Question 1048965
let m equal the number of matchsticks.
let s equal the number of sachets.


first verbal statement:


if one matchstick is put into each sachet, there are 2 matchsticks left.


in order for this to occur, you have to have 2 more matchsticks than sachets.


your equation is therefore m = s + 2.


subtract s from both sides of this equation to get m - s = 2.


second verbal statement:


if 2 matchsticks are put in each sachet, there are 24 sachets left.


in order for this to occur, you have to have 24 more sachets than half the number of marbles.


the equation for this is therefore s = m/2 + 24.


subtract m/2 from both sides of this equation to get s - m/2 = 24.


another way these can be derived which is a little more difficult to understand is as follows:


when you put one matchstick into each sachet, you are left with 2 matchsticks.


this is like a division.


you are dividing the number of sachets by the number of matches and your result is 1 with a remainder of 2.


in a division, this is shown as s/m = 1 + 2/m


the remainder is really the fraction of what's left over divided by the divisor.


so you start with s/m = 1 + 2/m
then you multiply both sides of the equation by m to get s = m + 2
then you subtract m from both sides of the equation to gets - m = 2.


the second one is a little trickier but should give you the correct result if you do it right.


the statement says that if you put 2 matchsticks in each sachet, then you have 24 sachets left.


this means that is you divide the number of sachets by the number of marbles, your result will be 1/2 + a remainder of 24 sachets.


the algebraic equation for this would be s/m = 1/2 + 24/m.


multiply both sides of this equation by m and you get s = 1/2 * m + 24
subtract 1/2 * m from both sides of this equation to get s - 1/2 * m = 24
this can be shown as s - m/2 = 24.


from these two equations, you can solve for the number of sachets and the number of marbles.


the two equations are:


m - s = 2
s - m/2 = 24


solve for m in the first equation to get m = s + 2
replace m with s + 2 in the second equation to get:


s - (s+2)/2 = 24


multiply both sides of this equation by 2 to get 2s - (s+2) = 48


simplify to get 2s - s - 2 = 48


combine like terms to get s - 2 = 48


add 2 to both sides of this equation to get s = 50


since m = s + 2, then m = 52.


your solution is:


m = 52
s = 50


your first equation of m = s + 2 is satisfied because 52 = 50 + 2.


your second equation of s - m/2 = 24 is satisfied because 50 - 52/2 = 50 - 26 = 24.


both equations are satisfied so the value of m must be 52 and the value of s must be 50.


best i can do.


hopefully it makes sense to you.


first equations wasn't that hard to derive.


second equation took a little more thought.