A carpenter purchased 50 ft of redwood and 80 ft of pine for a total cost of $277. A second purchase, at the same prices, included 90 ft of redwood and 60 ft of pine for a total cost of $381. Find the cost per foot of redwood and of pine.

Let R = cost per foot of redwood, and

let P = cost per foot of pine.

The 1st sentence gives us 60 ft * R + 90 ft * P = $303, and

the 2nd sentence gives us 90 ft * R + 60 ft * P = $357.

So this is two equations and two unknowns, so we should be able to solve these.

Solve the first equation for P: 60 ft * R + 90 ft * P = $303

Subtract 60 ft * R from both sides of the equation: 90 ft * P = $303 - 60 ft * R

Now divide both sides of the equation by 90 ft: P = ($303 - 60 ft * R) / 90 ft = $101 / 30 ft - 2/3 * R

Next, substitute P into the second equation: 90 ft * R + 60 ft * P = $357

so we then have 90 ft * R + 60 ft * [$101 / 30 ft - 2/3 * R] = $357.

Now solve for R (remember, parentheses, exponents, multiplication and division, addition and subtraction):

90 ft * R + [$202 - 40 ft * R] = 50 ft * R + $202 = $357, or 50 ft * R = $155; so R = $155 / 50 ft = $3.10 / ft.

Now we can substitute this value for R into the first equation and solve for P: 60 ft * R + 90 ft * P = $303

so we then have 60 ft * $3.10 / ft + 90 ft * P = $303, or $186 + 90 ft * P = $303.

Subtracting $186 from both sides of the equation, and then dividing by 90 ft, we have P = $1.30 / ft

So P = $1.30/ft, and R = $3.10/ft.

To check this answer, plug both into the second equation:

90 ft * R + 60 ft * P = $357

90 ft * $3.10/ft + 60 ft * $1.30/ft = $279 + $78 = $357

75 tickets have been sold so far.

step-by-step explanation:

in this question it is clearly mentioned that 45 children's tickets have been sold and this number is 60% of the total tickets sold.

now let's assume the total number of tickets sold be x.

then 60% of x = 45

⇒

⇒ x(.6) = 45

by dividing with .6 on both the sides of the equation

⇒ x = 75

therefore 75 tickets have been sold so far.

x = ± 6i

step-by-step explanation:

x^2 = -36

take the square root of each side

sqrt(x^2) = sqrt(-36)

we know sqrt(ab) = sqrt(a) sqrt(b)

x = ±sqrt(36) sqrt(-1)

we know that sqrt (-1) = i

x = ± 6i