How to find the solution to an equation

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ISEE Upper Level Quantitative Reasoning › How to find the solution to an equation

Questions 1 - 10

Define .

$(g \circ f) \left ( \frac{1}{2} \right ) = 6$

Which is the greater quantity?

(a)

(b)

(a) is the greater quantity

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

(a) and (b) are equal

(b) is the greater quantity

Explanation

, so, by substitution,

$f \left ( \frac{1}{2} \right ) = 6 \cdot \frac{1}{2} = 3$ .

By way of the definition of a composition of functions,

$(g \circ f) \left ( \frac{1}{2} \right ) = g \left [ f \left ( \frac{1}{2} \right ) \right ] = g(3 )$ .

Since $(g \circ f) \left ( \frac{1}{2} \right ) =6$ , it follows that .

Also, by substitution,

$f(3 )= 6 \cdot 3 = 18$ .

Therefore, .

$\left \lfloor N\right \rfloor$ refers to the greatest integer less than or equal to .

and are integers.

Which is greater?

(a) $\left \lfloor a\right \rfloor + \left \lfloor b \right \rfloor$

(b) $\left \lfloor a+ b \right \rfloor$

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

Explanation

(a) Since is an integer, $\left \lfloor a \right \rfloor = \left \lfloor x + 0.5 \right \rfloor = x$ .

Since is an integer, $\left \lfloor b \right \rfloor = \left \lfloor y + 0.5 \right \rfloor = y$ .

$\left \lfloor a\right \rfloor + \left \lfloor b \right \rfloor = x + y$

(b) By closure, is an integer, so

$\left \lfloor a+b \right \rfloor = \left \lfloor x + 0.5 +y+0.5 \right \rfloor = \left \lfloor x +y+1 \right \rfloor = x +y+1$ .

This makes (b) greater.

Which is the greater quantity?

(a)

(b)

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(a) is greater.

Explanation

Each can be rewritten as a compound statement. Solve separately:

$45 - t =- 60 \textrm{ or }45 - t = 60$

Similarly:

$45 - u = -50 \textrm{ or } 45 - u = 50$

Therefore, it cannot be determined with certainty which of and is the greater.

Solve for :

$t= \frac{100 + 3x}{y}$

$t= \frac{100 -3x}{y}$

$t= \frac{100 -x}{3y}$

$t= \frac{100 -y}{3x}$

$t= \frac{100 +y}{3x}$

Explanation

$\frac{yt}{y} = \frac{100 + 3x}{y}$

$t= \frac{100 + 3x}{y}$

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).

Given the line of the equation , which is the greater quantity?

(A) The -coordinate of the -intercept of the line

(B) The -coordinate of the -intercept of the line

(B) is greater

(A) is greater

(A) and (B) are equal

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

Explanation

The -coordinate of the -intercept of the line is 0, so to find the -coordinate, we set and solve for :

$4x-5 \cdot 0 = -20$

$4x \div 4 = -20 \div 4$

Similarly, to find the -coordinate of the -intercept, we set and solve for :

$4 \cdot 0 -5y = -20$

$-5y\div (-5) = - 20 \div (-5)$

(B), the -coordinate of the -intercept of the line, is greater.

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.