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6/12= 50%, which means that Lily ate half the ducks. 50% of 12 is 12 divided by 2, so 6. 

Lily ate 6 ducks, and 6 ducks remained.

2. If a,b, and c are integers from the set of positive integers less than 7 such that 

\(abc\equiv 1\pmod 7, 5c\equiv 2\pmod 7, 6b\equiv 3+b\pmod 7,\)

 then what is the remainder when a+b+c is divided by 7?

5*6=30=2 mod7     so  c=6

let ab=x

6x=1 mod 7

x=6

ab=6    (1*6) (2*3) there might be others as well....   6 is already used I don;t know if it can be used twice.

so try 2 and 3

check 2*3*6=36=1mod7  so at least that works

If b=2 then    6*2mod7=5mod7       3+2=5 mod7    so that does work

S0 a combination that works is

a=3,  b=2,    c=6

a+b+c = 3+2+6 = 11 = 4mod7   The remainder is 4

1. Find the least four-digit solution r of the congruence .  

\(r^2 + 4r + 4 \equiv r^2 + 2r + 1 \pmod{55}\)

I am not sure but I will give it a shot

\(\begin{align} r^2 + 4r + 4 &\equiv r^2 + 2r + 1 \pmod{55}\\ 2r &\equiv -3 \pmod{55}\\ 2r&=55x-3\\ 55x-2r&=3 \qquad x\in Z \qquad r \in Z \quad r \ge 1000\\ &\mbox{so if r=1000 then}\\ 55x-2000&=3\\ 55x&=2003\\ x&=37\\ \end{align}\\ \mbox{So I think that r=1000 works}\)

Yes, we used to have a huge amount of fun here.  ://

The rule for muliplying negatives and positives together is that if you have an even number of negative numbers, then your product is positive. If you have an odd number of negative numbers being multiplied, then your product will be negative. 

If you are just multiplying two negatives, then your answer should be positive, not negative.

in which function is x=2 mapped to 32

\(f(x) = x^{p} = 32 \)

\(x^{p} =32\)

\(x=2\) 

\(2^{p} =2^{5} \)

\(p=5\)

\(f(x)=x^{5}=32\)

56

in which function is x=2 mapped to 32

\(f(x) = 2^{x} = 32 \)

\(2^{x} =2^{5} \) 

\(2^{5} =32 \)

\(x=5\)

B=34.95(D) + .55(m)        in this case D=1

B=34.95 + .55(m)

You have MIXED UNITS in your question....the result will be

100 m/s  +  4480 m^2 / s^2

Multiplying:  Result is negative

Adding ONLY   :  Negative

Subtracting........ C0uld be negative or positive

I know this!

1.3e+3

1.2345e-2

4.26e-5

negative

The only way of not picking a red card after picking a blue or yellow card is to pick all the red cards first. The probability of doing this is q = 10!*35!/45!  i.e. (10/45)*(9/44)*...*(1/36)

Hence, the probability of picking a red card after picking a blue or yellow card is p = 1 - q

I've assumed the question intends all the cards to be picked; that it doesn't just refer to the 1st and 2nd cards picked.

54 = 76

Plugging in the values for the variables into the equation, you get h = -16(2^2)+48(2).

h = -16(2^2)+48(2) --->

h = -16(4) + 96 --->

h = -64 + 96 --->

h = 32

Phil is correct (as usual)   ...I do not know how I came up with 59 while I was on my bike ride doing this in my head this morning.....

HOWEVER    92 + 15  equals    107      NOT 106 as Phil added.

5. communitive property of multiplication (yx = xy)

6. associtative property of addition ((a + b) + c = a + (b + c))

-16/3 = (-5*3-1)/3 = -5  1/3

Here's a few....

1. What is the additive inverse of 2/3 ?   →    -2/3

2. What is the multiplicative inverse of 2/3 ?    → 3/2

3. Justify the statement, "Division is not commutative." → for example,   15/3  = 5  but  5/15  = 1/3

4. Is the set of whole numbers closed under subtraction? Justify your answer.

Whole numbers are 0,1,2,3,4.........

"Closed" in this case means that when we perform a certain operation with two set members, we get a third set member returned. The set of whole numbers is not closed under subtraction because, for example.....   0 - 4  = -4     and -4 is not a member of the set of whole numbers

7. What is the additive inverse of the expression -ab ?  →  ab  ....   because -ab + ab = 0

9. Is the set of non-zero integers closed under division? Justify your answer.
Definitely not.....example  .......22/7  = 3 +1/7    and this is not a non-zero integer

10. Show the set of non-zero rational numbers is closed under division.

Let m = a/b   and n = c/d    such that a/b and c/d ≠ 0. Them m/n  =  a/b ÷ c/d  =   [ad] / [ bc]   which is also a non-zero rational

1. Additive inverse is the number you add to the given number in order to get 0. So -2/3.

2. Muiltiplicatice inverse of a/b is b/a. So 3/2.

3. x-y ≠ y-x. Just plug in numbers - 1-2≠2-1, 3-4≠4-3, etc...

4. Set of closed number is closed under addition and multiplication. Not subtraction because x-y can be negative, and negative numbers are not whole numbers (example: 3-5 = -2).

7. ab

8. ab/-1

9. No. Integers are closed under addition, subtraction, and multiplication. (Example: 8/6 is 4/3, which is not an integer.)

Let N be the number of cookies

And we have that

N =  11m + 4

N =  7n + 1       subtracting the second equation form the first, we have

11m - 7n = -3   →  11m  = 7n - 3

And the  only possible values for {m,n}  are  {1, 2}  or (8, 13}

Thus N  could be either 15 or 92

And their sum = 106