# SAT II Math I : Other Mathematical Relationships

## Example Questions

### Example Question #111 : Sat Subject Test In Math I

Multiply in modulo 6:

Explanation:

In modulo 6 arithmetic, a number is congruent to the reainder of its division by 6.

Therefore, since  and ,

.

The correct response is 0.

### Example Question #111 : Sat Subject Test In Math I

Which is an example of a set that is not closed under addition?

The set of all positive even integers

The set of all negative integers

The set

All of the sets given in the other responses are closed under addition.

The set of all integers between 1 and 10 inclusive

The set of all integers between 1 and 10 inclusive

Explanation:

A set is closed under addition if and only if the sum of any two (not necessarily distinct) elements of the set is also an element of the set.

The set of all negative integers is closed under addition, since any two negative integers can be added to yield a third negative integer.

The set of all positive even integers is closed under addition, since any two positive even integers can be added to yield a third positive even integer.

The remaining set is the set of all integers between 1 and 10 inclusive. It is not closed under addition, as can be seen by this counterexample:

but

### Example Question #111 : Sat Subject Test In Math I

varies directly as the square root of .

If   then . To the nearest tenth, calculate  if .

Explanation:

varies directly as , which means that for some constant of variation ,

We can write this relationship alternatively as

where the initial conditions can be substituted on the left side and final conditions, on the right. We will be solving for  in the equation

### Example Question #71 : Mathematical Relationships

varies inversely as the square of  and directly as the cube of .

If  and , then . Calculate  if .

Explanation:

varies inversely as  and directly as the cube of . This means that for some constant of variation

We can write this relationship alternatively as

where the initial conditions can be substituted on the left side and final conditions, on the right. We will be solving for  in the equation

### Example Question #1 : Other Mathematical Relationships

Sarah notices her map has a scale of .  She measures between Beaver Falls and Chipmonk Cove.  How far apart are the cities?

Explanation:

is the same as

So to find out the distance between the cities

### Example Question #2 : Direct Proportionality

If an object is hung on a spring, the elongation of the spring varies directly as the mass of the object. A 20 kg object increases the length of a spring by exactly 7.2 cm. To the nearest tenth of a centimeter, by how much does a 32 kg object increase the length of the same spring?

Explanation:

Let  be the mass of the weight and the elongation of the spring. Then for some constant of variation

We can find  by setting  from the first situation:

so

In the second situation, we set  and solve for :

which rounds to 11.5 centimeters.

### Example Question #1 : Other Mathematical Relationships

Sunshine paint is made by mixing three parts yellow paint and one part red paint. How many gallons of yellow paint should be mixed with two quarts of red paint?

(1 gallon = 4 quarts)

Explanation:

First set up the proportion:

x =

Then convert this to gallons:

### Example Question #1 : Direct Proportionality

Sally currently has 192 books. Three months ago, she had 160 books. By what percentage did her book collection increase over the past three months?

Explanation:

To find the percentage increase, divide the number of new books by the original amount of books:

### Example Question #1 : Direct Proportionality

Find  for the proportion .

Explanation:

To find x we need to find the direct proportion. In order to do this we need to cross multiply and divide.

From here we mulitply 100 and 1 together. This gets us 100 and now we divide 100 by 4 which results in

### Example Question #1 : Basic Single Variable Algebra

On a map of the United States, Mark notices a scale of    . If the distance between New York City and Los Angeles in real life is  , how far would the two cities be on Mark's map?