Can you solve this without using algebra?

Maybe this is the difference between American English and British English, but you’ve lost me with this sentence: ” You have sufficient change in coins to buy the item in addition to various dollar denominations including a one dollar bill.” What does this mean?

One dollar?

Seems like however you’re thinking that through, it’s approximately the same method as using algebra.

@downtide , What I mean is that you could purchase the item just using coins, whose denominations are all less than a dollar . You also have paper money in various whole dollar amounts, including a one dollar bill.

@PhiNotPi , Yes the answer is a dollar. Can you explain why without using algebra?

@dabbler , Since algebra is just an abstraction of arithmetic, the arithmetic and algebraic explanations will have a similarity. What I am looking for is a very simple explanation that could, in principle, be understood by someone with no exposure to algebra.

A dollar because the change from the dollar plus the coins you already have will equal a dollar since you already had enough change to buy the item to begin with. If I understand the question right.

It’s not one dollar in coins, it’s 99 cents. The item costs less than one dollar.

So what is the least number of coins one must have to reach any combination up to 99 cents?

For 99 cents you need 3 quarters, 2 dimes, 4 four pennies.

For 94 cents you need 3 quarters, 1 dime, 1 nickel, and four pennies.

So for any price from 1 cent to 99 cents, you need a combination of 3 quarters, two dimes, one nickel, and four pennies. 10 coins, totaling $1.04.

https://googledrive.com/host/0B7lnqr4vhqgXSXYxdXNrRHZQMjA/dollar.png

One dollar, because buying something with a bill is the same as changing that bill for coins and then buying the item with those coins.

So when you buy something with the bill, it is per definition one dollar more, because you add 1 dollar in change to your collection of coins and then buy the item from that collection.

this is of course only true for the smallest denomination of the bill. $1 in casu, but $2 in Singapore, for instance.

Reread and understand the question…

I made the false assumption of determining the number of coins to pay exactly for the item. But the way the question is worded, one only needs one coin – a $1 coin, to buy the item. And the change from a $1 coin will be the same as if you paid with a $1 bill.

But no matter if you use variables in your head to solve this or not, you are using algebra rules in the end, since algebra is not something invented; only the way to express in algrebra is. I mean, no matter if one uses χ, ψ or whatever variable in their head to solve this or not, it is always algebric calculations in the end. The symbols are just words of a language. Or am I understanding this question wrong?

I think my explanation above doesn’t involve Algebra.

@whitenoise I am sorry but I think it does. Here is what you say:
[Money in coins]=[money in notes]
[Money in notes]= 1 => [money in coins]= 1
Thus the answer is 1.
You just used words instead of symbols but the rules used are algebra still.

That’s what I mean. Algebra describes facts. It doesn’t matter if you use words or symbols, in the end you use the same rules. In the intro to algebra at second grade this is what we learn first: how to translate words to symbols. Second graders solve simple problems here without using any χ or ψ (or x and y if you like) but by using words to describe the algebric rules. This does not mean it is not algebra anymore (in my opinion).

@whitenoise‘s explanation is what I had in mind. This is my version of it. In both cases the total amount of money after the transaction is the same. If you pay with a dollar, you will end up with a dollar less in paper money, which must be compensated for in coins.

@kounoupi , I would say that algebra involves the use of symbols to stand for unknown quantities. This is a major leap from simple arithmetic reasoning.

@kounoupi
Nope, that’s not what I said. That is an attempted translation into algrbra of what I said. That’s not the same.

Actually, with the exact wording of the question, you cannot answer it without knowing the cost of the item to be purchased. It asks, “How much more money in coins will you have if you pay using the dollar bill than if you paid using the coins?” That would be the difference between the price of the item and $1.00. So if the item costs $0.01 you would have $0.99 more, while a item costing $0.99 would net you $0.01 more. The answer of $1.00 is a correct answer to, “How much total money in coins will you have if you pay using the dollar bill than if you paid using the coins?, and not to “How much more money in coins will you have…”

@ETpro
I think you are mistaken.

@whitenoise Show me how.

@ETpro

See my answer above. It is about the amount of money in coins that you have more, when you pay with the 1 dollar bill. (Versus when you pay in coins.)

To know the total amount in coins you need to know how much you had in coins, to begin with.

@ETpro , I am not following your logic. To use your example, if you you start with 1 cent and use it to pay for the item, you end up without any coins. If you pay using the dollar, you end up with 99 cents in change plus the 1 cent you started with, which is a dollar more in coins than not having any coins.

For reasoning to be algebraic, you have to solve the problem working backwards from what is known. If I said that twice a number plus 3 is 15 and you subtract 3 and divide by 2, then that would be algebraic reasoning. @whitenoise is using forward reasoning, so I would not classify it as algebraic.

The question is “How much more money in coins will you have if you pay using the dollar bill than if you paid using the coins?” The answer to that would vary depending on the cost of the item. It’s very, very simple English. No algebra skills required.

The answer does not vary. It is always a dollar.

To do this algebraically:
Let v be the cost (value) of the item and c be the amount of change that you have in coins, assuming that c >= v.
case 1: You pay using the coins. The amount in coins that you end up with is c – v.
case 2: You pay using a dollar bill. You get back 1 – v in change. The total that you now have in coins is c + 1 – v.

The difference between the two amounts is (c + 1 – v) – (c-v) = 1

My non-algebraic way of explaining this is as follows.
If you pay using a dollar, you have a dollar less in paper money than if you paid using coins. Since the total amount that you end up with is the same in both cases, the dollar less in paper money must compensated for by an extra dollar in coins.

Note that one immediate consequence of this is that if you have the exact change and you pay with a dollar, you end up with at least a dollar in change. @whitenoise ‘s approach makes this even more obvious.

A similar result follows for avoiding an accumulation of pennies. If you use a penny to, for example, pay for something that costs so many dollars and 6 cents, you will end up with 5 fewer pennies than if you paid without using the penny.

Dear ET,

The core of the issue is that you buy something with the smallest denomination in notes and the item costs less than that note.

Now two options:
1) you pay with the coins you have.
2) you pay with the 1 dollar note.

Option 2 is the same as changing your 1 dollar bill into coins (adding 1 dollar in coins to your stash of coins) and then pay the item in coins.

Option 2 therefore always results in exactly 1 dollar more in coins.

I can see how you have exactly a dollar more in coins after you have made change for your bill but then you pay for the item and reduce your coin stock. And I agree we don’t know how much is paid for the item so we can’t tell how much value in coins is ultimately left.

@LostInParadise & @whitenoise I can see how you have exactly $1.00 in coins as well. But read the question. If necessary, read it over and ovef until you understand what it is asking. It is NOT asking how much you have left in coins. It is asking “HOW MUCH MORE MONEY IN COINS will you have…” MORE MONEY IN COINS means more than you had before the transaction, not the total after the transaction.

If you had @0.01 before the transaction and the item cost $0.99, you pay in a $1.00 bill and you would get back $0.99 MORE in coins. If you had $0.99 in coins and the item cost $0.99, you pay with a $1.00 bill and you would get back $0.00 MORE in coins. I’m pretty sure $0.98 and $0.00 are not the same amount, nor are they equal to $1.00.

The OP meant to ask, “how much in TOTAL will you have in coins?” That is not what the OP wrote, though.

@ETpro It’s not asking how much in coins will be added to your total in coins.

It’s asking how much more you’ll have in coins than you would have in coins if you had paid with only your coins.

Recall that the question states that you have enough to pay in coins, but you choose to use the dollar bill instead.

The answer is one dollar.

@dabbler
In both cases you would pay from your coin stack. Therefore the fact that you pay doesn’t matter on the remaining amount. Therefore you know that you will have 1 dollar more.

Think about it this way: in any financial transaction, it doesn’t matter which way I pay, because in the end I will spend the same amount of money regardless of payment method. Also, my total savings will be the same regardless of payment method. (Ignoring transaction fees, they are out of scope)

So, let’s say that wanted to purchase a small item worth a few cents, and my wallet consisted of two things: a $1 bill, and exactly the right number of coins.

In situation A, I could pay with the coins, in which case I would have no coins left and a one dollar bill. I have $1 in total.

In situation B, I could pay with the one dollar bill, and I would receive some change from the cashier. Look back at the principle at the top, and you will see that I must now also have a total of $1 remaining. Now, that $1 must be in coins, since I spent my only dollar bill (and the cashier only gave me coins in return).

In both situations, I end up with a total of $1 in savings. In the first case, however, I have $0 of coins, while I have $1 of coins in the second case. The question wants to know the difference between the two, which is one dollar.

@PhiNotPi , That is similar to the way I saw it. It does simplify things to consider the case where the total that you have in change is exactly the cost of the item.

@PhiNotPi In the scenario you lay out, I agree you end up with $1.00. However, the question as worded is not asking what you end up with. It asks how much MORE will you have in coins.That’s asking for a difference between your coins before and after the transaction, not a sum. Let’s assume the item sells for $0.05 and you have a nickel and a $1.00 bill. If you buy the item with the nickle, you will have $0.00 in coins, so you will have NO MORE in coins. In fact, you will have less. If you buy it with the $1.00 bill, you will have $0,95 MORE in coins.

Again, I know what @LostInParadise was trying to ask, but it’s not worded to get at the sought-after conclusion of $1,00. A+ on math but back to the basics for the two of you on basic English.

@ETpro
Please…. Reread and edit your answer…. Please…

It is not asking about the difference in coins between before and after the transaction!!!!

pi is asking about the difference in outcome between two scenarios: 1) you pay in coins 2) you pay with a 1$ bill.

@whitenoise “How much more money in coins will you have…” That means more than you already have.

Lat’s say you have $1000 in savings and you take a job that pays you $500. How much more money do you have after being paid for the job? The answer is $500, not $1500.

You left out the most important part of the sentence. The whole sentence is:

“How much more money in coins will you have if you pay using the dollar bill than if you paid using the coins?”

That is: “How much more A than if B.

In this case B doess not stand for “after the transaction”, but for “using a bill instead of the coins”.

@ETpro
Please come over here (I’m in Hong Kong now) so I can shake you by the ears….

@whitenoise I’m smart enough to know you’re not reading the OP correctly, and I’m way too smart to travel halfway around the world at my own expense so somebody can assault me. :-)

“You have sufficient change in coins to buy the item in addition to various dollar denominations including a one dollar bill. How much more money in coins will you have if you pay using the dollar bill than if you paid using the coins?”

The OP says a little experimentation, so let’s experiment.
I have 1 penny and $1.

The item costs 1 cent.
If I pay with my coin, I have 0 more money in coins.
If I pay with the bill, I have $0.99 more in coins.

Now let’s say I have three quarters, two dimes and four pennies plus my dollar bill.
Let’s say the item costs $0.99.
If I pay with my coins, I have 0 more money in coins, same as above.
If I pay with the bill, I have $0.01 more in coins.
, nowhere close to the same as the first test.

We can substitute other costs for the item and the results will always vary based on the value we assign to the item. They will not always be the same. What will always be the same is the total amount in coins we have after the transaction if we pay with the dollar bill. But that simply is not what the question, as worded, asks.

@ETpro
Sorry to say that you prove not smart enough. :-/

There have been enough explanations offered. You either willingly not read those or you are not smart enough.

Let’s postpone discussing this further, until you’re willing to address the possibility of your being mistaken.

Can we call a truce on this? At the very least I have to concede that I could have done a better job of presenting the question. My natural tendency is to be as terse as possible, which may not be the best way of being understood. What I take away from this is that my questions should be presented in such a way as to make it easy to understand what I am asking. The work done by the reader should be in answering the question, not in figuring out what is being asked.

@whitenoise I’d ask you to consider the same possibility.

@LostInParadise Agreed. Thanks for an interesting and engaging discussion.

@ETpro

I have proven again and again, on this forum, that my stupidity dwarfs my smartness.

You’re still mistaken though… :-)

@whitenoise I have done as the OP asked and experimented. You have not, except when ignoring the requirements of the OP. You have pretty much just told me I’m wrong. But when I experimented as requested by the OP, it verified what I was saying. Show me your experiments that actually duplicate what the OP asked and prove you case, and I will be glad to consider them and admit error if that’s what they indicate.

@ETpro you asked :-)

Experiment 1) item costs 23 cents on hand coins: $1.79

Paying with dollar bill: coins left = $2.56 ($1.79 + 77 cents change from the bill)
Paying with coins: coins left = $1.56 ($1.79 – 23 cents from the item)
Result: paying with bill leaves you with $1.00 more than paying with coins

Experiment 2) item costs 49 cents on hand coins: $1.22

Paying with dollar bill: coins left = $1.73 $1.22 + 51 cents change from the bill)
Paying with coins: coins left = $0.73 ($1.22 – 49 cents from the item)
Result: paying with bill leaves you with $1.00 more than paying with coins

Experiment 3) item costs 49 cents on hand coins: $1,500

Paying with dollar bill: coins left = $1,500.51 $1,500 + 51 cents change from the bill)
Paying with coins: coins left = $1,499.51 ($1,500 – 49 cents from the item)
Result: paying with bill leaves you with $1.00 more than paying with coins

So… Regardless of the amount of coins you have at the beginning, you end up with $1.00 more in coins, if you pay withe bill, than if you pay with the cojns you have on hand.

The reason I told you you were wrong wasn’t because I didn’t experiment, but because you were.

See here

@ETpro

Addressing your examples:

Example 1)
I have 1 penny and $1.
The item costs 1 cent.
If I pay with my coin, I have 0 more money in coins.
You have – (minus) 1 cent more in coins; you just spent one cent.

Example2
Now let’s say I have three quarters, two dimes and four pennies plus my dollar bill.
Let’s say the item costs $0.99.
If I pay with my coins, I have 0 more money in coins, same as above.

Again: you have – (minus) 99 cents more in coins; you just spent 99 cents.

@whitenoise You’re flailing around answering every question you can think of EXCEPT the one asked. It didn’t ask how much LESS you had if you paid with a coin, it asked how much MORE you have if you pay with a dollar bill. More means in addition to the coin you already had. Just leave the word “more” out of the OP, and then you get the result the OP wanted, that you always end up with the same amount.

Dear @ETpro,

I respect you very, very much. We will stop this discussion.

I understand what you are saying… Nevertheless I don’t agree with your interpretation. I explained why. (You have to look at the whole sentence of the question; the second half being a modifier to determine what ‘more’ is being referred to – not ‘more after the transaction’, but ‘more if you pay with the bill vs pay with the coins you have’.)

In all honesty I am totally frustrated by this situation. I feel your grass is blue and I have no way to proof to you it isn’t. After all ‘blue’ is what you see.

No harm intended. :-)

Good grief, @ETpro. Is it so difficult to simply admit that you misread the question?

@whitenoise I admire your patience.

Not trying to take sides but, personally, I read the OP the same way @ETpro has.
I found the OP ambiguous, and come to the same conclusions that the question is incorrectly written (and maybe was supposed to mean what several people are generously interpreting it to mean) or there is no definite answer.

… :-)

biting my tongue