# Math Help - inequality word problem

1. ## inequality word problem

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.

2. ## Re: inequality word problem

Originally Posted by pyreof88
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?
When authors use the word 'between' there is general agreement that is is strict.
So if $N$ is the number of marbles then $40.

You want to find two positive integers $j~\&~k$ such that $3j+2=N=5j+4$.

3. ## Re: inequality word problem

Originally Posted by pyreof88
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.
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.

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.
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.
"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?

4. ## Re: inequality word problem

Please thank you for your contributions to this post, being busy I need time to put this information together. I was so dragged into the inequality that I couldn't find the connection with the equation.

5. ## Re: inequality word problem

Originally Posted by pyreof88
Please thank you for your contributions to this post, being busy I need time to put this information together. I was so dragged into the inequality that I couldn't find the connection with the equation.
\begin{align*} 40< &3j+2<50\\38&<3j< 48\\13&\le j \le 15\end{align*}
and
\begin{align*} 40< &5j+4<50\\36&<5j<46\\7&\le k\le 9\end{align*}

Now there is only one possible outcome in that. What is it?

6. ## Re: inequality word problem

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.

7. ## Re: inequality word problem

Originally Posted by Plato
\begin{align*} 40< &3j+2<50\\38&<3j< 48\\13&\le j \le 15\end{align*}
and
\begin{align*} 40< &5j+4<50\\36&<5j<46\\7&\le k\le 9\end{align*}

Now there is only one possible outcome in that. What is it?
.

You confused me, why did you round your pareameters?

8. ## Re: inequality word problem

I now know that the number of marbles is 44, but only by trial and error.