# SAT Mathematics : Solving Linear Equations in Word Problems

## Example Questions

### Example Question #1 : Solving Linear Equations In Word Problems

Jackie plans to buy one video game and a number of hardcover books. The video game costs $40 and each book costs$30. If she must spend at least $120 in order to get free shipping, what is the minimum number of books she must buy in order to get free shipping? Possible Answers: 4 2 5 3 Correct answer: 3 Explanation: Since you know that Jackie will purchase exactly one video game for$40, you can set up your equation (or inequality) here as:

Where  represents the number of books she buys. Since we're looking for the minimum number of books she needs to buy in order to be greater than or equal to $120, we use the greater-than-or-equal-to inequality. Now you can subtract 40 from both sides: And when you divide both sides by 30 you'll see that you have a number between 2 and 3. Note that you do not have to do the full decimal calculation, because you cannot buy part of a book! So since she can't buy 2.67 books, the minimum number she can purchase is 3. ### Example Question #2 : Solving Linear Equations In Word Problems On the first day of the week, a bakery had an inventory of 450 loaves of bread. It bakes 210 loaves of bread and sells 240 loaves of bread each day that it is open, and then closes for a baking day when it runs out of loaves. How many days can it be open before it must close for a baking day? Possible Answers: 5 2 9 15 Correct answer: 15 Explanation: 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: B = 450 - 30d Where B represents the number of loaves left and d 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 B = 0, you get: 0 = 450−30d. Now you can solve: add 30d to both sides to get: 30d = 450 And then divide both sides by 30 to get d = 15 ### Example Question #3 : Solving Linear Equations In Word Problems Katharine currently has$10,000 saved for a down payment on purchasing her first house, which she can do when her savings has reached $50,000. Each month she earns$6,500 but incurs $4,000 in expenses. If her earnings and expenses remain constant, how many months will it take until she has reached her savings goal? Possible Answers: 12 16 18 14 Correct answer: 16 Explanation: 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 m to represent the number of months, an initial equation would look like:

$50,000 =$10,000 + $6,500m -$4,000m

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:

$40,000 =$2,500m

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

m = 16 months

### Example Question #4 : Solving Linear Equations In Word Problems

If 75 gallons of water were added to a pool that is half full, the pool would then be  full. How many gallons of water does the pool hold when it is full?

Possible Answers:

Correct answer:

Explanation:

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:

### Example Question #5 : Solving Linear Equations In Word Problems

A wood beam 48 inches in length is cut into two smaller beams. If one of the new beams is 20 inches longer than the other, what is the length of the longer beam?

Possible Answers:

Correct answer:

Explanation:

To translate this word problem into equations, first assign variables to the new beams. Since you know there will be a longer and a shorter beam you might use  for longer and  for shorter. You would then have:

(the total length is 48)

(the longer is 20 more than the shorter)

You can then substitute the second equation into the first, replacing  with . This then gives you:

So  and

But remember - the question asked you for the LONGER beam, so you'll need to plug that back in to one of the equations. Since  you can say that , so 34 is the correct answer.

### Example Question #6 : Solving Linear Equations In Word Problems

Three friends split their dinner bill such that Shay paid $7 less than half the total bill, Ashley paid$8 more than  the total bill, and Jenna paid the remaining $20. How much was the total bill? Possible Answers: Correct answer: Explanation: While it is tempting to set up and solve an equation for this problem, the algebra for this problem may be difficult to translate quickly and accurately. Because you are dealing with relatively simple numbers that add up to a total, you should see that this should be an easy problem to backsolve. Start with a total of ,$80. If Shay paid $7 less than half of the total, that means she paid , or . And if Ashley paid$8 more than  of the bill, that means that she paid , or $28. That means that Jenna must have paid the remainder, . However, since you know that Jenna paid$20, you should recognize that the total for answer choice $80 was too small and that the actual bill must have been slightly more. You can go through the same process with answer choice$84. If Shay paid $7 less than half of the total, she paid . And if Ashley paid$8 more than  of the bill, that means that she paid .

Since you know that Jenna paid $20, you should recognize that . Since the original total and the final sum match up, you can conclude that$84 is your answer, pick it, and move on.

Alternatively, you could set up the algebra:

While the problem setup here may seem a bit involved, if you are quick to combine like terms you should see that the algebra is very manageable here. You're given two fractions of the bill ( and  of the total) and three actual values (for Shay subtract 7, for Ashley add 8, and for Jenna add 20). So what you really have when you combine all of your like terms is:

(where  represents the total bill). Since $21 represents the missing of the bill, the bill is 4($21) = $84. ### Example Question #7 : Solving Linear Equations In Word Problems Eight times one-third of a number is one greater than five times one half of that number. What is that number? Possible Answers: Correct answer: Explanation: 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. ### Example Question #8 : Solving Linear Equations In Word Problems Of the 480 cars on the lot at a dealership, are hybrids. If of the hybrids were to be sold, what fraction of the total number of cars on the lot would hybrids then represent? Possible Answers: Correct answer: Explanation: 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: ### Example Question #9 : Solving Linear Equations In Word Problems Five years ago, Juliet was three times as old as Will. If Juliet is currently twice as old as Will, how old is Juliet? Possible Answers: Correct answer: Explanation: 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 . ### Example Question #10 : Solving Linear Equations In Word Problems On the same day, Devin bought a car for$40,000 that would then decrease in value by $1200 per year, and Anita purchased a piece of real estate for$6,000 that would then increase in value by \$500 per year. After how many years will Anita’s land be worth as much as Devin’s car?

Possible Answers:

Correct answer:

Explanation:

Mathematically you can express the value of Devin’s investment using the equation , where  represents the number of years. You can do the same for Anita’s using . And then to put them together, recognize that you’re looking for when Anita’s real estate is worth as much as Devin’s car, so you’ll want to use the equation

Then combine like terms by adding  and subtracting  on each side:

The math here reduces quickly, as you can first divide both sides by :

And you know that , so the answer then must be .