MHB Find the total number of red and blue beads

[anemone]

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...

 

[castor28]

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 ?

 

[Opalg]

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 $n$.

 

[anemone]

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.

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)

 

[anemone]

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 $5\times 20 =100$.

 

[anemone]

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)