Word Translations & Equation Interpretation - 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

The correct answer is “This relationship is exponential and doubles every hour.” Since the time variable is an exponent, we can narrow down our options and definitively state the function is exponential. The is an exponent with a base of , so as grows, the number of times is multiplied by itself also grows. Each multiplication of two is a doubling of the previous hour’s population (when , vs. when , ).

The correct answer is II. This is an exponential function, not a linear function. When we plug in for , . When we plug in for , , not .

The correct answer is . Maria’s apple harvest follows an exponential pattern while her peach harvest is linear. Since time is relative to any given start point, we can call 2003 the year where . Her apple harvest *grows* by so when we convert our percent back to a decimal, we get an exponent base of . Recall, that if her harvest was *losing* of produce yearly, we would use . Since her starting harvest in 2003 was bushels, we can model her total bushel count as a function of time through the following equation: .

Her peach harvest is a linear function that grows by bushels every years. Since time is measured in single years and our growth is given in periods of years, be sure to take this into account in your linear equation . Since her starting harvest in 2003 was bushels, we can model her total bushel count as a function of time through the following equation: .

When we plug into our equation from subtracting 2003 from 2018, we get bushels of apples and bushels of peaches. .

The correct answer is “This company is exponentially shrinking by .” The above function is an exponential function of t because is an exponent of a base in the function. A linear equation would utilize as a coefficient, not an exponent. The best way to differentiate between growth and shrinkage in an exponential function is to see if the exponent’s base is greater or less than . If the base is , there would be no change in regardless of the value of , but if the base is less than , the company’s value is decreasing. so a loss of of market value annually.

The correct answer is “Plant A shows exponential growth while Plant B is linear.” Each hour doubles the height of Plant A. Doubling is a characteristic of exponential growth . Each hour adds cm to the height of Plant B. Additive and subtractive trends are characteristic of linear growth .

The correct answer is . The graph is curved and does not have a defined slope, so it cannot be a linear function . Now, between our exponential functions, we can see that at , , so this must be a coefficient outside of the exponential base. This only leaves . It might be one’s first thought to see that at , , pointing to the choice, but this equation does not work at any other value of .

Both exponential equations should represent curved lines while the linear equations should be straight lines. shows exponential growth, so as increases, should increase. shows exponential decay, so as increases, should decrease. This narrows us down to Option 2 and 4. Within the linear equations, should be a steeper line than since the slope is greater, thus leaving us with Option 4 as the correct answer.

Since the total moth population is a percentage of the previous period’s population, we know this is an exponential function and not a linear one. We also know that the population has been decreasing. An exponent base of would represent no change in the population, while a base of less than one would represent a decrease, so this rules out the option with as the exponent base. The population has been decreasing by so we can say the original of the population loses , leaving us with of the original population for every years that pass. Also, we use because the population only decreases by every years. This leaves us with .

Exponential functions are in the form while linear are . Linear functions change at a constant rate per unit interval while exponential functions change by a common ratio over equal intervals.

When (the y-intercept), . The base of this equation is . Exponential functions never have x-intercepts unless they are in the form .

Note that in this equation, the number 54.1 is a constant: it isn't multiplied by anything. So 54.1 is a minimum number: a femur cannot have a minimum length, so even if were to equal zero, the estimated height would be 54.1. This tells you that 54.1 has to be the absolute minimum estimated height of a hominid using this method.

Remember: in word problems, units are always important! In this problem, thinking about units is a great place to start. Since you know that P is a number of pushups, then the other side of the equation needs to be a number of pushups, too, so that the units match. And three of the four items in the equation are given to you:

P = Number of pushups left
270 = Total number of pushups
d = Days that Trey has done pushups

That alone helps you use process of elimination, since "number of days he's already done pushups" and "number of pushups remaining" are already defined by other parts of the equation. And it should also help you think your way through it. Without using the test’s variables, you could set up your own equation:

Pushups left = Total - Pushups he’s already done

And since “pushups he’s already done” is represented mathematically as 45d, you can eliminate that answer since 45 itself doesn’t equal that overall value – it needs to be multiplied by d. Or, as before, you can look at the 45d portion as a recurring amount. d = the number of days, and if you multiply the number of days by the number of pushups per day, you’ll get the total amount of pushups over that period. So 45 is the number of pushups that he does each day.

Notice that the number 2.50 isn’t multiplied by anything, so it’s not a recurring charge of any kind. Even if m = 0, that 2.50 remains unchanged, so 2.50 is the charge that happens regardless of the distance. This tells you that 2.50 is the up-front fee that is charged regardless of the number of miles on the trip.

On occasion, the SAT will present you with problems like this one, in which you have to determine what must be true for up to three different statements. These tend to be a bit more time consuming, because they’re essentially three problems in one.

Looking at the given equation, , analyze the role of each term.

is the total number of bees at any given point of time.

says that for every month, the number of bees increases by approximately 2/3 (note that this is the beekeeper's model, not an exact number, so it's okay that the number of bees isn't always an integer)

is a constant, and if were 0 the total number of bees would be 212. So this must mean that there were 212 bees to start.

If you then look at the statements:

I is not true, as the number of bees is increasing with each month.

II is true; the beginning number was 212.

III is true; if the number of bees grows by 2/3 each month, and there are 12 months in a year, then the number of bees is growing by an estimated 8 bees each year.