Ratios

[homeschool mom]

I have a bag with red and blue marbles in it. At the moment, the ratio of blue marbles to red marbles is 5:4. I then add 4 blue marbles and 4 red marbles, and the ratio becomes 6:5. How many blue marbles were in the bag before I added more?

 

[MarkFL]

I would let \(R\) be the initial number of red marbles, and \(B\) be the initial number of blue marbles. Using the given ratio, can you write a relationship between \(R\) and \(B\)?

 

[homeschool mom]

B/R=5/4 ?

 

[MarkFL]

Yes. I would choose to write this as:

[MATH]4B-5R=0[/MATH]
Can you write a 2nd equation using the ratio after the marbles are added?

 

[homeschool mom]

yes, [MATH]5(B+4)-6(r+4)=0[/MATH]and then distribute to
[MATH]5B+20-6r-24=0[/MATH][MATH]5B-6r-4=0[/MATH]?

 

[MarkFL]

Okay, so we have:

[MATH]4B-5R=0[/MATH]
[MATH]5B-6R=4[/MATH]
And we are asked to find \(B\), so what if we multiple the first equation by -6 and the second by 5 to get:

[MATH]-24B+30R=0[/MATH]
[MATH]25B-30R=20[/MATH]
What do you get when you add the two equations?

 

[homeschool mom]

I don't know how to add the 2 equations

 

[MarkFL]

I don't know how to add the 2 equations

Have you not been taught about elimination as one way to solve a system of equations?

 

[homeschool mom]

no I haven't

 

[MarkFL]

How about substitution?

 

[homeschool mom]

I got
[MATH]-24b+30r=0[/MATH]
[MATH]30r=24b[/MATH]
[MATH]r=\frac{24}{30}b[/MATH]
[MATH]25b-30(\frac{24}{30}b)=20[/MATH]
[MATH]25b-24b=20[/MATH]
[MATH]b=20[/MATH]is this right?

 

[MarkFL]

Yes. That's the same result I got when adding the equations above:

[MATH](25B-24B)+(30R-30R)=20[/MATH]
[MATH]B=20[/MATH]

 

[homeschool mom]

thank you