How to solve this question... Ellie is dropping marbles into a box one at a time in the following order: red, white, white, blue, blue, blue; red, white, white, blue, blue, blue; ...How many marbles will be in the box after the 100th blue one is put in?
Find the period of this. What can you say?
What?
Can you see a regularity of dropping?
The same types of marbles in the same order?
Yes. Do you see a cycle?
Yes, red, white, white, blue, blue, blue
Nice. Can you tell me how many marbles will be in the box after the 10-th blue marble is put in?
Using this cycle.
16?
Think more. Not right.
22?
Did you just count?
Yes
it's not right?
It is right. But I want you to not to count but to think.
Oh, sorry
I don't really do it correctly if I don't put it on paper though...
Do you see that there are 3 blue marbles in one cycle? And this cycle has length 6.
Yes
Nice. Can you tell me how many marbles will be in the box after the 33-th blue marble is put in?
I want you to find the solution by yourself.
66?
Yes. Did you count or think?
I don't know what yo umean by think...
Ok. Did you count?
Yes
Now, watch. If 1 cycle of dropping passed. There are 6 marbles in the box and 3 of them are blue. 2: 12 and 6 3: 18 and 9 4: 24 and 12 ... what can you say about 99 cycle?
idk :( I see that the number of red and white marbles is always double the number of blue marbles and the number of the cycle (2,3, and 4) goes into the first number 6 times, but if that was true for all of it then 99 would go into 594 times and I know that's not right...
Did you understand the table? Cycle | All marbles | Blue marbles 2 | 12 | 6 3 | 18 | 9 4 | 24 | 12 5 | 30 | 15 ...
Ohh, I didn't know that was all of the marbles, but I'm still not getting the right answer...
Can you write a part of this table for \(n\)-th cycle?
n meaning all of the numbers after that or just 99?
No. \(n\) means any positive integer.
oh... 6 l 36 l 12 7 l 42 l 21 how's that?
Fill in the dots: Cycle | All marbles | Blue marbles \(n\) | ... | ...
Not a concrete number, but in general case.
I don't get it... :/
Do you understand this? Cycle | All marbles | Blue marbles \(n\) | \(6n\) | \(3n\)
ohh yea, I understand that
If you put any number instead of \(n\) you will get the quantity of all marbles and the quantity of blue marbles.
right
Can you use this table to solve your problem, when 33-th blue marble is put in the box?
Do not count. Just use the table.
Yes. Now make that with 99-th blue marble is put in the box.
Do you understand why I drew the table for you?
n l 6n l 3n 33 l 198 l 99? Yes, so that I could solve it by myself and without counting. It's a better method and it is quicker?
I want you to understand how to solve it. Please, try to understand. It is a very good problem. Do you realize what you've done, when you put 99 into the table?
sorry, idk why it did that
yes I understand it now, at first I was putting 99 as the 99th cycle when really it should've been the 99th blue marble.
You just put the number of blue marbles in the table and simply can get the whole number of marbles in the box! But it works only for full cycles.
Do you understand it?
Ohh, yea!
Does this mean that in answer to the question there will be 202 marbles when the 100th blue marble is put in?
Yes!!! I hope you understood everything I wrote. For checking this I'd like to know how many marbles will be in the box after 2012-th blue marble is put in?
You can use a calculator.
:o umm lesse n l 6n l 3n 670 l 4,020 l 2010 so that means that there will be 4,025 marbles all together after you have the 2012th blue marble? ok, using a calculator, 2012th blue marble with not complete a full cycle, and n is 670
So good of you!
Yes! Thanks so much :D
Do you think you could help me with one or two more?
Sure! You are welcome. I'd like to help if you are interested in understanding!
Yes, I am most definitely interested in understanding!
If 1 < 3x - 5 < 2, what is one possible value for x?
I'm not exactly sure how any number can be more than one and less than two unless it's like 1 and three quarters or 1 and a half
Sorry for this question, but how old are you? I'd like to know what could you know in math.
I'm in 8th grade trying to work on SATs and get a head start, these are all the questions I skipped since I didn't know how to do them yet
Ok. Come back to the problem. You want to solve an inequality: \(1<3x-5<2\) What is the main problem for you?
I don't know how to do it at all, I see the problem and I understand what they want in the answer, but I don't know how to get the answer
Ok. What can you say about \(x\) in this case (example): \(x-2<5\)
well in that case, x could be any number from 5 and infinity right?
Going to the left anyway
Hm.. Next question. How do solve an equation like \(x-2=5\)
How do you solve equations like this? By steps.
x could only be 7
oh, well x - 2 = 5 Whatever x is, it has to equal 5 when 2 is subtracted from it. So x - 2 = 5 + 2 + 2 x = 7 so... 7 - 2 = 5 Which is correct
Yes. The same thing is with inequalities. Try to use this method for \(x-2<5\)
Yes. This always works with addition or subtraction of any number to the left and right parts of inequality. It can be written as If \(a<b\), then \(a+c<b+c\). Where \(c\) is any number.
Do you understand this?
Yes
Do you know what is rational numbers? Like \(\frac 12, \frac{123}{456},...\)
Yes, rational numbers are any number that can be a fraction and the denominator doesn't equal 0 right?
Yes. And tell one more thing. Have you learned how to work with inequalities?
Yea and I keep losing connection and it keeps logging me off, just so you know
Solve an inequality: \(-3x\leqslant6\)
is that a less than or equal to sign?
Less than or equal. \(\le\)
ohh ok I haven't done this kind of inequality yet but I'll try anyway \[-3x \le 6\] \[+3 (space) + 3\] and then... \[x \le 9\] Is that right?
That is not right.
Solve an equation \(-3x=6\)
By steps.
-3x = 6 + 3 + 3 x = 9... is that right?
No! \(-3\cdot9\ne6\)
Why do you add 3 to both parts?
I thought that was what you do in inequalities...and yea I should've double checked that ...
Now just solve equation: \(-3\cdot x=6\)
x should be -2 then
Yes. Show the solution by steps.
OH! OH wait! I get it now! That's why I got it wrong, I should have been dividing! -3x = 6 /-3 = /-3 x = -2 3 X -2 = 6 which is correct
-3*
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