Need help with math!! I tried everything!!?

Question

Here are the questions. Please, I've tried everything to figure out these problems!

((1.)) Nine consecutive odd whole numbers sum to 243. Find the sum of the first and last odd whole numbers in the nine-addend set.

((2.)) Tommy invited some friends and several cousins over for his birthday. Each friend is 9 years old, and each cousin is 11 years old. None of his cousins attend his school. If the sum of the ages of Tommy, his friends, and his cousins add up to 122, how many of Tommy's friends and how many cousins attended his party?

((3.))Which set of factors of the number 420 has the least possible sum? Which set of factors of 420 has the greatest possible sum? Be sure that the two sets of factors both have a product of 420.

((4.)) Find the greatest whole number that meets all the following conditions:

a. It is greater than 100.
b. It is less than 200.
c. It is 20 greater when rounded to the nearest 100 than when rounded to the nearest 10.

Please help! And thanks!;)

Answer 1

1.

nine consecutive odd numbers. awesome. so let the first number be n. then, by definition of odd, the next odd number will be n + 2, and the next n + 4 ....
so
n + (n + 2) + (n + 4) + (n + 6) + (n + 8) + (n + 10) + (n+12) + (n + 14) + (n + 16) = 243

9n + 72 = 243
subtract 72
9n = 171
divide by 9
n=19

so the first number is 19 the last is 19 + 16 = 35. so, there sum is 19 + 35 which is:
54.

2 ok, there is not enough information to solve this problem. if we can ASSUME that tommy is the same age as his friends. then 9 ( x) + 11 (y) = 122
where x is the number of friends of tommy's + 1 ( for tommy) and y is the number of cousins, we can graph this as y= -9/11 x + 122/11

the only answer that works is 4 friends ( 5 9 year olds if you include Tommy) and 7 cousins.

3. ok. so we can factor 420 OR we can set up an equation
x * y = 420 and x + y =? so, let x = 420/y

420/y + y= ?

now, when you graph this, it is continuously decreasing. so the maximum sum is 421 ( 1 + 420). the minimum sum is 61 ( 20 + 41)

4. 180. i don't know how to explain this.

Answer 2

#1. It is important to remember that when you are adding consecutive odd numbers that you must increase the number by two each time. If you were to add one, the number would become even. For example 3 + 1 = 4, and 3 + 2 = 5. Therefore, chose a variable to represent your unknown odd number, e.g., X. So, your first odd number is X. The next one would be X + 2; then x + 4, X + 6, X + 8, X + 10, X + 12, X + 14, and finally (#9), X + 16. The sum of these numbers, according to your problem, is 243. If you add the numbers together you get 72. Adding the X's gives you 9X. Now, your problem becomes 9X + 72 = 243. Next you solve it with algebra, first subtracting 72 from both sides of your equation. This gives you 9X = 171. When you divide both sides of your equation by 9, you find that X = 19. So, your first odd number in this sequence is 19. The last would be 35, and I'll let you find their sum. Hope this is helpful.

Answer 3

To solve some of these, it's easiest to write formulas.

1. x+x+2+x+4+x+6+x+8+x+10+x+12+x+14+x+16=243
From there, you can find x. The smallest number is x, the largest is x+16.

2. Let x= the number of friends.
Let y= the number of cousins.
9x+11y+9=122
The nine is Tommy's age.
Because there's no more information, from there I'm afraid you'll have to guess and check. Just plug in values until you find something close to 122, and then from there you can find an equal.

3. List the factors of 420. Go from 1*420 to 420*1. From there, you can find which has the greatest sum and which has the least.

4. The number is between one hundred and two hundred.
If it is twenty greater when rounded to the nearest hundred than when rounded to the nearest ten, it would be rounded to two hundred. Two hundred minus twenty is 180. What is the largest number that can be rounded to the nearest ten to equal 180?

Hope that helps. =)

Answer 4

let x=lowest odd number
x+(x+2)+(x+4)+(x+6)+(x+8)+(x+10)+(x+12)+(x+14)+(x+16)=243
9x+72=243
9x=171
x=19

so the lowest number is 19. The highest number is x+16, or 19+16, which is 35. so the total of the smallest and largest
would be 35+19=54

2. 9x+11y+9=122
9x+11y=113

by trial and error I found that 4 nine year olds and 7 eleven year olds, plus Tommy's 9 years add up to 122.

Answer 5

1. 19+21+23+25+27+29+31+33+35=243
19+35=54

Answer 6

1.
Let W= first whole number

W+(W+2)+(W+4)+(W+6)+(W+8)+(W+10)+(W+12)+(W+14)+(W+16)=243

9W+72=243
-72

9W=171
divided by 9

W=19
last number=19+16=35

2. not sure u can be completel;y accurate with given info.

3.20 and 21=41 least (use numbers clostest to square root for this)
1 and 420=421 most

4.280

Answer 7

1.) 19, 21, 23, 25, 27, 29, 31, 33, 35

2.) I can't figure it right now

3.) least possible sum: 20 + 21 = 42
greatest possible sum: 420 + 1 = 421

4.) 184

Answer 8

Solve -14 = 8 + j. -14 - 8 = j subtract 8 from both sides -22 = j negative and negative give you more negative -------------------------- Solve x - 6 = -18. x = - 18 + 6 add 6 to both sides x = - 12