This problem can be solved algebraically but is likely solved more quickly by simply back solving. Your algebra would set up as:

(eight times one-third of a number) (is one greater than) (five times one half of that number).

Those who are algebraically inclined could of course find common denominator, combine the terms, and solve (more on that below). But others might be quicker looking at the answer choices. Since one of the first steps the problem tells you is that the number has to be divided by three ("eight times one-third of a number"), you'll want to plug in a number divisible by three for an easy start. And since you'll also have to divide that number by 2 ("five times one half that number"), you'll want an even one.

So start with 6. Eight times one-third of 6 is 8(2) = 16. Is that one greater than five times one half of 6? It is. Five times one half of 6 is 5(3) = 15. The relationship works, so 6 is correct.

An important concept when calculating discounts and percent reductions is that if something is discounted by 25%, the other 75% (the portion necessary to add to the original 100%) is what's left. Essentially, 25% off means 75% "on." So here Olivia doesn't pay 25% of the price, meaning that she does pay the other 75%. So the price she pays for the telescope is .

Then she is responsible for the sales tax. And note that the question asks only about the amount of sales tax she pays, not the total amount. For that reason, you'll multiply by and not by . The "1" would represent the fact that you keep the total amount (her price, plus tax), but if you only care about the smaller amount of tax she pays, you don't keep the amount of the sale. That makes your answer .

When dealing with percent discounts, recognize that when the discount is subtracted from the total of 100%, you have the actual amount paid. So if George got a 20% discount, that means that he paid 80% of the price. So George's cost for the pizza can be represented as .

Then you need to account for the tip. An important thing to note is that the question asks for George's total cost: the pizza plus the tip. For that reason, you'll multiply by 1 to account for the price he paid for the pizza, plus the 0.15 to account for the 15% tip. That's why you multiply by 1.15; had it only asked for the tip, you wouldn't need the 1, but the 1 represents "he paid for the pizza, plus a percentage of it."

Therefore the answer is .

If the bakery bakes 210 loaves of bread and sells 240 loaves of bread, then that means that total, it loses 30 loaves of bread per day. Since you know that it starts with 450 loaves of bread, you can use this information to write a linear equation relating the number of loaves of bread left with how many days it has been since the bakery has closed for a baking day. It should look like:

Where represents the number of loaves left and represents the number of days since the bakery’s last baking day.

In order for the bakery to need to close for a baking day, the number of loaves left must equal 0. If you substitute in , you get: .

Now you can solve: add to both sides to get:

And then divide both sides by 30 to get

You're given that the length of the picture frame is , and you know that the perimeter of a rectangle can be expressed as 2(Length + Width), so you're halfway there. Next you have to account for the way to represent the width. If the width is 4 inches shorter than the length, that means that the width is .

Then you need to plug in those values to the formula 2(Length + Width). That gives you , which reduces to and then to , the correct answer.

Note that on these questions that ask you to algebraically express a relationship, you also have the opportunity to pick numbers and then test the answer choices. If you were to say that the length of the frame is 6 and the width is 2 (holding of course to the rule from the question that the length is 4 inches longer than the width), you could say then that the perimeter is 6 + 6 + 2 + 2 = 16. Then plug in to the answer choices to see which one gives you 16. Only the correct answer does: in that situation, would be 24 - 8 = 16, proving the right answer.

To translate this word problem into equation form, note that your unknown is the capacity of the pool. You know that the pool is currently half full and that with the proposed addition it would be 2/3 full, but you need to assign those fractions to an actual value. So your variable, , will represent the total capacity of the pool. Then you can set up your problem as:

From here it's an algebra problem. To get your terms together, subtract from both sides:

Then you'll need to find a common denominator of 6, and rewrite what you have to reflect that common denominator:

You can then combine like terms on the right-hand side of the equation:

And then multiply both sides by 6 to finish the problem:

This word problem gives you two totals: one related to a total number of items sold, and the other related to the total revenue.

To turn the number of items sold into an equation, just add the number of boxes of popcorn and the number of sodas and set that equal to the total number of items: .

For the revenue equation, note that the revenue is equal to the price per item times the number of items sold. That means that the revenue from popcorn would be and the revenue from sodas would be , and the total revenue of $100 would come from adding those two pieces: .

To begin on this problem, first, take of 480 to determine that there are currently 120 hybrid cars. So if half were to be sold, then that would leave 60 hybrids.

Importantly, note that if 60 hybrids are sold, that means that 60 of the total of 480 cars are sold. That affects both the number of hybrids AND the number of total cars, which is now reduced to 420. So your new proportion of hybrids is the 60 hybrids out of the 420 total cars:

Age problems derive much of their difficulty from the fact that people are often careless in accounting for time shifts. The current age equation here should be easy to set up: . But the "5 years ago" equation takes a bit of thought: 5 years ago, they were BOTH five years younger, so you need to account for that on both sides of the equation. Juliet, five years ago, was three times as old as Will, five years ago:

If you then substitute in for , you have:

Adding 15 to both sides and subtracting from both sides leaves:

And since the problem asked for Juliet's age, not Will's, you'll plug in to to get .

To turn this problem into an equation, you can start by putting Katharine's goal of $50,000 on one side of the equation, and then arranging all of the inputs to that goal (ways she earns money) and impediments (ways she loses money) on the other. If you use to represent the number of months, an initial equation would look like:

Which, qualitatively, is that her goal of $50,000 is equal to the $10,000 she has plus the $6,500 she earns each month, minus the $4,000 she loses each month.

Now you can combine like terms to simplify the equation:

And then divide 40000 by 2500 (which simplifies to 400 divided by 25) to solve:

months