Chairs and tables - Part 2
At a coffee shop that I visit, the tables each has only 1 leg, and the chairs each have 4 legs. If there are a total of 68 legs, and there are at most 4 chairs at each table, what is the most number of chairs in the coffee shop?
Note: Every chair must be at a table.
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10 solutions
Nice and simple solution, thank you!
It's really a good solution
I think you are really smart I just guessed because I couldn't be bothered to do math
Assuming that a chair must "belong" to a table, we need to maximize the ratio of chairs to tables. Since we can have 0 , 1 , 2 , 3 or 4 chairs at a table, the number of legs at a table/chairs "set" can be 1 , 5 , 9 , 1 3 or 1 7 .
So we need to solve the equation a + 5 b + 9 c + 1 3 d + 1 7 e = 6 8 with the goal of maximizing e then d and on down the line. But since 1 7 ∗ 4 = 6 8 we don't have to go that far, as this gives us a maximum of 1 6 chairs.
@Brian Charlesworth Could u explain the 2nd part of your solving method please? i solved this quickly but that's because of the option!
@Chung Kevin I like this question since it can easily be made more difficult by changing the total number of legs to a less "friendly" number. If there must be only 6 4 to 6 7 legs I'm getting maximums of 1 5 chairs. From 6 0 to 6 3 legs I'm getting a maximum of 1 4 chairs. The next variable would be to restrict the number of possible tables. I'm sure that coffee shop owners think about this sort of thing all the time. :P
You might want to specify that each chair must "belong" to a table, for as presently worded we could just ditch the tables and have all the legs belonging to chairs, in which case the answer would be 1 7 chairs. It would make for a lame coffee shop, (and perhaps some burnt laps), but some people might interpret it this way without any further clarification.
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Thanks Brian, I added that just in case.
There is a much easier solution which tells you directly that the maximum number of chairs is ⌊ 1 7 4 L ⌋ . This agrees with the values 60-63 and 64-67 that you stated.
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Ah, yes, good point. I guess I was thinking ahead to solving for the number of configurations that would yield each possible chair total, from the maximum down to zero, (all tables), rather than just focusing on the question at hand.
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@Brian Charlesworth – I was thinking about "is there some number of legs in which the configuration forces a different value?". As it turns out, in the case where the table has 1 leg, the answer is no, because we could always add more tables without consequence, to fill up the total number of legs.
It gets interesting when the table has more legs. For example, if the table has 3 legs and the chairs have 5 legs. Then, working out the case for 93 legs, a max table configuration has 3 + 4 × 5 = 2 3 legs, and so we might expect there to be 4 such configurations, or 16 chairs. However, as we can't have 1 table and 18 chairs, we must have 6 tables and 13 chairs!
I'm trying to see if there is an interesting question to be posed here. If I can't get at a deeper question, I might post this "simple" version.
14 chairs has 56 legs and 14 tables has 14 legs for a total of 70 legs but the total legs in the problem is only 68 ....if there are 16 chairs , that is 64 legs and 4 tables only ?
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Yes, there are 1 6 chairs, 4 at each of 4 tables, which results in 1 6 ∗ 4 + 4 ∗ 1 = 6 8 legs as specified. If there were 1 7 chairs then there would need to be another table to accommodate the 17th chair, so then there would be 1 7 ∗ 4 + 5 ∗ 1 = 7 3 legs, in excess of the given number of legs, so 1 6 is the maximum number of chairs.
x is the number of tables, y is the number of chairs. There are a total of 68 legs: x + 4 y = 6 8 → x = 6 8 − 4 y Because there are at most 4 chairs at each table: y ≤ 4 x → y ≤ 4 ( 6 8 − 4 y ) → y ≤ 1 6 Therefore the most number of chairs in the coffee shop is 16.
Very nice solution Loc Pham!
What about all the tables have only 1 chair? Or each table has a variety of chairs, ranging from 0 to 4. Your statement of the problem did not preclude those instances.
since a table must have 4 chairs, then 4 times 4 equals to 16. let's plus the number of legs the table has:16+1=17. 68 divided by 17=4. 4+1=5. 5-1=4 4*4=6. Im actually just 12 so don't blame for the a little bit blurry info. Sorry.
it is not true that "a table must have 4 chairs".
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I did almost the same thing (I didn't add 1 then less one from the quotient). It's true that it wasn't mentioned that a table must have 4 chairs, but since it is the maximum number of chairs a table can have, shouldn't you try to know how many tables have 4 chairs? It just happened that all the tables had 4 chairs in them.
The maximum amout of legs a group can have is 17 (four chairs each having four legs gives you 16 legs and the table has one leg so that gives ou a total of 17 legs per group) 68 / 17 is 4 meaning that you can have four of the groups sice that will give you 68 legs. If you can have four of those groups each with four chairs, ( 4 ) ( 4 ) = 1 6
There are 16 legs in total for the maximum number of chairs with a table. If we include the number of legs with the table, there are 17 legs .
Then from this we can work out the number of tables, which is 68 / 17 = 4 and if 68 is divisible by the amount of legs above, then we can assume each table has the maximum amount of chairs. Hence we multiply the amount of tables by the maximum amount of chairs for a table which is 4 x 4 = 16 Hence the answer is 16 chairs
How do we know that we cannot do better?
The question states that there is only one table. "At a coffee shop that I visit, the table has only 1 leg ..." (Both the noun and the verb are singular, making it unlikely that the intended phrase was "tables have".) Therefore, there can be at most 4 chairs. The rest of the legs are, as the accompanying photograph shows, on stools, at least one of which must have only 3 legs.
I edited the problem to indicate that each table has 1 leg.
In the photograph, the chairs have 4 legs.
14 * 4=56 -- it means there will be 14 legs remain and hence 14 tables. So 14 chairs and 14 tables not possible. 16 * 4=64 -- it means there will be 4 tables and 16 chairs with each table having 4 chairs. So it is right answer.
Every chair has 4legs.But there is at least a table with one leg.so,if there was 17 chair then total legs are 68.But there is at least one table with one leg.So,there is 16 chair.
4 tables ( 4 legs) and 1 6 chairs ( 6 4 legs) for the largest number of chairs in the coffe shop
How do we know that we can't do better?
in a group there are 4 chairs and 1 table.... It means that there are ( 4 ) ( 4 ) + 1 = 1 7 legs in each group. To determine the number of groups, 1 7 6 8 = 4 and the number of chairs is, ( 4 ) ( 4 ) = 1 6