Set up a system of linear equations based on our data:

40,000P + 45,000A = 40,415,000

P/A = 4/3

To make things easiest, solve the second equation for A in terms of P:

A = (3/4) P

Replace this value into the first equation:

40,000P + 45,000 * (3/4)P = 40,415,000

Simplify:

40,000P + 33,750P = 40,415,000

73,750P = 40,415,000

P = 548 (The number of professors)

To show that c is of the profit of the transaction, we must represent the profit as the difference between the price and the value of the car, or "(p – v)"

To show that Abby's commission in dollars is a percentage of the profit, we use 0.01 * c to convert the commission she earns to a percent.

To shift the earnings from Abby to the dealership (which is what the question requires of us), we must take 1 – 0.01c since this will accommodate for the remaining percentage. For example, it shifts 75% (0.75) to 25% (1 – 0.75 or 0.25).

Putting this all together, we get a final expression of:

(p – v)(1 – 0.01c) = dealership earnings

Check answer with arbitrary values: letting p = 300, v = 200, and c = 20, we get a value of 80 which makes sense as the $100 profit must be distributed evenly between Abby ($20) and the dealership ($80).

To simplify the word problem, express the ages in terms of variables in a system of equations. Note that we want to compare S with D:

S = A – 2

D = T + 5

A = T + 6

By substituting A for T in the first equation, we can get S in terms of T, which will let us directly compare the values of S and D.

S = (T + 6) – 2 = T + 4

If D = T + 5, and S = T + 4, D must be the greater value and Daisy is one year older than Sally.  B is the correct answer.

First let's solve for y using the second equation, x – y = 6.

x = 6 + y.  Then plug this in to the other equation.

3 (6 + y) + 4y = 5

18 + 3y + 4y = 5

18 + 7y = 5

7y = –13

y = –13/7. Now plug this value back into x = 6 + y.

x = 6 – 13/7 = 29/7.  x is positive and y is negative, so clearly x is larger, so Quantity A is bigger.

Because there are two different equations for the two variables (x and y), you are able to solve for the value of each. You can rearrange the first equation to show that

y = 12 – x by subtracting x from both sides.  

Then you can substitute this equation into the second equation to give you

2x – (12 – x) = 6.  

This new equation can be simplified to

2x – 12 + x = 6

3x =18

x = 6

Filling this back into the first equation, we get 6 + y = 12 which means y must also equal 6.  Because x and y are equal, we choose the option both quantities are equal.

Let c = number of kids tickets sold. Then (548 – c) adult tickets were sold. The revenue from kids tickets is $5c, and the total revenue from adult tickets is $10(548 – c). Then,

5c + 10(548 – c) = 3750

5c + 5480 – 10c = 3750

5c = 1730

c = 346. 

We can check to make sure that this number is correct:

$5 * 346 tickets + $10 * (548 – 346) tickets = $3750 total revenue

Let s stand for the sick tree and h for the healthy tree. The beginning information about cutting the sick tree and the healthy tree growing is actually not needed to solve this problem! We know that the healthy tree is 4 feet taller than the sick tree, so h = s + 4.

We also know that the two trees are 28 feet tall together, so s + h = 28. Now we can solve for the two tree heights.

Plug h = s + 4 into the second equation: (s + 4) + s = 28. Simplify and solve for h: 2s = 24 so s = 12. Then solve for h: h = s + 4 = 12 + 4 = 16.

1. Add 7 to both sides

3(z + 4)³ – 7 + 7= 17 + 7

3(z + 4)³ = 24

2. Divide both sides by 3

(z + 4)³ = 8

3. Take the cube root of both sides

z + 4 = 2

4. Subtract 4 from both sides

z = –2

For this type of problem, we use the formula:

When Karen catches up with Jen, their distances are equivalent. Thus,

We then make a variable for Jen's time, . Thus we know that Karen's time is  (since we are working in hours).

Thus,

There's a logical shortcut you can use on "catching up" distance/rate problems. The difference between the faster (Karen at 60mph) and slower (Jen at 45mph) drivers is 15mph.  Which means that every one hour, the faster driver, Karen, gains 15 miles on Jen.  We know that Jen gets a 1/2 hour head start, which at 45mph means that she's 22.5 miles ahead when Karen gets started.  So we can calculate the number of hours (H) of the 15mph of Karen's "catchup speed" (the difference between their speeds) it will take to make up the 22.5 mile gap:

15H = 22.5

So H = 1.5.

Bill builds  toys an hour. Bob builds  toys an hour. Together, their rate of building is . Together they can build  toys in 2 hours. They would be able to build  toys in 8 hours.