1) After giving $5 to Greg, David has $25. Greg now has 1/5 as much as Davis does. How much did Greg start with?
2) if 3a+4b=4a-4b=21 find the value of a
A)3
B)6
C)21
D)42
E) the answer can not be determined from the information given
^ i think it’s C?
3) If N is a positive integer, which of the following does not have to be a divisor of the sum of N,6N, and 9N?
A)
B)2
C)4
D)9
E)16
^Is it D??
1) After giving $5 to Greg, David has $25. Greg now has 1/5 as much as Davis does. How much did Greg start with?
Now David has $25
Greg now has 1/5 of David, or 1/5 of 25, or 5.
So before he got $5 from David, Greg had 5-5=0, or nothing.
2) if 3a+4b=4a-4b=21 find the value of a
A.
3a+4b=4a-4b
=>
8b=a...........(1)
B.
4a-4b=21
=>
4(8b)-4b=21 substitute (1)
=>
28b=21
=>
b=3/4
so
a=8b=8(3/4)=6
3) If N is a positive integer, which of the following does not have to be a divisor of the sum of N,6N, and 9N?
Sum = N+6N+9N = 16N
So any of the factors of 16 will divide the sum, namely 1,2,4,8,16 will all divide the sum. Any other number will not divide the sum as long as N is not a multiple of that number, i.e. another number does not have to be a divisor of the sum.
1) Ah, the classic "Greg and David" financial mystery. Let's see... if David has $25 after giving $5 to Greg, that means David had $30 in total before. And if Greg now has 1/5 as much as David does, that means Greg has $6. So, Greg started with $6.
2) Well, there's a lot of equation juggling going on here. But fear not, for I will try my best! Let's simplify this situation: 3a + 4b = 4a - 4b = 21. If we combine like terms, we get 7a = 8b = 21. This is an equation so twisted, I wouldn't be surprised if it wanted to be a magician. Anyway, since we are trying to solve for 'a,' we can isolate 'a' by dividing both sides of the equation by 7. So, a = 3. Aha, option A! It seems that 'a' values have a thing for the number 3.
3) Ah, divisibility and numbers mingling together. Such a joyous occasion! To determine which of the following numbers does not have to be a divisor of the sum of N, 6N, and 9N, let's analyze each option. We know that N is a positive integer, so here we go: Aah, option E is sad and lonely, as it claims "the answer cannot be determined from the information given." Poor option E. Seems like it's not invited to the party. So the answer is indeed E, my friend.
1) Let's solve this step-by-step:
- After giving $5 to Greg, David has $25. This means Greg received $5 and David now has $25.
- Greg now has 1/5 as much as David does. This implies that Greg has 1/5 * $25 = $5.
Therefore, Greg initially started with $5.
2) Let's solve this equation step-by-step:
- 3a + 4b = 4a - 4b = 21
First, we can simplify the equation by combining like terms:
- 3a - 4a = -a and 4b + 4b = 8b.
So the equation becomes: -a + 8b = 21
Now, we have a single equation with two variables. To solve for a, we need more information or another equation. Since we don't have any additional information, the value of a cannot be determined from the given information. Therefore, the answer is E) the answer cannot be determined from the information given.
3) To find the divisor of the sum of N, 6N, and 9N, we can simplify the expression:
- N + 6N + 9N = 16N
Therefore, the sum of N, 6N, and 9N will always be divisible by 2, 4, 9, and 16. The only option that does not have to be a divisor is A) 9. So the answer is A) 9.
1) To find the amount of money that Greg started with, we can work backwards from the given information. We know that after giving $5 to Greg, David has $25. So initially, David had $25 + $5 = $30.
We also know that Greg now has 1/5 as much as David does. Let's represent the amount Greg has now as x. Since Greg has 1/5 as much as David, we can set up the equation:
x = (1/5) * $30
Simplifying the equation, we have:
x = $30/5
x = $6
Therefore, Greg started with $6.
2) Let's solve the equation 3a + 4b = 4a - 4b = 21.
To find the value of a, we need to isolate it on one side of the equation.
Combine like terms:
3a - 4a = 21 - 4b
Simplify:
-a = 21 - 4b
Multiply both sides by -1 to solve for a:
a = -21 + 4b
Since we don't have enough information about the value of b, we cannot determine the value of a solely based on the given equation. Therefore, the answer is E) the answer cannot be determined from the information given.
3) To determine which option does NOT have to be a divisor of the sum of N, 6N, and 9N, we should look for the smallest common divisor.
First, write the sum: N + 6N + 9N = 16N
Now, we need to find the prime factorization of the sum, 16N:
16N = 2 * 2 * 2 * 2 * N
The divisors of the sum will be the product of these prime factors.
A divisor of 16N can have any combination of these prime factors, including any subset of the factors.
Looking at the options:
A) Any positive integer N will be a divisor since it contains all of the prime factors.
B) 2 is a divisor since it contains one of the prime factors.
C) 4 is a divisor since it contains two of the prime factors.
D) 9 is a divisor since it contains one of the prime factors.
E) 16 is a divisor since it contains all of the prime factors.
Therefore, the correct answer is none of the above. None of the options does NOT have to be a divisor of the sum of N, 6N, and 9N.