Describing Roles of Variables & Constants - SAT Math

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.

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."

As you can see from the answer choices to the first question, you need to make two important determinations:

• Is this an exponential decrease, or a linear decrease?
• Should the number in question be 73 or 27? When you’re determining whether a change is exponential or linear, look to see if there is an exponent. The exponent, , just outside the parentheses tells you that this is an exponential function. What does that mean? That the loss of 27% per month compounds each month. Exponential increases and decreases mean that the increase or decrease is taken as a percent of the new value each time; linear increases and decreases mean that the same amount is increased or decreased each month, because that percent change is taken from the original value. So this is an exponential change, meaning that the answer must be one of the percentages, not one of the numbers.

The fact that this is a decrease tells you what is happening inside the parentheses. For an exponential decrease, the formula is:

New value = (Original Value)(1 - rate)(number of periods)

The rate is a percentage expressed as a decimal, and in a decrease problem that rate is always subtracted from 1. This is because you’re calculating the number of items (in this case trees) remaining, so you need to convert the decrease into a function of the number that you keep, not the number that you’re losing. Think of it like a discount at a store: if an item is 20% off, you pay the other 80% of the price that’s remaining. So here, the expression shows that 73% of the trees remain each month, meaning that the forest is losing 27% of its trees each month (1−0.27=0.73). Therefore, the correct answer is "the number of trees is decreasing by 27% each month."

When you see formulas like this that express the total cost for a number of items, there are some standard components you should recognize. Whenever there is a plain number added, with no variable next to it (as you see with + 10 here), that is a fixed charge that does not depend on the number of items purchased. Where you do see a number multiplied by a variable, like with 12s here, that represents the amount of money that depends on the number of items purchased: the numerical coefficient is the cost-per-item, and the variable is the number of items. Which should make logical sense - the shopper gets to choose the number of items to buy, so that number can vary customer to customer, making a "variable" appropriate there. Here you're asked about the number 12, which is multiplied by the variable s - that means that 12 is the price per shirt, with s being the number of shirts Shawn purchases.

In this equation you know that is the total number of pizzas and that is the number of football games. Think about what happens: if increases, there are more pizzas ordered. tells us how many extra pizzas are ordered for each game that is being televised. The pizzeria would sell 10 pizzas if no games were on TV, 13 if one is on, 16 if two is on, etc. So is the number of additional pizzas expected to be ordered per football game on TV.

An exponential change calculation generally has three components: the initial amount, the percent change, and the number of periods that have undergone that change. The typical structure of such a calculation is:

New Amount = Original Amount(1 +/- Percent Change As A Decimal)Number of Periods

Here we're dealing with a 12.8% decrease, so within the parentheses the percentage is subtracted, and has been identified as the number of periods. is the original amount that gets multiplied by the percent change taken to its exponent, so the correct answer is the original amount, which in this problem is defined as "The number of square kilometers of ice in Greenland as of September 30, 1979."

In this problem, Sarah's income is quoted as an annual amount but her expenses are quoted as a monthly amount. In order to calculate her savings, she needs those units to both be in either "per year" or "per month." Since the variable also represents a number of months, it is necessary for her to convert her annual salary into a monthly amount so that she can properly subtract expenses from income to reflect the amount of money saved. Therefore, dividing by 12 is how she makes that conversion.

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.

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."

As you can see from the answer choices to the first question, you need to make two important determinations:

• Is this an exponential decrease, or a linear decrease?
• Should the number in question be 73 or 27? When you’re determining whether a change is exponential or linear, look to see if there is an exponent. The exponent, , just outside the parentheses tells you that this is an exponential function. What does that mean? That the loss of 27% per month compounds each month. Exponential increases and decreases mean that the increase or decrease is taken as a percent of the new value each time; linear increases and decreases mean that the same amount is increased or decreased each month, because that percent change is taken from the original value. So this is an exponential change, meaning that the answer must be one of the percentages, not one of the numbers.

The fact that this is a decrease tells you what is happening inside the parentheses. For an exponential decrease, the formula is:

New value = (Original Value)(1 - rate)(number of periods)

The rate is a percentage expressed as a decimal, and in a decrease problem that rate is always subtracted from 1. This is because you’re calculating the number of items (in this case trees) remaining, so you need to convert the decrease into a function of the number that you keep, not the number that you’re losing. Think of it like a discount at a store: if an item is 20% off, you pay the other 80% of the price that’s remaining. So here, the expression shows that 73% of the trees remain each month, meaning that the forest is losing 27% of its trees each month (1−0.27=0.73). Therefore, the correct answer is "the number of trees is decreasing by 27% each month."

When you see formulas like this that express the total cost for a number of items, there are some standard components you should recognize. Whenever there is a plain number added, with no variable next to it (as you see with + 10 here), that is a fixed charge that does not depend on the number of items purchased. Where you do see a number multiplied by a variable, like with 12s here, that represents the amount of money that depends on the number of items purchased: the numerical coefficient is the cost-per-item, and the variable is the number of items. Which should make logical sense - the shopper gets to choose the number of items to buy, so that number can vary customer to customer, making a "variable" appropriate there. Here you're asked about the number 12, which is multiplied by the variable s - that means that 12 is the price per shirt, with s being the number of shirts Shawn purchases.

In this equation you know that is the total number of pizzas and that is the number of football games. Think about what happens: if increases, there are more pizzas ordered. tells us how many extra pizzas are ordered for each game that is being televised. The pizzeria would sell 10 pizzas if no games were on TV, 13 if one is on, 16 if two is on, etc. So is the number of additional pizzas expected to be ordered per football game on TV.

An exponential change calculation generally has three components: the initial amount, the percent change, and the number of periods that have undergone that change. The typical structure of such a calculation is:

New Amount = Original Amount(1 +/- Percent Change As A Decimal)Number of Periods

Here we're dealing with a 12.8% decrease, so within the parentheses the percentage is subtracted, and has been identified as the number of periods. is the original amount that gets multiplied by the percent change taken to its exponent, so the correct answer is the original amount, which in this problem is defined as "The number of square kilometers of ice in Greenland as of September 30, 1979."

In this problem, Sarah's income is quoted as an annual amount but her expenses are quoted as a monthly amount. In order to calculate her savings, she needs those units to both be in either "per year" or "per month." Since the variable also represents a number of months, it is necessary for her to convert her annual salary into a monthly amount so that she can properly subtract expenses from income to reflect the amount of money saved. Therefore, dividing by 12 is how she makes that conversion.