parmen

Problem 1

Case 1: All three numbers are positive. Then ∣a∣=a, ∣b∣=b, and ∣c∣=c, so the expression is equal to [a/a + b/b + c/c + abc/abc = 1 + 1 + 1 + 1 = \boxed{4}.]

Case 2: Two of the numbers are negative and one is positive. Without loss of generality, assume that a and b are negative and c is positive. Then ∣a∣=−a, ∣b∣=−b, and ∣c∣=c, so the expression is equal to [\frac{-a}{-a} + \frac{-b}{-b} + \frac{c}{c} + \frac{-abc}{-abc} = 1 + 1 + 1 - 1 = \boxed{2}.]

Case 3: All three numbers are negative. Then ∣a∣=−a, ∣b∣=−b, and ∣c∣=−c, so the expression is equal to [\frac{-a}{-a} + \frac{-b}{-b} + \frac{-c}{-c} + \frac{-abc}{-abc} = 1 + 1 + 1 + 1 = \boxed{4}.]

Therefore, the possible values of the expression are 2 and 4.

To make an equation have no solutions, the two sides of the equation must be unequal no matter what values are plugged in for the variables. One way to do this is to have the same variable on both sides of the equation, but with different coefficients. For example, the equation x+2x=3x+1 has no solutions, because if we plug in any value for x, the two sides of the equation will never be equal.

Another way to make an equation have no solutions is to have a contradiction on one side of the equation. For example, the equation x+1=0 has no solutions, because it is impossible for any number plus 1 to equal 0.

To fill in the blanks in the equation $___ + ___ = ___ + ____$ so that it has no solutions, we can use either of these methods.

Method 1:

4p + 3p = 7p + 0

This equation has no solutions because it has the same variable on both sides of the equation, but with different coefficients.

Method 2:

p + 1 = 0

This equation has no solutions because it is impossible for any number plus 1 to equal 0.

Therefore, the completed equation is either:

4p + 3p = 7p + 0

or:

p + 1 = 0

We can solve this problem by considering the cases. There are four cases for the number of cards that have the same number: 2, 3, 4, and 5.

Case 1: 2 cards have the same number.

There are seven different numbers, so there are seven ways to choose which number will be shared by the two cards. Then, there are four ways to choose the color of the first card, and four ways to choose the color of the second card. However, we have double-counted the cases where the two cards are the same color, so we must divide by 2. Finally, there are four ways to choose the color of the third card, three ways to choose the color of the fourth card, and two ways to choose the color of the fifth card.

Therefore, the probability that exactly two cards have the same number is:

P(2 cards) = 7 * 4 * 4 / 2 * 4 * 3 * 2 = 840 / 5040 = 7/42

Case 2: 3 cards have the same number.

There are seven different numbers, so there are seven ways to choose which number will be shared by the three cards. Then, there are four ways to choose the color of the first card, four ways to choose the color of the second card, and three ways to choose the color of the third card. However, we have double-counted the cases where two of the cards are the same color, so we must divide by 3. Finally, there are three ways to choose the color of the fourth card, and two ways to choose the color of the fifth card.

Therefore, the probability that exactly three cards have the same number is:

P(3 cards) = 7 * 4 * 4 * 3 / 3 * 3 * 2 * 2 = 1920 / 5040 = 4/11

Case 3: 4 cards have the same number.

There are seven different numbers, so there are seven ways to choose which number will be shared by the four cards. Then, there are four ways to choose the color of the first card, four ways to choose the color of the second card, three ways to choose the color of the third card, and two ways to choose the color of the fourth card. However, we have double-counted the cases where two of the cards are the same color, so we must divide by 6. Finally, there is one way to choose the color of the fifth card.

Therefore, the probability that exactly four cards have the same number is:

P(4 cards) = 7 * 4 * 4 * 3 * 2 / 6 * 1 = 2520 / 5040 = 3/5

Case 4: 5 cards have the same number.

There are seven different numbers, so there are seven ways to choose which number will be shared by the five cards. Then, there are four ways to choose the color of the first card, four ways to choose the color of the second card, three ways to choose the color of the third card, two ways to choose the color of the fourth card, and one way to choose the color of the fifth card.

Therefore, the probability that exactly five cards have the same number is:

P(5 cards) = 7 * 4 * 4 * 3 * 2 / 1 = 1728 / 5040 = 2/3

Total probability:

To find the total probability that exactly two cards have the same number, we must add up the probabilities of each case:

P(exactly 2 cards) = P(2 cards) + P(3 cards) + P(4 cards) + P(5 cards)

= 7/42 + 4/11 + 3/5 + 2/3

= 77/210

a) Probability that the first card has an even number and the second card has an odd number

There are 14 even cards and 14 odd cards in Janice's deck. So, the probability that the first card Ella draws has an even number is 14/28.

Once Ella draws the first card, there are only 27 cards left in the deck. Since Ella is not replacing the first card, there are now 13 even cards left in the deck. So, the probability that the second card Ella draws has an odd number is 13/27.

Therefore, the probability that the first card Ella draws has an even number and the second card Ella draws has an odd number is:

(14/28) * (13/27) = 13/54

b) Probability that the first card has an even number and the second card has a multiple of 3

There are 14 even cards in Janice's deck, and 7 of them are multiples of 3 (2, 4, 6, 8, 10, 12, and 14). So, the probability that the first card Ella draws has an even number and is a multiple of 3 is 7/28.

Once Ella draws the first card, there are only 27 cards left in the deck. Since Ella is not replacing the first card, there are now 6 even cards left in the deck, and 3 of them are multiples of 3 (4, 8, and 12). So, the probability that the second card Ella draws is a multiple of 3 is 3/27.

Therefore, the probability that the first card Ella draws has an even number and the second card Ella draws has a multiple of 3 is:

(7/28) * (3/27) = 1/36.

We can use the following equation to relate the number of painters, the area of the wall, and the time to paint the wall:

Painters * Time = Wall area / Paint rate

We know that 50 painters can paint a 3600 square foot wall in 20 minutes, so we can calculate the paint rate as follows:

Paint rate = Wall area / (Painters * Time) = 3600 sq ft / (50 painters * 20 minutes) = 7.2 sq ft/painter-minute

Now, we can use the paint rate to calculate the amount of time it would take for 25 painters to paint a 4000 square foot wall:

Time = Wall area / (Painters * Paint rate) = 4000 sq ft / (25 painters * 7.2 sq ft/painter-minute) = 22.2 minutes

Therefore, it would take 22.2 minutes for 25 painters to paint a 4000 square foot wall.

We can use the following formula to find the weight of the berries after two days:

New weight = Original weight * (1 - Water loss)

We know that the original weight of the berries was 200 kg and that the water loss was 99% - 95% = 4%. Therefore, the new weight of the berries is:

New weight = 200 kg * (1 - 0.04) = 200 kg * 0.96 = 192 kg

Therefore, the total weight of the berries after two days is 192 kg.

Let Alice's initial amount of money be a and Bob's initial amount of money be b.

If Alice receives x dollars from Bob, then she will have a+x dollars and Bob will have b−x dollars. Therefore, a+x is 2×(b−x), so 3x=2b−a.

If Alice gives y dollars to Bob, then she will have a−y dollars and Bob will have b+y dollars. Therefore, a−y is 3×(b+y), so 4y=3a−3b.

Subtracting the first equation from the second equation, we get y=a−2b. Substituting this into the first equation, we get 3(a−2b)=2b−a, so 5a=7b.

Therefore, the ratio of Alice's amount of money to Bob's amount of money is 7:5​.