Word Translations - SAT Math

First, translate the problem into three equations. The statement, "Joule is three years older than twice Newton's age" is mathematically translated as

where represents Joule's age and is Newton's age.

The statement, "Newton is Toby's age younger than eleven years" is translated as

where is Toby's age.

The third statement, "Toby is one year younger than Joule" is

.

So these are our three equations. To figure out the age of these dogs, first I will plug the third equation into the second equation. We get

Plug this equation into the first equation to get

Solve for . Add to both sides

Divide both sides by 3

So Joules is 9 years old. Plug this value into the third equation to find Toby's age

Toby is 8 years old. Use this value to find Newton's age using the second equation

Now, we have the age of the following dogs:

Joule: 9 years

Newton: 3 years

Toby: 8 years

Since this is a long word problem, it might be easy to confuse the two behaviors and come up with the wrong answer. Let's avoid this problem by turning each behavior into a variable. If we call "sitting quietly" and "completing assignments" , then we can easily construct a simple system of equations,

and

.

We can multiply the first equation by to yield .

This allows us to cancel the terms when we add the two equations together. We get , or .

A quick substitution tells us that . So, sitting quietly is worth 3 tokens and completing an assignment on time is worth 7.

Let be the number of adult tickets sold. Then the number of child tickets sold is .

The amount of money raised from adult tickets is ; the amount of money raised from child tickets is . The sum of these money amounts is , so the amount of money raised can be defined by the following equation:

To find the number of adult tickets sold, solve for :

adult tickets were sold.

6,035 total tickets were sold, and the total number of tickets is the sum of the adult and child tickets, .

Therefore, we can say .

The amount of money raised from adult tickets is $11 per ticket mutiplied by tickets, or dollars; similarly, dollars are raised from child tickets. Add these together to get the total amount of money raised:

These two equations form our system of equations.

This system can be solved a variety of ways, including graphing. To solve algebraically, write an equation for each of the different trains. We will use y to represent the distance from San Francisco, and x to represent the time since 8AM.

The blue train travels 80 miles per hour, so it adds 80 to the distance from San Francisco every hour. Algebraically, this can be written as .

The green train starts 380 miles away from San Francisco and subtracts distance every hour. This equation should be .

To figure out where these trains' paths will intersect, we can set both right sides equal to each other, since the left side of each is .

add to both sides

divide both sides by 140

Since we wrote the equation meaning time for , this means that the trains will cross paths after 2.714 hours have gone by. To figure out what time it will be then, figure out how many minutes are in 0.714 hours by multiplying . So the trains intersect after 2 hours and about 43 minutes, so at 10:43AM.

To figure out how far from San Francisco they are, figure out how many miles the blue train could have gone in 2.714 hours. In other words, plug 2.714 back into the equation , giving you an answer of .

Let's call "" the cost of one jersey (this is the value we want to find)

Let's call the amount of money Jack starts with ""

Let's call the amount of money Aaron starts with ""

We know Jack buys a glove for and bats for each, and then has left over after. Thus:

simplifying, so Jack started with

We know Aaron buys hats for each and jerseys (unknown cost "") and spends all his money.

The last important piece of information from the problem is Aaron starts with dollars more than Jack. So:

From before we know:

Plugging in:

so Aaron started with

Finally we plug into our original equation for A and solve for x:

Thus one jersey costs

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

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:

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 .

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:

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 .

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.

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:

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: .

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 .

What gives this problem its difficulty is its abstraction--all the answer choices are variables--but you can simplify that abstraction in one of two ways:

1. Write out the relationship in words that you know. You know that the amount of money you earn per hour times the number of hours is your total pay. For Alexis, we know that's . And we know that her total pay is at least (which means greater than or equal to) . So by methodically listing out the concepts, not just the variables, we can get to:

2. You could also pick simple numbers to set up the math, since most people are much more comfortable with numbers than with variables. If you were to say that (she earns $20/hour) and that she works 10 hors per week (), you would know to set up her total pay as . Well, working back to variables, you know that that's . And then if you were to say that that total is at least , that's .

Since we know that stands for weeks, the answer has to have something to do with the weeks. This eliminates "the number of candles she has remaining for the month." Also, we can eliminate "the number of weeks that she has sold candles this month" because that would be our value for , not what we'd multiply by. The correct answer is, "the number of candles that she sells each week."

First, translate the problem into three equations. The statement, "Joule is three years older than twice Newton's age" is mathematically translated as

where represents Joule's age and is Newton's age.

The statement, "Newton is Toby's age younger than eleven years" is translated as

where is Toby's age.

The third statement, "Toby is one year younger than Joule" is

.

So these are our three equations. To figure out the age of these dogs, first I will plug the third equation into the second equation. We get

Plug this equation into the first equation to get

Solve for . Add to both sides

Divide both sides by 3

So Joules is 9 years old. Plug this value into the third equation to find Toby's age

Toby is 8 years old. Use this value to find Newton's age using the second equation

Now, we have the age of the following dogs:

Joule: 9 years

Newton: 3 years

Toby: 8 years