unable to solve these 3 word problems

3. If I made a list of every seven-digit whole number greater than 1 million which has exactly six of its digits equal to 9, how many different numbers would be on my list?
Click to expand...
The formula: 6 digits x9+ the greatest digit x8=62
Click to expand...
62 is the correct answer, but your explanation is not very clear.

Given a 7 digit number greater than 1 million and with exactly 6 digits equal to 9, how many places can you put the different digit?
7, obviously. The different digit can be in the first position, or the second, or the third .....

Ignoring the first position, how many digits different from 9 can you put into the position with the different digit?
9, namely 0 through 8.
So, ignoring the first position, the desired numbers = (7 - 1) * 9 = 6 * 9 = 54.

How many digits different from 9 can you put in the first position of a seven digit number and stay over a million?
8, namely 1 through 8, 0 does not work.

So the answer is 8 + 54 = 62.

Frequently explaining how you got your answer is just as important as the answer itself.

By the way, you should (1) ask 1 question per thread, (2) put puzzles under Odds and Ends (more people look there I suspect), and (3) make sure to explain that you are in fifth grade each time so that people will not give you some solution based on integral calculus or category theory
 
JeffM said:
62 is the correct answer, but your explanation is not very clear.

Given a 7 digit number greater than 1 million and with exactly 6 digits equal to 9, how many places can you put the different digit?
7, obviously. The different digit can be in the first position, or the second, or the third .....

Ignoring the first position, how many digits different from 9 can you put into the position with the different digit?
9, namely 0 through 8.
So, ignoring the first position, the desired numbers = (7 - 1) * 9 = 6 * 9 = 54.

How many digits different from 9 can you put in the first position of a seven digit number and stay over a million?
8, namely 1 through 8, 0 does not work.

So the answer is 8 + 54 = 62.

Frequently explaining how you got your answer is just as important as the answer itself.

By the way, you should (1) ask 1 question per thread, (2) put puzzles under Odds and Ends (more people look there I suspect), and (3) make sure to explain that you are in fifth grade each time so that people will not give you some solution based on integral calculus or category theory
Click to expand...

To JeffM

Thank you so much! I will keep your advices in mind.

My mom said you are such a good mentor. I hope you can be the coach of our math club.

I will keep working on those unsloved words problems I got from the math club during this summer.
 
Strawberry

First, I thank you and your mom for the kind words.

Second, I doubt I am the right person to run a math club; I am neither a mathematician nor a teacher.

Third, there are many very good tutors at this site such as denis, subhotosh khan, mmm, and soroban who look at threads posted under Odds and Ends. It is very important, however, to post puzzles as puzzles because many tutors will not do homework problems and to say that you are in fifth grade so that you get an answer that does not involve mathematical concepts that you are not yet familiar with.

Fourth, I have a suggestion for the parents who run the math club. If they prepare in advance for the math club, they too can use this site. In their case, they need to explain that they are parents volunteering to run a math club for fifth graders and need help in solving and explaining some problems in an age-appropriate way. Some tutors here seem to have been teachers below the college level and can probably give the parents very helpful advice.

Finally, I have a story and a puzzle for you. It is said of a very famous mathematician named Gauss that, when he was about 5 years old, his teacher needed to keep Gauss quiet for a half hour or so and told him to add up all the numbers from 1 through 100. This was more than 250 years before there calculators; things were done with paper and pencil. Gauss gave the teacher the correct answer in a matter of minutes. My puzzle for you is: how did Gauss do that without a calculator or a computer?
 
JeffM said:
Finally, I have a story and a puzzle for you. It is said of a very famous mathematician named Gauss that, when he was about 5 years old, his teacher needed to keep Gauss quiet for a half hour or so and told him to add up all the numbers from 1 through 100. This was more than 250 years before there calculators; things were done with paper and pencil. Gauss gave the teacher the correct answer in a matter of minutes. My puzzle for you is: how did Gauss do that without a calculator or a computer?
Click to expand...

To JeffM,

1+2+3+4+5+6+7+8+9+10=(1+9)+(2+8)+(3+7)+(4+6)+10+5=10x5+5=55
11+12+13+14+15+16+17+18+19+20=10x10+(1+2+3+4+5+6+7+8+9+10)=10x10+55
21+22+23+24+25+26+27+28+29+30=20x10+55
31+32+33+34+35+36+37+38+39+40=30x10+55
41+42+43+44+45+46+47+48+49+50=40x10+55
51+52+53+54+55+56+57+58+59+60=50x10+55
61+62+63+64+65+66+67+68+69+70=60x10+55
71+72+73+74+75+76+77+78+79+80=70x10+55
81+82+83+84+85+86+87+88+89+90=80x10+55
91+92+93+94+95+96+97+98+99+100=90x10+55

Therefore,
55+(55+10x10)+(55+20x10)+(55+30x10)+(55+40x10)+(55+50x10)+(55+60x10)+(55+70x10)+(55+80x10)+(55+90x10)
=55x10+(10+20+30+40+50+60+70+80+90)x10
=55x10+450x10=5050

Is it how Gauss did? Wow! He was so smart that he was only 5.

After I have done this. I start thinking: how about 100 to 1000?
101+102+103+104+105+106+107+108+109+110=55+100x10
111+112+113+114+115+116+117+118+119+120=55+110x10
and so on.

Them it would be :
(100+110+120+130+140+150+160+170+180+190+200+210+220+230+240+250+260+270+280+290+300+310+320+330+340+350+360+370+380+390+400+410+420+430+440+450+460+470+480+490+500+510+520+530+540+550+560+570+580+590+600+610+620+630+640+650+660+670+680+690+700+710+720+730+740+750+760+770+780+790+800+810+820+830+840+850+860+870+880+890+900+910+920+930+940+950+960+970+980+990)x10+55x900
=(10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38+39+40+41+42+43+44+45+46+47+48+49+50+51+52+53+54+55+56+57+58+59+60+61+62+63+64+65+66+67+68+69+70+71+72+73+74+75+76+77+78+79+80+81+82+83+84+85+86+87+88+89+90+91+92+93+94+95+96+97+98+99)x100+55x990=[10+(55+10x10)+(55+20x10)+(55+30x10)+(55+40x10)+(55+50x10)+(55+60x10)+(55+70x10)+(55+80x10)+(45+90x10)]x100+55x900=
(10+55x8+45+45x100)x100+55x900

Am I lost?

This pattern doesn't look nice.

How about 1,000 to 10,000? I definately can not use this method.

I am suck. Is the first pattern of 1 to 100 not good enough?
 
What Gauss realized (according to one version of the legend) was this:
Code: 
1 + 100 = 101

2 + 99   = 101

3 + 98   = 101
.
.
.
47 + 54 = 101

48 + 53 = 101

49 + 52 = 101

50 + 51 = 101

then we have

1 + 2 + 3 + .... + 98 + 99 + 100 = 50 * 101 = 5050

The problems that you are thinking about can be done this way. Another way to do this is to use the following formula:

Sum of a series = [(first number) + (last number)] * (number of numbers in the series) / 2

for Gauss's problem

first number = 1

last number = 100

number of numbers = 100

then

Sum = [1 + 100] * (100)/2 = 101 * 50 = 5050
 
See. Lots of great tutors here. You now have met Denis and Subhotosh Khan.

And you have learned a story (unfortunately probably not true) about Gauss, one of the great mathematicians. There is a true story about Gauss though. He secretly tutored by mail a girl who loved math but whose parents would not let her take classes in it because they thought it unfeminine.
 
Subhotosh Khan said:
Sum of a series = [(first number) + (last number)] * (number of numbers in the series) / 2
Click to expand...

To Subhotosh,

Thank you so much for your reply!
I see. This formula is so clever.
 
JeffM said:
See. Lots of great tutors here. You now have met Denis and Subhotosh Khan.

And you have learned a story (unfortunately probably not true) about Gauss, one of the great mathematicians. There is a true story about Gauss though. He secretly tutored by mail a girl who loved math but whose parents would not let her take classes in it because they thought it unfeminine.
Click to expand...

I am so glad that my mom found this website for me.
Thank all the great tutors who worked with me.
It is so much fun!!
I love the story that Gauss secretly tutored by mail a girl.
I feel I am as lucky as that girl ^_^!
 
*looks at the three math problems wearily* Now I'm know I'm stupid when I'm not able to solve even a "simple" fifth grade problem.
 
SuckAtMath said:
*looks at the three math problems wearily* Now I'm know I'm stupid when I'm not able to solve even a "simple" fifth grade problem.
Click to expand...

That attitude needs to go - no body is stupid unless they choose to act like one.

The problems in this thread are very advanced (although posted by fifth grader) - many 12 th grader in AP class will be stumped by these.
 
You must log in or register to reply here.
Top