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ISEE Upper Level Quantitative : How to find the solution to an equation

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Example Questions

Example Question #111 : How To Find The Solution To An Equation

Define f (x)= 2x+ 5 and g (x) = 2x+3 .

Evaluate $\left ( g \circ f \right )(3)$ .

Possible Answers:

$\left ( g \circ f \right )(3) = 20$

$\left ( g \circ f \right )(3) = 33$

$\left ( g \circ f \right )(3) = 28$

$\left ( g \circ f \right )(3) = 23$

$\left ( g \circ f \right )(3) = 25$

Correct answer:

$\left ( g \circ f \right )(3) = 25$

Explanation:

$\left ( g \circ f \right )(3) = g (f(3))$

f (x)= 2x+ 5 , so

$f (3)= 2 \cdot 3 + 5 = 6 + 5 = 11$

g (x) = 2x+3 , so

$g (f(3)) = g (11) = 2 \cdot 11 + 3 = 22 + 3 = 25$ ,

which is the correct response.

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Example Question #112 : How To Find The Solution To An Equation

Define f (x)= 2x+ 5 and g (x) = 2x+3 .

If $\left ( f \circ g\right ) (N) = 100$ , evaluate .

Possible Answers:

None of the other responses gives a correct answer.

$N = 21 \frac{3}{4}$

$N = 22 \frac{3}{4}$

$N = 21 \frac{1}{4}$

$N = 22 \frac{1}{4}$

Correct answer:

$N = 22 \frac{1}{4}$

Explanation:

$\left ( f \circ g\right ) (x) = f(g(x)) = f(2x+3) = 2 (2x+3) +5 = 4x + 6 + 5 = 4x + 11$

Therefore, if

$\left ( f \circ g\right ) (N) = 100$ ,

we solve for in the equation

4N + 11 = 100

4N + 11- 11 = 100 - 11

4N = 89

$4N \div 4 = 89 \div 4$

$N = 22 \frac{1}{4}$

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Example Question #113 : How To Find The Solution To An Equation

For all real numbers and , define an operation $\Leftrightarrow$ as follows:

$a\Leftrightarrow b = \frac{a + b}{ ab+1}$

is a positive number. Which is the greater quantity?

(A) $t \Leftrightarrow 4t$

(B) $2t \Leftrightarrow 2t$

Possible Answers:

(B) is greater

(A) is greater

(A) and (B) are equal

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

Correct answer:

(A) is greater

Explanation:

Substitute for both in the expression:

$a\Leftrightarrow b = \frac{a + b}{ ab+1}$

$t\Leftrightarrow 4t = \frac{t + 4t}{ t \cdot 4t +1} = \frac{5t}{4t^{2}+1}$

$2t\Leftrightarrow 2t = \frac{2t + 2t}{ 2t \cdot 2t +1} = \frac{4t}{4t^{2}+1}$

The quantities share a denominator that is positve, being one greater than for times a square of a positive. Therefore, we need only compare numerators. For any positive , 5t > 4t . Therefore,

$\frac{5t}{4t^{2}+1} > \frac{4t}{4t^{2}+1}$

and

$t\Leftrightarrow 4t > 2t\Leftrightarrow 2t$

making (A) greater.

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Example Question #114 : How To Find The Solution To An Equation

and are both positive.

20 added to four-thirds of is equal to . Which is the greater quantity?

(A)

(B)

Possible Answers:

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

(B) is greater

(A) is greater

(A) and (B) are equal

Correct answer:

(B) is greater

Explanation:

The statement is equivalent to

$y = \frac{4}{3}x + 20$

If is positive, then

$\frac{4}{3} x > x$

and

$y = \frac{4}{3} x + 20 > \frac{4}{3} x > x$

and y > x ,

making (B) greater.

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Example Question #115 : How To Find The Solution To An Equation

For all real numbers and , define an operation $\oplus$ as follows:

$a \oplus b = \frac{ab+ 1}{a + b + 1}$

is a positive integer. Which is the greater quantity?

(A) $2 t \oplus 4t$

(B) $3t \oplus 3t$

Possible Answers:

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

(A) and (B) are equal

(A) is greater

(B) is greater

Correct answer:

(B) is greater

Explanation:

Substitute for both in the expression:

$a \oplus b = \frac{ab+ 1}{a + b + 1}$

$2 t\oplus 4 t = \frac{2 t \cdot 4t + 1}{2t + 4t + 1} = \frac{8 t ^{2}+ 1}{6t + 1}$

$3t \oplus 3t = \frac{3t \cdot 3t + 1}{3t + 3t + 1}= \frac{9t^{2}+ 1}{6t + 1}$

Since is positive, so is common denominator 6t + 1 , so we need only compare numerators. Since $t^{2}$ must be positive,

$8 t ^{2}< 9t ^{2}$

$8 t ^{2}+ 1 < 9t ^{2} + 1$

$\frac{8 t ^{2}+ 1}{6t + 1} < \frac{9t ^{2}+ 1}{6t + 1}$

and

$2 t\oplus 4 t < 3t\oplus 3 t$

making (B) greater.

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Example Question #116 : How To Find The Solution To An Equation

and are both positive.

20 subtracted from five-sevenths of is equal to . Which is the greater quantity?

(A)

(B)

Possible Answers:

(A) and (B) are equal

(B) is greater

(A) is greater

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

Correct answer:

(A) is greater

Explanation:

The statement is equivalent to

$\frac{5}{7}x -20 = y$ or, alternatively,

$y = \frac{5}{7}x -20$

$y+ 20 = \frac{5}{7}x -20 + 20$

$y+ 20 = \frac{5}{7}x$

Since is positive and $\frac{5}{7}x > y$ , we can derive:

$y < y+ 20 = \frac{5}{7}x < x$

making greater than . This makes (A) greater.

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Example Question #117 : How To Find The Solution To An Equation

Define $f (x) = \sqrt{14 - 2x}$ and $g(x) = \sqrt{3x - 21}$

What is the domain of the function f + g ?

Possible Answers:

$\varnothing$

$(- \infty,\infty )$

None of the other responses gives a correct answer.

$[7,\infty )$

$(- \infty, 7]$

Correct answer:

None of the other responses gives a correct answer.

Explanation:

$f (x) = \sqrt{14 - 2x}$ has as its domain the set of values of for which its radicand is nonnegative; that is,

$14 - 2x \geq 0$

$14 - 2x- 14 \geq 0 - 14$

$- 2x \geq - 14$

$- 2x \div (-2) \leq - 14 \div (-2)$

$x \leq 7$ or $(- \infty, 7]$

Similarly, $g(x) = \sqrt{3x - 21}$ has as its domain the set of values of for which its radicand is nonnegative; that is,

$3x - 21 \geq 0$

$3x - 21+ 21 \geq 0 + 21$

$3x \geq 21$

$3x \div 3 \geq 21 \div 3$

$x \geq 7$ or $[7,\infty )$

The domain of the sum of two functions is the intersection of the domains of the two individual functions. This intersection is

$(- \infty, 7] \cap [7,\infty ) = \left \{ 7\right \}$

The domain of f + g is the single value $\left \{ 7\right \}$ , which is not among the choices.

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Example Question #118 : How To Find The Solution To An Equation

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

Possible Answers:

Correct answer:

Explanation:

To solve this problem, you can translate the question into an equation. It should look like: x+x+1+x+2=36 . 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 3x+3=36 . 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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Example Question #119 : How To Find The Solution To An Equation

Column A Column B

The value of x in The value of x in

$\frac{x}{5}+2=18$ 4x-3=13

Possible Answers:

The quantity in Column A is greater.

The quantities in both columns are equal.

The quantity in Column B is greater.

The relationship between the columns cannot be determined.

Correct answer:

The quantity in Column A is greater.

Explanation:

First, solve for x in Column A. Subtract 2 from both sides so that $\frac{x}{5}=16$ . Multiply by 5 on both sides so that X equals 80. Then, solve for x in Column B. Add 3 to both sides so that 4x=16 . X equals 4 in this equation. Therefore, Column A is greater.

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Example Question #120 : How To Find The Solution To An Equation

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

Possible Answers:

Correct answer:

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.

$\frac{60}{100}=\frac{x}{70}$

Given that $\frac{60}{100}=\frac{6}{10}=\frac{3}{5}$ , it follows that

$\frac{3}{5}=\frac{x}{70}$

Next, we cross multiply, which gives us:

5x=210

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

x=42

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : How to find the solution to an equation

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #111 : How To Find The Solution To An Equation

Example Question #112 : How To Find The Solution To An Equation

Example Question #113 : How To Find The Solution To An Equation

Example Question #114 : How To Find The Solution To An Equation

Example Question #115 : How To Find The Solution To An Equation

Example Question #116 : How To Find The Solution To An Equation

Example Question #117 : How To Find The Solution To An Equation

Example Question #118 : How To Find The Solution To An Equation

Example Question #119 : How To Find The Solution To An Equation

Example Question #120 : How To Find The Solution To An Equation

All ISEE Upper Level Quantitative Resources

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