Are You Smarter Than a Sixth Grader?

120, 174.

I think that Kilgore has it. The difference is increasing by 3 each stage.

The difference increasing by three with the reduced equation doesn't make sense, unless that isn't what you are talking about...

8,12,18,27,...

12-8 = 4

18-12 = 6

27-18 = 9

That is not a difference increasing by 3...

The difference is D2 = D1 +1 [9 = (6-4) + 1]in Kilgores case, but you would think that there would be more trials, because there are only two examples in Kilgores case. You'd think there would be more sampling if that was the case.

Thanks Just an Engineer--

The difference between 24 and 36 is 12 then the difference is 18 which increased by 6--then the difference is 27 which increased by 9---then increased by 12 and then 15...........

If it is lineal.

I think that you were on the right track. I think it just sounded more work than it was because you simplified it (by dividing by 3). Seeing as in the original equation:

N2 = N1 *1.5 (as you said in the simplified version)

81 * 1.5 = 81 + 40.5 = 121.5

121.5 * 1.5 = 121.5 + 60.75 = 182.25

I think that 6th graders should be able to handle increments of 1/4. I think that (presumably) your kid might have been mistaken. I have thought long and hard and I can't see anything besides that relationship.

Best Wishes!

I realized that reducing was not required right after I sent it in...

so we all agree,

N2=N1*1.5

N1 = 9 + 1x3 = 12,

N2 = N1 + 2x3 = 18,

N3 = N2 = 3x3 = 27,

N4 = N3 + 4x3 = 39,

N5 = N4 + 4x3 = 54,

12+N1 = 24. 24+N2 = 32. 32+N3 = 81. 81+N4 = 120. 120+N5 = 174.

ANSWER 120 and 174.

Regards JD.

N5 = N4 + 5X3 = 54. Regards JD.

You are right.

No decimals are needed.

Series becomes 36, 54, 81, 120, 174, 246, 339, 456, 600

54= 36 + (6*3)

81= 54 + ((6+3)*3)........... 3

81 + (9+4)*3 = 120 .......... 4

120 + (13+ 5)*3 = 174 ....... 5

174 + (18 + 6)*3 = 246 ....... 6

246 + (24 + 7)*3 = 339 ....... 7

339 + (31 + 8)*3 = 456 ....... 8

456 + (39 + 9)*3 = 600 9

<N5 = N4 + 4x3 = 54,>

This shd have been :

N5= N5+5X3 =54

Also-- 24+N2 = 32---- shd be 42

Also ---32+N3 is not 59---but= 81

So 81+N4 is = 120. and 120+N5 is = 174.

ANSWER is indeed 120 and 174. All's well that ends well.

That is correct, & I would be willing to bet that it is also the correct method, which the school is trying to show the kids.

Thank you kilowatt0, it seemed a lot easier than long verbal explanations.

GA - I came, I saw, I concurred.

This makes more sense to me:

Don't you agree? If so, give a me a good answer. I'll take a sixth grader's score!

Sorry, I can't add! That last set of add 5 should be:

18 X 13 = 234

18 X 18 = 324

Interesting, the numbers you are multipying by (8, 12, 18, 27 [except for the 8] are the difference between the numbers). ...and i agree with your child, they cannot be looking for decimals or decimals would have been in the format.

You are Correct

You are right as each number is 1.5 multiple of previous number.

Anthony Gozeppa

This is a Geometric Progression.

l=ar^(n-1)

l=value of number

a=first number

r=Common Ration

n=place of the number you are looking for

r=36/24=1.5

l(5)=24*1.5^(5-1)=121.5

l(6)=182.25

ChazL

forgot to log in

24/2=12+24=36/2=18+36=54

N1/2+N1=N2

N2/2+N2=N3

N3/2+N3=N4

N4/2+N4=N5 WHICH IS 81/2=40.5+81=121.5

N5/2+N5=N6 121.5/2=60.75+121.5=182.25

N6/2+N6=N7 182.25/2=91.125+182.25=273.375

TELL KID HE SHOULD NOT GO BEYOND 81 0R IGNORE DECIMAL FIGURE

JB

No decimals is the condition, see my post 18 above

CaptnPea it is back to school for you,

i just worked it out on a piece of paper 12, 18, 27 there is an increase of 6, 9 and 12.

Don't rely on calculters so much.

Just a suggestion.

Here is a different way to look at it. Remember there are often many ways to make a progression.

A. 2*2*2*3=24

B. 2*2*3*3=36

C. 2*3*3*3=54

D. 3*3*3*3=81

E. 3*3*3*4=108

F. 3*3*4*4=144

There are four multipliers and for each set and one multiplier is increased by one on each step.

Interesting?

Regards JD.

Good Answer! This is the correct solution!

I think post # 20 was what the teached had in mind - No decimal places, no fractions, not a true geometric progressions, but a pattern nonetheless:

(There are other answers that may be correct, but smack of parental assistance.) Another look at #20's pattern:

2x2x2x3 =24
2x2x3x3 =36
2x3x3x3 =54
3x3x3x3 =81
3x3x3x4 =108
3x3x4x4 =144

Just my $0.02 -

interesting solutions, all.

my money is on the geometric progression.

...athough i agree that high-school is an irrational place, where decimals are considered hard (!), whereas complex fractions are considered easier (!!). i always throught that 6th grade excercises were harder than university, because of the stupid restrictions on methodology...

tell your kid not to sweat over it.

best regards

There seem to be a number of solutions.

Here is another one. Sorry I can't write 2 to the power 3

24 = 2 exp3 x 3 exp1

36 = 2 exp2 x 3 exp2

54 = 2 exp1 x 3 exp3

81 =2 exp 0 x 3 exp4

therefore

2 exp(-1) x 3 exp5 = 1/2 x 243 = 121.5

2 exp (-2) x 3 exp6 = 1/4 x 729 = 182.25

agreeing with CaptnPea, except that the derivation is slightly different.

Watching at the series:

24 = 2³ x 3¹

36 = 2² x 3²

54 = 2¹ x 3³

81 = 2⁰ x 3⁴

So following in the series we'll have (decreasing the power of 2 and increasing the power of 3):

121.5 = 2^-1 x 3⁵

182.25 = 2^-2 x 3⁶

And so on

I guess the continuation is 120,174

Reason

24 36 54 81 120 174

Divide by 3 8 12 18 27 40 58

Substract each 4 6 9 13 18

Again Substract 2 3 4 5

You're right. Apparently there is no other factor than '1.5' that defines this series, unless some complex mathematical expression (which would be overwhelming for a 6th grader). You filled the series with correct answers, in my opinion.

"Apparently there is no other factor than '1.5' that defines this series, ..."

So, the 10th of the series then is 922.640625, and the 20th is 53204.1076812744? Will this not overwhelmed our 6th grader?

Oftentimes, teachers will stop series if they become overly complicated numbers. The problem with stopping at two additional numbers is that there could be multiple options. The N1 = N2*1.5 seems reasonable after two additional numbers because they are 1/4 denominations WHICH for a 6th grader shouldn't be THAT overwhelming. This would make sense that the numbers would cease after 2, because then you go into more complex denominations.

As for the whole numbers, it is true that lots of series need whole numbers, but their way of answering just seems to complex for a sixth graders. After all, if a sixth grader can get that arithmetic, shouldn't they be able to get the N1 = N2*1.5 (decimals and all)?

I want to know what the teacher was thinking. I also think that it is somewhat funny that this problem became so controversial. I bet that the teacher never thought that about 472+ smart engineers (and other handymen :D) would be weighing in to find an answer. Tell us what the answer was!

The progression is the previous number multiplied by 1.5

24 X 1.5 = 36 X 1.5 = 54 X 1.5 = 81 X 1.5 =

This is already there in post 3

Captain,

Please be sure to let us know what the teachers answer looks like. Others have have posted what I think is the answer, that is, the missing numbers in the series are 120 and 174.

24, 36, 54, 81, 81, 81

2+4, 3+6, 5+4, 8+1, 8+1, 8+1,

(6) * (9) = 54

(9) * (9) = 81

(9) * (9) = 81

Sum the digits, then multiply by the next sum,

A typical problem in a "High IQ" quiz book.

Answer:

24 = 3*(2*2^2)

36 = 3*(3*2^2)

54 = 3*(2*3^2)

81 = 3*(3*3^2)

so the pattern follows...

3*(2*4^2) = 96

3*(3*4^2) = 144

Answer: 96, 144

24,36,54,81,108,144;

divide them by 3 to be : 8,12,18,27,36,48,

8=2x2x2

12=2x2x3

18=2x3x3

27=3x3x3

36=3x3x4, 48=3x4x4.

N/2 + N works perfectly:

• 24/2 = 12+24 = 36
• 36/2 = 18+36 = 54
• 54/2 = 27+54 = 81
• 81/2 = 40.5+81 = 121.5
• 121.5/2 = 60.75+121.5 = 182.25

Sequence is 24,36,54,81,121.5,182.25

Mastering Technical Mathematics

By Bob Goodman, Stan Gibilisco, Norman H. Crowhurst

An exerpt from page 269:

Right out of a text book.

108 and 144

24 to 36 is a difference of 12

36 to 54 is a difference of 18

54 to 81 is a difference of 27

The difference of the difference from 12 to 18 is 6

The difference of the difference from 18 to 27 is 9

The next difference of the difference would be 12

The final difference of the difference would be 15

27 + 12 = 39, 39 + 81 = 120

39 + 15 = 54, 54 + 120 = 174

121.5...182.25

There are several possibilities. The one most 6th graders would probably find is that each number in the series is equal to the previous number plus 1/2 of the previous number.

I like your reasoning and gave you an attaboy for it.

I'm a great test taker too... although I always have this internal dialogue going on with the nitwit who made up the test.

Here is the real sequence:

24,36,54,81,25,37,55,82,26,37,55,83...

We just didn't happen to know what the test writer was thinking at the time.

I have exactly the same pet peeve as you do. I also have a pet peeve in general with ambiguously worded questions, questions that presume a certain amount of ignorance on the part of the student, and teachers who mark wrong any number of right answers the come from different perspectives. Good teachers, will say, "How cool! That's another way of interpreting the question." Too many teachers, instead, look in the answer guide and say "Wrong."

Too many times, my kids have had math teachers who knew only one way to solve a problem, and would mark wrong the answers of kids who knew more than one way. I remember being marked wrong for failing to "show my work" on a math question, despite having answered correctly. There was no work involved.

In the classic physics test question, "How would you find the height of a tall building by using a barometer?" the kid who answers "Take the barometer into the basement, find the building super and say "I'll give you this nice barometer if you tell me the height of this building" should certainly get extra credit. But even if he does not get the extra credit on the test, he will get along just fine in life.

"I'll give you this nice barometer if you tell me the height of this building" would be MY method, and I actually HAVE gotten along rather nicely in life, too!

Interestin! So far, playing strictly by the rules (no fractional or decimal answers, whole numbers only) there are three possible (defensible) solutions, proposed each by several posters, the first instance of which is listed below:

#1 - 120, 174

#2 - 108, 144

#3 - 96, 144

Are they all correct? Not likely, but which one IS correct? Since all appear defensible mathematically, I am not choosing, but the teacher presumably will...

Post #32 is also defensible. It is also the simpler of the solutions.

ChazL

I don't know what happened to my list - it originally said #1, #20, and #34...

I like your list, and have I mentioned that you are a handsome man? LOL

Anyway, Thank you.

Ok ...
I'm only bothering 'cos I don't want to be left out.

The Q is of course ridiculous...
There are an infinite number of answers...

I assume the sequence is in hexadecimal ..

Thus my answer would be...oh, I just can't be bothered anymore...

BTW dunno if anyone has said this yet...but I wonder how a teacher would mark a graphically estimated answer?....plot the data, interpolate the results I'm a visual cat .. stuff the maths

Del

Oh go on Del, it's fun......howzabout a Fibonacci-additive-multpyded answer ;

0......1.....1.....2.....3.....5.....8.....13.....21.....34

.....................1.....2.....3.....4......5.......6.......7

+ ..................3.....5.....8....12.....18.....27.....41

X3..............................24...36.....54.....81....123

between 24 and 36 is 12, 3x4. between 36 and 54 is 18, 3x6. between 54 and 81 is 27, 3x8, so logically the next # is 3x10 added to 81 is 111 nad the next # is 3x12 added to 111 is 147, and so forth.

It is a sixth grade question! I am sure this is getting completely overcomplicated.

Taz, I am sorry to point out but "between 54 and 81 is 27, 3x8," is not 3x8 but 3x9. The logic you are showing would indicate that the next number after 54 should be 78 (54+(3x8)). but it is not logical to step by 2 in the first two and then step by one in the next several.

Yes, I know I let my 11 year old answer the question and I did not check. SORRY

I've given you a GA - it's pretty cool to have your 11 year old have a go, and you both deserve commendation.

The correct answer to your question is: Yes, there are many other possible solutions that you do not see. The only way to know for sure is to ask your kid's teacher what criteria he wants used to arrive at a solution. If your kid is right about not using decimals (fractions) then obviously there is another way to look at it.

we just have to wait to Captain Pea gives the answer.

The kids teacher will say which answer is supposed to the correct one.

Some observations, not offered as a solution, just interesting:

24 = 3 x 2²

36 = 3² x 2²

54 = 3³ x 2

81 = 3² x 3²

120 = 3 x 2³ x 5

174 = 3⁴ x 2

Factors and exponents are 2 and 3 or a sum of some combination of 2 and 3.

N1=24

N2= (2+4)^2=36

N3=(N1+30)=54

N4=(5+4)^2=81

N5=(N3+30)=84

N6=(8+4)^2=144

24, 36, 54, 81, 84, 144

A sixth grader would immediately recognize the 36 and the 81 as being 6 squared and 9 squared. Then they would realize that the numbers proceeding the squares are in fact 2+4=6 and 5+4=9. They would then guess that the next number would have to be 8+4=12 and that 12 squared is 144. They would then conclude that they are right because the 2, 5, and 8 in the 24, 54, and 84 are separated by 3 and the 6^2, 9^2, and 12^2 are all also separated by 3. It makes since to a sixth grader because he has no knowledge of geometric series or Taylor/Maclaurin series to cloud his judgement.

You are all guilty of making it more difficult than what a sixth grader would be required to know. The objective is to recognize a pattern in the numbers not to figure out whether or not it is some series as encountered in college level calculus classes.

KISS (Keep It Simple Stupid)

I'm still waiting for the final answer. Teachers are slow with homework. I'm also confused on how one person who answered before the other with the exact same answer only received 1 "good answer" and the next person who copied his answer receives 2 "good answer" comments.

Confusing. Must be more popular.

Another possibility: 135, 225,...

The next "Factor" to shift into the series is 5 not 4.

(Prime Factors)

2 x 2 x 2 x 3 = 24
2 x 2 x 3 x 3 = 36
2 x 3 x 3 x 3 = 54
3 x 3 x 3 x 3 = 81
3 x 3 x 3 x 5 = 135
3 x 3 x 5 x 5 = 225
3 x 5 x 5 x 5 = 375
5 x 5 x 5 x 5 = 625

Any other suggestions?

of course it may be 24, 36, 54, 81, 96, 114.9

but thats just an uneducated guess.

just sent the teacher an E-mail... hopefully we'll have our answer soon.

Here it is straight from the teacher...

You would get the A! The correct answer as given by the book was 121.5 & 182.25, although I did tell the kids that for all I know there may be different patterns out there that could work. That's exactly why I made that one optional! I like the community effort, and hopefully it didn't keep anybody awake night. :)

What happened to NO DECIMALS?????

Oh, I feel dirty!

I know how you feel

Hello,

Here is my attempt.

24, 36, 54, 81, 96, 144, 150, 216, 225, 294, 324, 441----------------------------

And here is equation

(7x2² - 2²)=24

(7x2²+2x2²)=36

(7x3² - 3²)=54

(7x3²+2x3²)=81

(7x4² - 4²)=96

(7x4²+2x4²)=144

(7x5² - 5²)=150

(7x5²+2x5²)=225

(7x6² - 6²)=216

(7x6²+2x6²)=324

(7x7² - 7²)=294

(7x7²+2x7²)=441

Regards

Here is further simplification

This is mix of two series one is 6n² and another is 9n²

6 X2² = 24

9 x 2^{2 =} 36

6 X3^{2 =} 54

9 x3^{2 =} 81

6 X4^{2 =} 96

9 x 4^{2 =} 144

6 X 5^{2 =} 150

9 x 5² = 225

6 X 6^{2 =} 216

9 x 6² = 324

6 X 7² = 294

9 x 7² = 441

My first attempt at finding the pattern gave me 120, 174. This is how I worked it out on paper. The other two solutions also seem to be mathematically correct: 96, 144 & 108, 144

24---------36---------54---------81----------120---------174-----------246

------12---------18--------27---------39------------54----------72

-4*3-------6*3-------9*3------13*3---------18*3--------24*3

--------2----------3--------4-----------5------------6

The dashes are just there too keep the spacing between the numbers.

Thanks for the cerebral exercise :)

i am a 6th grade student anyone thgat did 5th grade should know there are decIimal THAT IS THE ONLY WAY

You know what, here is something that can help you: I just found answers to my homework questions on TutorAnswer.com!