MHB Find the total number of red and blue beads

  • Thread starter Thread starter anemone
  • Start date Start date
Click For Summary
The discussion revolves around a problem involving the transfer of red and blue beads between two containers. Container A starts with 500 beads, with 40% being blue, while Container B has 300 beads with 20% blue. Participants debate whether the problem can be solved without algebra, with some arguing that moving beads will not achieve the desired proportions in both containers. A method using visual representation is proposed, showing that transferring specific amounts can meet the required ratios. Ultimately, it is concluded that moving a total of 100 beads achieves the desired outcome.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
There are some red and blue beads.
The beads were packed into 2 containers.
At first, Container A contained 500 beads and $\dfrac{2}{5}$ of them were blue beads. Container B contained 300 beads and $\dfrac{1}{5}$ of them were blue.

Find the total number of red and blue beads that must be moved from Container A to Container B such that $\dfrac{3}{5}$ of the beads in Container A are red and $\dfrac{1}{4}$ of the beads in Container B are blue.

I am wondering if this problem can be solved without using any algebra method...
 
Mathematics news on Phys.org
Hi anemone,

Are you sure the problem is correctly stated ?

As I read it, you should still end up with $\dfrac25$ blue beads in container A. As the proportion does not change, you must move $3$ red beads for every $2$ blue.

However, this can only decrease the proportion of blue beads in container B, and it cannot increase from $\dfrac15$ to $\dfrac14$.

Did I miss something ?
 
Mathematics news on Phys.org
Initially, 60 of the 300 beads in container B are blue. After adding $5n$ beads, $2n$ of which are blue, the proportion will be $\dfrac{60+2n}{300+5n}$, which could increase towards a limit of $\dfrac25$ for large enough $n$.
 
castor28 said:
Hi anemone,

Are you sure the problem is correctly stated ?

Thanks for the reply, castor28! Yes, I checked again with the source from where I got the problem, there isn't any typo.

castor28 said:
As I read it, you should still end up with $\dfrac25$ blue beads in container A. As the proportion does not change, you must move $3$ red beads for every $2$ blue.

However, this can only decrease the proportion of blue beads in container B, and it cannot increase from $\dfrac15$ to $\dfrac14$.

Did I miss something ?
(Mmm) I actually don't see why not...(Crying)
 
The following method (without using algebra but the Singapore model method) works, by separating one part of fraction in container A into 5 equal subparts and one part of fraction in container B into 3 equal subparts, where each subpart represents 20 beads as shown below:

Before:
[TIKZ]
\coordinate[label=left: Container A] (A) at (0,5.3);
\filldraw [fill=cyan, thin, draw=black] (0.3,5) rectangle (0.6,6);
\filldraw [fill=cyan, thin, draw=black] (0.6,5) rectangle (0.9,6);
\filldraw [fill=cyan, thin, draw=black] (0.9,5) rectangle (1.2,6);
\filldraw [fill=cyan, thin, draw=black] (1.2,5) rectangle (1.5,6);
\filldraw [fill=cyan, thin, draw=black] (1.5,5) rectangle (1.8,6);
\filldraw [fill=cyan, thin, draw=black] (1.8,5) rectangle (2.1,6);
\filldraw [fill=cyan, thin, draw=black] (2.1,5) rectangle (2.4,6);
\filldraw [fill=cyan, thin, draw=black] (2.4,5) rectangle (2.7,6);
\filldraw [fill=cyan, thin, draw=black] (2.7,5) rectangle (3,6);
\filldraw [fill=cyan, thin, draw=black] (3,5) rectangle (3.3,6);
\filldraw [fill=magenta, thin, draw=black] (3.3,5) rectangle (3.6,6);
\filldraw [fill=magenta, thin, draw=black] (3.6,5) rectangle (3.9,6);
\filldraw [fill=magenta, thin, draw=black] (3.9,5) rectangle (4.2,6);
\filldraw [fill=magenta, thin, draw=black] (4.2,5) rectangle (4.5,6);
\filldraw [fill=magenta, thin, draw=black] (4.5,5) rectangle (4.8,6);
\filldraw [fill=magenta, thin, draw=black] (4.8,5) rectangle (5.1,6);
\filldraw [fill=magenta, thin, draw=black] (5.1,5) rectangle (5.4,6);
\filldraw [fill=magenta, thin, draw=black] (5.4,5) rectangle (5.7,6);
\filldraw [fill=magenta, thin, draw=black] (5.7,5) rectangle (6,6);
\filldraw [fill=magenta, thin, draw=black] (6,5) rectangle (6.3,6);
\filldraw [fill=magenta, thin, draw=black] (6.3,5) rectangle (6.6,6);
\filldraw [fill=magenta, thin, draw=black] (6.6,5) rectangle (6.9,6);
\filldraw [fill=magenta, thin, draw=black] (6.9,5) rectangle (7.2,6);
\filldraw [fill=magenta, thin, draw=black] (7.2,5) rectangle (7.5,6);
\filldraw [fill=magenta, thin, draw=black] (7.5,5) rectangle (7.8,6);
\draw [very thick] (0.3,5) rectangle (1.8,6);
\draw [very thick] (1.8,5) rectangle (3.3,6);
\draw [very thick] (3.3,5) rectangle (4.8,6);
\draw [very thick] (4.8,5) rectangle (6.3,6);
\draw [very thick] (6.3,5) rectangle (7.8,6);
\coordinate[label=left: Container B] (B) at (0,5.-1);
\filldraw [fill=cyan, thin, draw=black] (0.3,3.6) rectangle (0.6,4.6);
\filldraw [fill=cyan, thin, draw=black] (0.6,3.6) rectangle (0.9,4.6);
\filldraw [fill=cyan, thin, draw=black] (0.9,3.6) rectangle (1.2,4.6);
\filldraw [fill=magenta, thin, draw=black] (1.2,3.6) rectangle (1.5,4.6);
\filldraw [fill=magenta, thin, draw=black] (1.5,3.6) rectangle (1.8,4.6);
\filldraw [fill=magenta, thin, draw=black] (1.8,3.6) rectangle (2.1,4.6);
\filldraw [fill=magenta, thin, draw=black] (2.1,3.6) rectangle (2.4,4.6);
\filldraw [fill=magenta, thin, draw=black] (2.4,3.6) rectangle (2.7,4.6);
\filldraw [fill=magenta, thin, draw=black] (2.7,3.6) rectangle (3,4.6);
\filldraw [fill=magenta, thin, draw=black] (3,3.6) rectangle (3.3,4.6);
\filldraw [fill=magenta, thin, draw=black] (3.3,3.6) rectangle (3.6,4.6);
\filldraw [fill=magenta, thin, draw=black] (3.6,3.6) rectangle (3.9,4.6);
\filldraw [fill=magenta, thin, draw=black] (3.9,3.6) rectangle (4.2,4.6);
\filldraw [fill=magenta, thin, draw=black] (4.2,3.6) rectangle (4.5,4.6);
\filldraw [fill=magenta, thin, draw=black] (4.5,3.6) rectangle (4.8,4.6);
\draw [very thick] (0.3,3.6) rectangle (1.2,4.6);
\draw [very thick] (1.2,3.6) rectangle (2.1,4.6);
\draw [very thick] (2.1,3.6) rectangle (3,4.6);
\draw [very thick] (3,3.6) rectangle (3.9,4.6);
\draw [very thick] (3.9,3.6) rectangle (4.8,4.6);
\draw [<->] (0.3,6.2) -- (0.6,6.2);
\node at (0.4,6.5) {20 beads};
[/TIKZ]

After:
[TIKZ]
\coordinate[label=left: Container A] (A) at (0,5.3);
\filldraw [fill=cyan, thin, draw=black] (0.3,5) rectangle (0.6,6);
\filldraw [fill=cyan, thin, draw=black] (0.6,5) rectangle (0.9,6);
\filldraw [fill=cyan, thin, draw=black] (0.9,5) rectangle (1.2,6);
\filldraw [fill=cyan, thin, draw=black] (1.2,5) rectangle (1.5,6);
\filldraw [fill=cyan, thin, draw=black] (1.5,5) rectangle (1.8,6);
\filldraw [fill=cyan, thin, draw=black] (1.8,5) rectangle (2.1,6);
\filldraw [fill=cyan, thin, draw=black] (2.1,5) rectangle (2.4,6);
\filldraw [fill=cyan, thin, draw=black] (2.4,5) rectangle (2.7,6);
\filldraw [fill=cyan, thin, draw=black] (2.7,5) rectangle (3,6);
\filldraw [fill=cyan, thin, draw=black] (3,5) rectangle (3.3,6);
\filldraw [fill=magenta, thin, draw=black] (3.3,5) rectangle (3.6,6);
\filldraw [fill=magenta, thin, draw=black] (3.6,5) rectangle (3.9,6);
\filldraw [fill=magenta, thin, draw=black] (3.9,5) rectangle (4.2,6);
\filldraw [fill=magenta, thin, draw=black] (4.2,5) rectangle (4.5,6);
\filldraw [fill=magenta, thin, draw=black] (4.5,5) rectangle (4.8,6);
\filldraw [fill=magenta, thin, draw=black] (4.8,5) rectangle (5.1,6);
\filldraw [fill=magenta, thin, draw=black] (5.1,5) rectangle (5.4,6);
\filldraw [fill=magenta, thin, draw=black] (5.4,5) rectangle (5.7,6);
\filldraw [fill=magenta, thin, draw=black] (5.7,5) rectangle (6,6);
\filldraw [fill=magenta, thin, draw=black] (6,5) rectangle (6.3,6);
\filldraw [fill=magenta, thin, draw=black] (6.3,5) rectangle (6.6,6);
\filldraw [fill=magenta, thin, draw=black] (6.6,5) rectangle (6.9,6);
\filldraw [fill=magenta, thin, draw=black] (6.9,5) rectangle (7.2,6);
\filldraw [fill=magenta, thin, draw=black] (7.2,5) rectangle (7.5,6);
\filldraw [fill=magenta, thin, draw=black] (7.5,5) rectangle (7.8,6);
\draw [very thick] (0.3,5) rectangle (1.8,6);
\draw [very thick] (1.8,5) rectangle (3.3,6);
\draw [very thick] (3.3,5) rectangle (4.8,6);
\draw [very thick] (4.8,5) rectangle (6.3,6);
\draw [very thick] (6.3,5) rectangle (7.8,6);
\coordinate[label=left: Container B] (B) at (0,5.-1);
\filldraw [fill=cyan, thin, draw=black] (0.3,3.6) rectangle (0.6,4.6);
\filldraw [fill=cyan, thin, draw=black] (0.6,3.6) rectangle (0.9,4.6);
\filldraw [fill=cyan, thin, draw=black] (0.9,3.6) rectangle (1.2,4.6);
\filldraw [fill=magenta, thin, draw=black] (1.2,3.6) rectangle (1.5,4.6);
\filldraw [fill=magenta, thin, draw=black] (1.5,3.6) rectangle (1.8,4.6);
\filldraw [fill=magenta, thin, draw=black] (1.8,3.6) rectangle (2.1,4.6);
\filldraw [fill=magenta, thin, draw=black] (2.1,3.6) rectangle (2.4,4.6);
\filldraw [fill=magenta, thin, draw=black] (2.4,3.6) rectangle (2.7,4.6);
\filldraw [fill=magenta, thin, draw=black] (2.7,3.6) rectangle (3,4.6);
\filldraw [fill=magenta, thin, draw=black] (3,3.6) rectangle (3.3,4.6);
\filldraw [fill=magenta, thin, draw=black] (3.3,3.6) rectangle (3.6,4.6);
\filldraw [fill=magenta, thin, draw=black] (3.6,3.6) rectangle (3.9,4.6);
\filldraw [fill=magenta, thin, draw=black] (3.9,3.6) rectangle (4.2,4.6);
\filldraw [fill=magenta, thin, draw=black] (4.2,3.6) rectangle (4.5,4.6);
\filldraw [fill=magenta, thin, draw=black] (4.5,3.6) rectangle (4.8,4.6);
\filldraw [fill=cyan, thin, draw=black] (4.8,3.6) rectangle (5.1,4.6);
\filldraw [fill=cyan, thin, draw=black] (5.1,3.6) rectangle (5.4,4.6);
\filldraw [fill=magenta, thin, draw=black] (5.4,3.6) rectangle (5.7,4.6);
\filldraw [fill=magenta, thin, draw=black] (5.7,3.6) rectangle (6,4.6);
\filldraw [fill=magenta, thin, draw=black] (6,3.6) rectangle (6.3,4.6);
\draw [very thick] (0.3,3.6) rectangle (1.2,4.6);
\draw [very thick] (1.2,3.6) rectangle (2.1,4.6);
\draw [very thick] (2.1,3.6) rectangle (3,4.6);
\draw [very thick] (3,3.6) rectangle (3.9,4.6);
\draw [very thick] (3.9,3.6) rectangle (4.8,4.6);
\draw [very thick, dotted] (4.8,3.6) rectangle (6.3,4.6);
\draw (2.7,5) --(3,6);
\draw (3,5) --(3.3,6);
\draw (6.9,5) --(7.2,6);
\draw (7.2,5) --(7.5,6);
\draw (7.5,5) --(7.8,6);
[/TIKZ]

As we can see, after moving 2 subparts of blue beads and 3 subparts of red beads from container A into container B, we get what we wanted in the end, $\dfrac{3}{5}$ of the beads in Container A are red and $\dfrac{1}{4}$ of the beads in Container B are blue.

Therefore, the total number of red and blue beads that must be moved from container A to container B is $5\times 20 =100$.
 
You're welcome, morgancol!

You're also encouraged to fire away with whatever math problems that you can't solve and wish to get some help at our forum! Just so you know, for problem involving fraction/ratio/percentage like this one, it can be solved using algebra or without algebra way. (Happy)
 

Similar threads

MHB A bag contains x beads. Calculate the value of x.
Replies
1
Views
1K
MHB S.aux.26 our-sided die has three blue faces, and one red face.......
Replies
2
Views
4K
MHB There are z red marbles. How many green marbles are there?
Replies
2
Views
1K
MHB Find the amount of sand in each bag
Replies
1
Views
1K
MHB Solve Bracelet Problem w/14 Beads (Red, White, Blue)
Replies
1
Views
2K
I Question on analytic mechanics
Replies
3
Views
1K
How to find the amplitude of oscillations of a string with 5 beads?
2
Replies
69
Views
4K
MHB How Many Blue and Red Socks Are in the Drawer?
Replies
1
Views
3K
B More convenient mathematical notation for a simple use case
Replies
6
Views
2K
MHB Could use help, know the answers but with the process and equations
Replies
2
Views
3K