ISEE Upper Level Quantitative Reasoning - Algebraic Concepts

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).

N is a positive integer; . Which is greater?

(a) $N^{2} - 4N + 4$

(b)

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

(a) and (b) are equal.

Explanation

If , then $N^{2} - 4N + 4 = 2 ^{2} - 4 \cdot 2 + 4= 4 - 8 + 4 = 0$ .

If , then $N^{2} - 4N + 4 = 3 ^{2} - 4 \cdot 3 + 4= 9 - 12 + 4 = 1$ .

Therefore, at least two possibilities can be demonstrated.

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.

Factor completely:

$3x^{2} - x -14$

$\left ( x+2 \right ) \left (3x- 7 \right )$

$\left ( x-2 \right ) \left (3x+ 7 \right )$

$\left ( x-2 \right ) \left (3x- 7 \right )$

$\left ( x+7 \right ) \left (3x-2 \right )$

$\left ( x-7 \right ) \left (3x+2 \right )$

Explanation

A trinomial whose leading term has a coefficent other than 1 can be factored using the -method. We split the middle term using two numbers whose product is and whose sum is . These numbers are , so:

$3x^{2} - x -14$

$= 3x^{2} - 7x +6x -14$

$= \left (3x^{2} - 7x \right ) +\left ( 6x -14 \right )$

$=x \left (3x- 7 \right ) +2 \left (3x- 7 \right )$

$=\left ( x+2 \right ) \left (3x- 7 \right )$

Define $f(x) = \frac{1}{x}$ .

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

$f(x) = \frac{1}{x}$

$f(A) = \frac{1}{A}=10$

$\frac{1}{A} \cdot \frac{A}{10} =10 \cdot \frac{A}{10}$

$\frac{1} {10} =A$

Also,

$f(10) = \frac{1}{10} = B$

Therefore, .

Which is the greater quantity?

(a) $\frac{1}{x}$

(b) $\frac{1}{y}$

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

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

Explanation

From these two scenarios, it can be seen that either (a) or (b) can be the greater.

Case 1: . Then .

$\frac{1}{x} =\frac{1}{3} = \frac{2}{6}$

$\frac{1}{y} =\frac{1}{2} = \frac{3}{6}$

$\frac{1}{y} > \frac{1}{x}$

Case 2: $y = - \frac{1}{2}$ . $x = y+1 = - \frac{1}{2} + 1= \frac{1}{2}$

$\frac{1}{x} =\frac{1}{ \frac{1}{2}} = 2$

$\frac{1}{y} =\frac{1}{- \frac{1}{2}} =-2$

$\frac{1}{x} > \frac{1}{y}$

Therefore, it cannot be determined which is greater.

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

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1 }}$

For Line A, $x_{1} =-5, y_{1} =0, x_{2} = 0, y_{2} = 9$ . Substitute in the slope formula.

The slope is

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{9-0}{0-(-5)} = \frac{9}{5}$

For Line B, $x_{1} =-7, y_{1} =0, x_{2} = 0, y_{2} = 9$ . Substitute in the slope formula.

The slope is

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{9-0}{0-(-7)} = \frac{9}{7}$

Since

$\frac{9}{5} = \frac{63}{35}> \frac{45}{35} = \frac{9}{7}$ ,

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

Factor completely:

$3x^{2} - x -14$

$\left ( x+2 \right ) \left (3x- 7 \right )$

$\left ( x-2 \right ) \left (3x+ 7 \right )$

$\left ( x-2 \right ) \left (3x- 7 \right )$

$\left ( x+7 \right ) \left (3x-2 \right )$

$\left ( x-7 \right ) \left (3x+2 \right )$

Explanation

A trinomial whose leading term has a coefficent other than 1 can be factored using the -method. We split the middle term using two numbers whose product is and whose sum is . These numbers are , so:

$3x^{2} - x -14$

$= 3x^{2} - 7x +6x -14$

$= \left (3x^{2} - 7x \right ) +\left ( 6x -14 \right )$

$=x \left (3x- 7 \right ) +2 \left (3x- 7 \right )$

$=\left ( x+2 \right ) \left (3x- 7 \right )$

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