💡Okay, so Sarah has seven children, and I need to find the probability that exactly half of them are girls. Hmm, wait a minute, seven is an odd number, so exactly half would mean 3.5 girls, which doesn't make sense because you can't have half a child. Maybe the question means approximately half? Or perhaps it's a trick question. But I think the intended meaning is that exactly half of the children are girls, which would mean either 3 or 4 girls since you can't have 3.5. Alright, so I should calculate the probability of having exactly 3 girls and exactly 4 girls, and then add those probabilities together. That makes sense because both scenarios would be considered "exactly half" in a practical sense, even though mathematically, half of seven is 3.5.To calculate this, I remember that the probability of having a certain number of girls in a given number of children follows a binomial distribution. The formula for the probability of having exactly k successes (in this case, girls) in n trials (children) is:[ P(k) = binom{n}{k} p^k (1-p)^{n-k} ]Where:- (binom{n}{k}) is the binomial coefficient, which represents the number of ways to choose k successes from n trials.- (p) is the probability of success on a single trial, which is 0.5 for having a girl.- (1-p) is the probability of failure, which is also 0.5 for having a boy.So, for Sarah's case, n is 7, and p is 0.5. I need to calculate this for k = 3 and k = 4.First, let's calculate the binomial coefficient for k = 3:[ binom{7}{3} = frac{7!}{3!(7-3)!} = frac{7!}{3!4!} ]Calculating the factorials:- 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040- 3! = 3 × 2 × 1 = 6- 4! = 4 × 3 × 2 × 1 = 24So,[ binom{7}{3} = frac{5040}{6 × 24} = frac{5040}{144} = 35 ]Now, plug this into the binomial formula for k = 3:[ P(3) = 35 × (0.5)^3 × (0.5)^{7-3} = 35 × (0.5)^3 × (0.5)^4 ]Simplify the exponents:[ (0.5)^3 × (0.5)^4 = (0.5)^{3+4} = (0.5)^7 ]Calculate (0.5)^7:[ (0.5)^7 = frac{1}{128} ]So,[ P(3) = 35 × frac{1}{128} = frac{35}{128} ]Now, let's do the same for k = 4.First, calculate the binomial coefficient:[ binom{7}{4} = frac{7!}{4!(7-4)!} = frac{7!}{4!3!} ]We already know that 7! is 5040, 4! is 24, and 3! is 6.[ binom{7}{4} = frac{5040}{24 × 6} = frac{5040}{144} = 35 ]Interesting, the binomial coefficient is the same for k = 4 as it was for k = 3. That makes sense because of the symmetry in the binomial coefficients.Now, plug this into the binomial formula for k = 4:[ P(4) = 35 × (0.5)^4 × (0.5)^{7-4} = 35 × (0.5)^4 × (0.5)^3 ]Again, simplify the exponents:[ (0.5)^4 × (0.5)^3 = (0.5)^{4+3} = (0.5)^7 = frac{1}{128} ]So,[ P(4) = 35 × frac{1}{128} = frac{35}{128} ]Now, to find the total probability of having exactly half girls (either 3 or 4), we add these two probabilities together:[ P(text{half girls}) = P(3) + P(4) = frac{35}{128} + frac{35}{128} = frac{70}{128} ]Simplify the fraction:[ frac{70}{128} = frac{35}{64} ]So, the probability that exactly half of Sarah's seven children are girls is 35/64.Wait, let me double-check my calculations to make sure I didn't make a mistake. The binomial coefficients for k = 3 and k = 4 are both 35, which seems correct because of the symmetry in the binomial distribution when p = 0.5. Then, multiplying each by (0.5)^7 gives 35/128 for each, and adding them together gives 70/128, which simplifies to 35/64. Yeah, that seems right.I think I've got it. The probability is 35/64.

💡First, I need to determine the time it took Louisa to travel 240 miles on the first day at an average speed of 60 miles per hour.Using the formula for time, which is distance divided by speed, I can calculate the time for the first day:Time₁ = 240 miles / 60 mph = 4 hoursNext, I'll let ( x ) represent the number of miles Louisa traveled on the second day. Since she traveled at the same average speed of 60 mph, the time for the second day's trip is:Time₂ = ( x ) miles / 60 mphAccording to the problem, the first day's trip took 3 hours less than the second day's trip. This gives me the equation:Time₁ + 3 = Time₂Substituting the known values:4 hours + 3 hours = ( x ) / 60 mph7 hours = ( x ) / 60 mphTo find ( x ), I'll multiply both sides of the equation by 60 mph:( x ) = 7 hours * 60 mph = 420 milesTherefore, Louisa traveled 420 miles on the second day.

💡First, I need to determine how many dimes Melanie had in her bank before receiving any additional dimes from her parents.Melanie received 39 dimes from her dad and 25 dimes from her mother. To find the total number of dimes she received from both parents, I will add these two amounts together.Next, I know that after receiving the additional dimes, Melanie has a total of 83 dimes. To find out how many dimes she had initially, I will subtract the total number of dimes she received from her parents from her current total.This calculation will give me the number of dimes Melanie had in her bank before her parents gave her more dimes.

💡First, I'll define the number of green buttons as ( G ).Since there are 10 more yellow buttons than green buttons, the number of yellow buttons is ( G + 10 ).There are 5 fewer blue buttons than green buttons, so the number of blue buttons is ( G - 5 ).The total number of buttons is the sum of green, yellow, and blue buttons, which equals 275. This gives the equation:[G + (G + 10) + (G - 5) = 275]Combining like terms:[3G + 5 = 275]Subtracting 5 from both sides:[3G = 270]Finally, dividing both sides by 3:[G = 90]Therefore, the tailor bought 90 green buttons.

💡First, I need to determine the number of red pens in Maria's desk drawer. Let's denote the number of red pens as ( R ).According to the problem, there are 10 more black pens than red pens. Therefore, the number of black pens can be expressed as ( R + 10 ).Similarly, there are 7 more blue pens than red pens. Thus, the number of blue pens is ( R + 7 ).The total number of pens is given as 41. This means the sum of red, black, and blue pens equals 41. I can set up the equation as follows:[R + (R + 10) + (R + 7) = 41]Combining like terms, the equation simplifies to:[3R + 17 = 41]To isolate ( R ), I'll subtract 17 from both sides of the equation:[3R = 24]Finally, I'll divide both sides by 3 to find the value of ( R ):[R = 8]Therefore, there are 8 red pens in Maria's desk drawer.

💡First, I need to simplify the expression inside the ceiling function: (left(frac{15}{8}right)^2 cdot frac{-34}{4}).1. Calculate (left(frac{15}{8}right)^2):[left(frac{15}{8}right)^2 = frac{225}{64}]2. Calculate (frac{-34}{4}):[frac{-34}{4} = -8.5]3. Multiply the results:[frac{225}{64} cdot -8.5 = frac{-1912.5}{64} approx -29.890625]4. Apply the ceiling function to (-29.890625):[leftlceil -29.890625 rightrceil = -29]Next, evaluate the expression inside the floor function: (frac{15}{8} cdot leftlfloor frac{-34}{4} rightrfloor).1. Calculate the floor of (frac{-34}{4}):[leftlfloor -8.5 rightrfloor = -9]2. Multiply by (frac{15}{8}):[frac{15}{8} cdot -9 = frac{-135}{8} = -16.875]3. Apply the floor function to (-16.875):[leftlfloor -16.875 rightrfloor = -17]Finally, subtract the two results:[-29 - (-17) = -12]