5. If we do not use a 1 cent coin, the sum will be a multiple of 5 since we will add two multiples of 5 (e.g. 5 + 10 = 15) We cannot represent 2 cents (1+1) or 6 cents (1+5) or 11 cents (1+10) in 4 coins but we can represent 26 cents (1+25) 26 = (1 + 5 + 10 + 10) Answer (C) GMAT-Club-Forum-qk0mf6ci.png

Sudeep16011995 · Answer

Thank You so much for the easiest explanation. Kudos!

VatsalParikh · Answer

I don't whether I was the only one but I found the wordings of the question to be very confusing. At first, it seemed like he has 4 coins each of the 4 denominations mentioned. There should have been "either" mentioned in the quastion right? either 0.01,0.05, 0.1 or 0.25.

HarshavardhanR · Answer

https://gmatclub.com/forum/download/file.php?id=81413 (1) We can quickly see that S1 is insufficient. Multiple combinations, multiple values. (2) I am not sure there is a neat formulaic way to extract every possible value here (there may be). As far as Statement 2 is concerned, quick examples can tell us that multiple values are possible (in my case, I observed 20 and 30). (1) and (2) - is where things get interesting. # If a>=1, the value of the change, a + 5b + 10c + 25d cannot end with 0 or 5. Because -> the other terms (exc. a) will end with 0 or 5 ->the sum of these other terms will end with 0 or 5 -> then, along with a, the total sum cannot in any scenario end with 0 or 5. # To get a sum that does not end with 0 or 5, from 2 coins, one of those coins has to be 1. Now, our job is easy. We just have to check for all possible 2-coin combinations that contain 1 -> can the same value be obtained from 4 coins? 1) 1 + 1 = 2. Impossible with 4 coins. 2) 1 + 5 = 6. Again, impossible with 4 coins. 3) 1+10 = 11. Again, impossible with 4 coins. 4) 1+25 = 26. This is possible. 1(1) + 5(1) + 10(2) Value of change has only one possibility here -> 26. Thus, using statements 1 and 2 together, we are able to determine what is needed. Thus, choice C . --- Harsha GMAT-Club-Forum-e6kb84q3.png

shisingh · Answer

The questions is worded in a very confusing manner. it seems that person had EXACTLY 4 (implied distinct )coins as change and then option 2 says COULD HAVE 2 coins

Usernamevisible · Answer

Read the question carefully. The actual change consists of exactly 4 coins, and each coin must be either 1, 5, 10, or 25 cents. The goal is to determine the total value.

---

Statement (1): One of the coins is a 1-cent coin.

The remaining three coins can still be any allowed denominations, leading to many possible totals. Therefore, statement (1) alone is insufficient.

Statement (2): The same total value can also be made using exactly 2 coins.

Do not think about specific 4-coin combinations yet. Instead, think about possible totals that have both a 2-coin representation and a 4-coin representation. For example, 20 can be made as 10 + 10 or as 5 + 5 + 5 + 5. Likewise, 30 can be made as 25 + 5 or as 10 + 10 + 5 + 5. Since multiple totals satisfy this condition, statement (2) alone is insufficient.

---

Now combine both statements.

From statement (1), the actual 4-coin combination contains a 1-cent coin. Therefore, its total cannot be a multiple of 5, because adding a single 1-cent coin to the remaining coins (which are all multiples of 5) produces a total ending in 1 or 6, not 0 or 5.

Hence, in the 2-coin representation from statement (2), one of the two coins must also be a 1-cent coin. The possible 2-coin totals are 1 + 1 = 2, 1 + 5 = 6, 1 + 10 = 11, and 1 + 25 = 26.

Now check which of these can also be formed using exactly 4 allowed coins. Total 2, 6, and 11 cannot be made with exactly 4 coins. However, 26 can be made as 10 + 10 + 5 + 1.

Since only one total survives, the answer is C.

---

Transferable learning:

Whenever a Data Sufficiency question says the same value can be represented in different ways, avoid generating random combinations. First, identify every possible representation implied by one statement. Then use the second statement to eliminate possibilities. If only one value remains, the statements together are sufficient. This approach is much faster and more reliable than brute-forcing coin combinations.

Yvonne purchased an item and the total cost, including tax, was less.

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

GMAT Data Sufficiency (DS) Questions

Yvonne purchased an item and the total cost, including tax, was less.

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia