# Math Help - Chairs and Stools

1. ## Chairs and Stools

The seats ‘R’ Us factory produces chairs with four legs and stools with three legs. The style of seats and legs are all the same. This year, if all the leftover legs were put with the leftover seats to make chairs, one seat would be left over without any legs. However, if all the leftover legs were put on the leftover seats to make stools, one leg would be left. How many more legs than seats are in the leftover inventory?

2. Originally Posted by aznmartinjai
The seats ‘R’ Us factory produces chairs with four legs and stools with three legs. The style of seats and legs are all the same. This year, if all the leftover legs were put with the leftover seats to make chairs, one seat would be left over without any legs. However, if all the leftover legs were put on the leftover seats to make stools, one leg would be left. How many more legs than seats are in the leftover inventory?
Hello,

let l be the number of legs
let c be the number of seats
let s be the number of stools

then you get the equations.

4(c-1) = l
3s + 1 = l

Eliminate l:

4c - 4 = 3s + 1 ===> c = (3/4)s + (5/4) = (1/4)(3s + 5)

c must be an integer; therefore you are looking for a value of s so that (3s+5) is a multiple of 4. The first possible value is s = 1 then c = 2 and you had 4 legs left over.

Make a table to see the pattern behind this problem:
Code:
      c       s        l
--------------------------
2       1        4
5       5        16
8       9        28
11      13       40
:       :       :
:       :       :
2+3n    1+4n     4+12n
EB

3. ## Alternative Solution to Chairs and Stools Problem

I would solve this in a similar manner:
If we make all chairs there will be one seat left over. How many legs go with a chair? 4

So the number of chairs that "L" legs can make is equal to L/4. If you have 16 legs, you have enough to make 4 chairs.

They tell us that we end up with 1 extra seat if we make chairs only, so we can say:

equation 1: L/4 = s-1

In words: we will use one less seat than S (the total number of seats) if all legs are used to make chairs.

We do the same thing if we are making stools only. There are 3 legs per stool. This one is a bit trickier, because instead of having one extra seat we have one extra leg. The number of legs you use to make stools, is one less than the total number of legs : L-1

We set it up the same way:

(L-1)/3 represents the number of stools we have legs to make. Since there is no surplus of seats in this case, the right side of the equation is simply "S".

Equation2: (L-1)/3 = S

Now you just need to solve the system of equations (Equation 1 and Equation 2)

Equation 1 becomes
L = 4s -4

Equation 2 becomes
L=3s+1

Set them equal to each other and you get s=5 or the number of seats is 5.

Substitute it back in and find the total number of legs - I'll let you do that on your own.

Remember that your final answer is the difference between the total number of legs and the total number of seats.