# SAT Mathematics : Translating Words to Linear Equations

## Example Questions

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### Example Question #1 : Translating Words To Linear Equations

Jason is 10 years older than Alexa. Using  to represent Jason's age and  to represent Alexa's age, which of the following demonstrates the correct relationship between Jason's and Alexa's ages?

Explanation:

Many words translate directly to mathematical symbols, and "is" is a great example: "is" means "equals" (if you were to say x is 6, then you could express that as x = 6, for example).

That helps here, because when you're told that "Jason is 10 years older than Alexa" you can put an equals sign right were "is" appears:  10 years older than Alexa. Then to account for the rest of the equation, recognize that we need more than Alexa's current age to equal Jason's, so we'll use a + sign there, arriving at:

Note that if you see a question like this on the SAT, it's helpful to choose numbers that work with the situation to check your answer. If Jason is 10 years older than Alexa, he might be 15 while she is 5. If you plug those in to the correct answer, you will see that it works:

15 = 10 + 5

### Example Question #1 : Translating Words To Linear Equations

Riley and four friends went to dinner and agreed to split the bill evenly. If the bill came to  dollars and the group left a 20% tip, which of the following represents the amount that each person paid?

Explanation:

One important thing to identify in this problem is that Riley and her four friends went to dinner, meaning that there were 5 total people at dinner, not 4. Then you can see that the price of the dinner, , plus 20% as a tip, would come out to . Because each person only pays their equal share, we divide that total amount of money,  by 5 to get our answer for what each of the five people had to pay:

### Example Question #2 : Translating Words To Linear Equations

At a medical practice, Dr. Ajit sees  patients per day and works 4 days per week, and Dr. Benton sees  patients per day and works 5 days per week. Which of the following expresses the number of patients that the two doctors, combined, see each week?

Explanation:

In this word translation, it can be helpful to recognize what multiplication really is: a faster way to express repetitive addition. For Dr. Ajit, seeing  patients each day for 4 days would mean that he would see  patients, but you can more simply express that as . The same ideology holds for Dr. Benton, whose total number of patients would be . Their sum, then, is .

### Example Question #4 : Translating Words To Linear Equations

Julia's window is 24 feet above the street. From her window, she releases a helium balloon that rises at a constant rate of 6 feet per second. Which of the following represents the height, , that the balloon will have risen after  seconds?

Explanation:

The height of the balloon begins at 24 feet above the street level, and only gets higher every second. So the first term in the expression should be the constant 24.  Then each second the balloon rises 6 additional feet. This could be expressed as 6 + 6 + 6 + 6... for as many seconds as we're looking for, but since we don't know how many seconds the question wants, it just gives us the variable . So we have to multiply 6 by the number of seconds, , to get our equation:

### Example Question #5 : Translating Words To Linear Equations

Olivia bought a telescope that was marked at a 25% discount off of the retail price. If the retail price before the discount was  dollars and Olivia had to pay a 9% sales tax on the price that she paid, which of the following represents the amount of sales tax that Olivia paid?

Explanation:

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 .

### Example Question #1 : Translating Words To Linear Equations

George uses a coupon to get a 20% discount on a pizza, which normally costs  dollars. He then gives the delivery driver a 15% tip on the price that he paid for the pizza. Which of the following represents George's total cost for the pizza, inclusive of tip?

Explanation:

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."

### Example Question #7 : Translating Words To Linear Equations

The length of a rectangular picture frame is 4 inches longer than the width of the frame. If the length is represented as , which of the following expresses the perimeter of the picture frame, in inches?

Explanation:

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.

### Example Question #8 : Translating Words To Linear Equations

During a recycling drive, Meghan and Phil combined to collect 900 recyclable bottles, and Meghan collected 200 more cans than Phil did. If the number of cans that Meghan collected is represented by  and the number that Phil collected is represented by , which of the following is a system of equations that could be used to solve for their individual totals?

Explanation:

This word problem provides you with two ways to look at the situation. The first is a total number of bottles, 900, that is created by adding the totals of Meghan and Phil. That means that . The other is that Meghan collected 200 more than Phil. That means that you would need to add 200 to Phil's total to equal Meghan's:

### Example Question #9 : Translating Words To Linear Equations

A movie theater snack bar charges $4 for each box of popcorn and$2.50 for each soda. On a particular day the snack bar sold a total of 31 items and earned a total of $100. Which of the following systems of equations could be used to solve for the number of boxes of popcorn, , and number of sodas, , the snack bar sold that day? Possible Answers: Correct answer: Explanation: 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: .

### Example Question #10 : Translating Words To Linear Equations

A skating rink can be rented for private parties. For the rental, the rink charges a $100 setup fee plus$3 per minute of the duration of the party. Which of the following expresses in dollars the rental cost, , of a party that lasts for  minutes?

The initial cost of the party is $100, and that part of the cost does not increase or decrease based on the number of minutes. So the correct equation must have a standalone as part of the cost. Then the additional cost is$3 for each minute. That means you need to multiply \$3 times the number of minutes, .  This makes the proper equation .