Hey community I'm having issues with the following inequality word problem:
A jar contains between 40 and 50 marbles. If the marbles are taken out of the jar three at a time, two marbles will be left in the jar. If the marbles are taken out the jar five at a time, four marbles will be left in the jar. How many marbles are in the jar?
Now I know that the number of marbles is between 40 & 50. Considering such a quantity and thinking logically I assumed the number of marbles in the jar would have to be 48 if there was only to left after after several deductions of a set of three. But then if deductions of 5 were made until there was a remainder of 4 left then there must be 49 marbles in the jar, right? I'm confused.
No, in addition to not being "logical", that is bad arithmetic! (Maybe just a typo.) 3 divides evenly into 48- there is no "two" left over. Multiples of 3 around 40 to 50 are 39, 42, 45, 48, and 51. Adding two to those we have possible values of 41, 44, and 47 actually between 40 and 50.
Let N be the number of marbles in the jar. "If the marbles are taken out of the jar three at a time, two marbles will be left in the jar." So N= 3k+ 2 for some integer k where "k" is the number of times we took out three marbles.But then if deductions of 5 were made until there was a remainder of 4 left then there must be 49 marbles in the jar, right? I'm confused.
"If the marbles are taken out the jar five at a time, four marbles will be left in the jar." So N= 5j+ 4 for some number j where "j" is the number of times we took out five marbles. Then we must have N= 3k+ 2= 5j+ 4 or 3k- 5j= 2.
Of course, 5- 3= 2 so one solution to that is k= -1, j= -1. Of course, k and j, here, have to be positive integers so that is not a valid answer. However, it is also true that k= -1+ 5m, j= -1+ 3m is a solution for any integer, m: 3(-1+ 5m)- 5(-1+ 3m)= -3+ 15m+ 5- 15m= 2 for all m. The total number of marbles in the jar was, remember, N= 3k+ 2= 5j+ 4, so N= 3(-1+ 5m)+ 2= 15m- 1.
Now, for what values of m is N between 40 and 50?
Damn, I'm back on thread. Sorry for delay but I'm also doing studies in electronics and building my first circuit. I erred at changing certain staements into algebraic statements. You're contributions have been of greatly help, it got me through this chapter.
Word problems kill me. Its just that sometimes I have a problem translating the statements in the word problem into proper algebraic expressions.