Equations

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ISEE Upper Level Quantitative Reasoning › Equations

Questions 1 - 10
1

Define .

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal

It cannot be determined which of (a) and (b) is greater

(a) is the greater quantity

(b) is the greater quantity

Explanation

Also,

Therefore, .

2

and are both negative.

Which is the greater quantity?

(a)

(b)

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

Explanation

, so either

or

Since is negative, is the only possibility.

, so either

or

is negative, so neither value can be eliminated.

. If , then ; if , then is the greater quantity. Therefore, it cannot be determined which is the greater.

3

On a 70-question exam, Lisa answered 60 percent correctly. How many answers did she get right?

Explanation

If Lisa answered 60 percent of the questions on a 70-question exam correctly, the following equation can be used to determine how many quesitons she got right. is equal to the number of questions she answered correctly.

Given that , it follows that

Next, we cross multiply, which gives us:

Now, we divide each side by 5, resulting in:

4

One-third of the sum of a number and sixty is ninety-three. What is the number?

Explanation

If we let be the number, "the sum of a number and sixty" can be written as

"One-third of the sum of a number and sixty" can be written as

Set this equal to ninety-three and solve:

5

Two lines have -intercept . Line A has -intercept ; Line B has -intercept . Which is the greater quantity?

(A) The slope of Line A

(B) The slope of Line B

(A) is greater

(B) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

Explanation

To get the slope of each line, use the slope formula

For Line A, . Substitute in the slope formula.

The slope is

For Line B, . Substitute in the slope formula.

The slope is

Since

,

Line A has the greater slope, and (A) is greater.

6

Which of the following is a true statement?

Explanation

,

so

Using two substitutions:

The correct choice is .

7

When is divided by , the remainder is . What is the remainder when is divided by ?

Cannot be determined

Explanation

Pick a number for that satisfies the condition for division by 15 and see what happens when it is divided by 7.

33 divided by 15 leaves a remainder of 3. 33 divided by 7 leaves a remainder of 5.

Let's try another number as well.

48 divided by 15 leaves a remainder of 3. 48 divided by 7 leaves a remainder of 6, which is different from the remainder left by 33.

Similarly, 63 divided by 15 leaves a remainder of 3. 63 divided by 7 leaves no remainder at all. Therefore, the answer cannot be determined.

8

Which of the following makes the equation true:

Explanation

To find the answer, we will solve for x. So, we get

9

Solve the equation for :

Explanation

First, simplify the equation as much as possible:

Now we can take the of both sides:

10

Three consecutive integers add up to 36. What is the greatest integer of the three?

Explanation

To solve this problem, you can translate the question into an equation. It should look like: . Since we don't know the first number, we name it as x. Then, we add one to each following integer, which gives us x+1 and x+2. Then, combine like terms to get . Solve for x and you get 11. However, the question is asking for the greatest integer of the set, so the answer is actually 13 (because it is the x+2 term).

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