Question

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?

Answer 1

Find the period of this. What can you say?

Answer 2

What?

Answer 3

Can you see a regularity of dropping?

Answer 4

The same types of marbles in the same order?

Answer 5

Yes. Do you see a cycle?

Answer 6

Yes, red, white, white, blue, blue, blue

Answer 7

Nice. Can you tell me how many marbles will be in the box after the 10-th blue marble is put in?

Answer 8

Using this cycle.

Answer 9

16?

Answer 10

Think more. Not right.

Answer 11

22?

Answer 12

Did you just count?

Answer 13

Yes

Answer 14

it's not right?

Answer 15

It is right. But I want you to not to count but to think.

Answer 16

Oh, sorry

Answer 17

I don't really do it correctly if I don't put it on paper though...

Answer 18

Do you see that there are 3 blue marbles in one cycle? And this cycle has length 6.

Answer 19

Yes

Answer 20

Nice. Can you tell me how many marbles will be in the box after the 33-th blue marble is put in?

Answer 21

I want you to find the solution by yourself.

Answer 22

66?

Answer 23

Yes. Did you count or think?

Answer 24

I don't know what yo umean by think...

Answer 25

Ok. Did you count?

Answer 26

Yes

Answer 27

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?

Answer 28

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...

Answer 29

Did you understand the table?
Cycle | All marbles | Blue marbles
2 | 12 | 6
3 | 18 | 9
4 | 24 | 12
5 | 30 | 15
...

Answer 30

Ohh, I didn't know that was all of the marbles, but I'm still not getting the right answer...

Answer 31

Can you write a part of this table for \(n\)-th cycle?

Answer 32

n meaning all of the numbers after that or just 99?

Answer 33

No. \(n\) means any positive integer.

Answer 34

oh...

6 l 36 l 12
7 l 42 l 21
how's that?

Answer 35

Fill in the dots:
Cycle | All marbles | Blue marbles
\(n\) | ... | ...

Answer 36

Not a concrete number, but in general case.

Answer 37

I don't get it... :/

Answer 38

Do you understand this?
Cycle | All marbles | Blue marbles
\(n\) | \(6n\) | \(3n\)

Answer 39

ohh yea, I understand that

Answer 40

If you put any number instead of \(n\) you will get the quantity of all marbles and the quantity of blue marbles.

Answer 41

right

Answer 42

Can you use this table to solve your problem, when 33-th blue marble is put in the box?

Answer 43

Do not count. Just use the table.

Answer 44

Yes. Now make that with 99-th blue marble is put in the box.

Answer 45

Do you understand why I drew the table for you?

Answer 46

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?

Answer 47

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?

Answer 48

sorry, idk why it did that

Answer 49

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.

Answer 50

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.

Answer 51

Do you understand it?

Answer 52

Ohh, yea!

Answer 53

Does this mean that in answer to the question there will be 202 marbles when the 100th blue marble is put in?

Answer 54

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?

Answer 55

You can use a calculator.

Answer 56

: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

Answer 57

So good of you!

Answer 58

Yes! Thanks so much :D

Answer 59

Do you think you could help me with one or two more?

Answer 60

Sure! You are welcome. I'd like to help if you are interested in understanding!

Answer 61

Yes, I am most definitely interested in understanding!

Answer 62

If 1 < 3x - 5 < 2, what is one possible value for x?

Answer 63

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

Answer 64

Sorry for this question, but how old are you? I'd like to know what could you know in math.

Answer 65

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

Answer 66

Ok. Come back to the problem. You want to solve an inequality:
\(1<3x-5<2\)
What is the main problem for you?

Answer 67

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

Answer 68

Ok. What can you say about \(x\) in this case (example):
\(x-2<5\)

Answer 69

well in that case, x could be any number from 5 and infinity right?

Answer 70

Going to the left anyway

Answer 71

Hm.. Next question. How do solve an equation like
\(x-2=5\)

Answer 72

How do you solve equations like this? By steps.

Answer 73

x could only be 7

Answer 74

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

Answer 75

Yes. The same thing is with inequalities. Try to use this method for
\(x-2<5\)

Answer 76

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.

Answer 77

Do you understand this?

Answer 78

Yes

Answer 79

Do you know what is rational numbers? Like \(\frac 12, \frac{123}{456},...\)

Answer 80

Yes, rational numbers are any number that can be a fraction and the denominator doesn't equal 0 right?

Answer 81

Yes. And tell one more thing. Have you learned how to work with inequalities?

Answer 82

Yea and I keep losing connection and it keeps logging me off, just so you know

Answer 83

Solve an inequality:
\(-3x\leqslant6\)

Answer 84

is that a less than or equal to sign?

Answer 85

Less than or equal. \(\le\)

Answer 86

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?

Answer 87

That is not right.

Answer 88

Solve an equation
\(-3x=6\)

Answer 89

By steps.

Answer 90

-3x = 6
+ 3 + 3
x = 9...
is that right?

Answer 91

No! \(-3\cdot9\ne6\)

Answer 92

Why do you add 3 to both parts?

Answer 93

I thought that was what you do in inequalities...and yea I should've double checked that ...

Answer 94

Now just solve equation:
\(-3\cdot x=6\)

Answer 95

x should be -2 then

Answer 96

Yes. Show the solution by steps.

Answer 97

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

Answer 98

-3*