Find the total number of red and blue beads

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Discussion Overview

The discussion revolves around a problem involving the movement of red and blue beads between two containers, focusing on the proportions of beads in each container after the transfer. Participants explore whether the problem can be solved without algebra and examine the implications of moving beads on the proportions of colors in both containers.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question the correctness of the problem statement, suggesting that moving beads would not allow for an increase in the proportion of blue beads in Container B.
  • One participant argues that the proportion of blue beads in Container A should remain at $\dfrac{2}{5}$ after the transfer, implying a specific ratio of red to blue beads moved.
  • Another participant proposes a method using visual representation to illustrate the movement of beads and the resulting proportions, indicating a potential limit to the proportion of blue beads in Container B.
  • There is uncertainty expressed about whether the problem can be solved without algebra, with some participants feeling conflicted about the feasibility of the proposed solutions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the problem's correctness or the feasibility of the proposed solutions. Multiple competing views remain regarding the implications of moving beads and the resulting proportions.

Contextual Notes

Participants highlight potential limitations in the problem's assumptions and the definitions of proportions, as well as the unresolved mathematical steps involved in the proposed solutions.

anemone
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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...
 
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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 ?
 
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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)
 

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