Full Length ISEE Upper Level Mathematics Practice Test

Solve for \(x\).
\(3 \ \leq \ 2 \ x \ - \ 3 \ < \ 13⇒\) (add \(3\) all sides) \(3 \ + \ 3 \ \leq \ 2 \ x \ - \ 3 \ + \ 3 \ < \ 13 \ + \ 3⇒\)
\(6 \ \leq \ 2 \ x \ < \ 16⇒\) ⇒ (divide all sides by \(2\))
\(3 \ ≤ \ x \ < \ 8\)
\(x\) is between \(3\) and \(8\).
Choice C represent this inequality.

A linear equation is a relationship between two variables, \(x \) and \(y\), and can be written in the form of \(y= m \ x \ + \ 𝑏\).
A non-proportional linear relationship takes on the form \(y= m \ x \ + \ 𝑏\), where \(b ≠ 0\) and its graph is a line that does not cross through the origin.

The correct answer is \(576 \) cm
Write the proportion and solve for missing side.
\(\frac{Smaller \ triangle \ height}{Smaller \ triangle \ base }= \frac{Bigger \ triangle \ height}{Bigger \ triangle \ base}⇒\)
\(\frac{100 \ cm}{162 \ cm} = \frac{100 \ + \ 250 \ cm}{x }⇒ x=576 \) cm

The correct answer is \(- \ 7, - \ \frac{1}{2}\)
\(x{1,2} = \frac{− \ b \ ± \ \sqrt{b^2 \ − \ 4 \ a \ c}}{ 2a}\)
\(a \ x^2 \ + \ b \ x + \ c = 0\)
\(2 \ x^2 \ + \ 15 \ x \ + \ 7 = 0 ⇒\) then:
\(a = 2, b = 15\) and \(c = 7\)
\(x = \frac{− \ 15 \ + \ \sqrt{15^2 \ − \ 4 \ . \ 2 \ . \ 7}}{ 2 \ . \ 2 }== \frac{− \ 15 \ + \ 13}{ 4 }= \frac{− \ 2}{4}= - \ \frac{1}{2}\)
\(x =\frac{ − \ 15 \ − \ \sqrt{15^2 \ − \ 4 \ . \ 2 \ . \ 7}}{ 2 \ . \ 2}=\frac{ − \ 15 \ − \ 13}{4}=\frac{ − \ 28}{4}= – \ 7\)

The correct answer is \(2\)
\(|1 \ – \ (12 \ ÷ \ | 1 \ - \ 5 |) | =|1 \ – \ (12 \ ÷ \ | - \ 4 |) | =|1 \ – \ (12 \ ÷ \ 4) | = |1 \ – \ 3 | = | – \ 2 | = 2\)

The correct answer is \(350\)
The ratio of boy to girls is \(7:4\).
Therefore, there are \(7\) boys out of \(4\) students.
To find the answer, first divide the total number of students by \(11\), then multiply the result by \(7\).
\(550 \ ÷ \ 11 = 50 ⇒ 50 \ × \ 7 = 350\)

Translated \(5 \) units down and \(4\) units to the left means: \((x.y) ⇒ (x \ − \ 4, y \ − \ 5)\)

The correct answer is \(7.685 \ × \ 10^{–7}\)
\(0.0000007685 = \frac{7.685}{10000000 }⇒7.685 \ × \ 10^{–7}\)

The correct answer is \(x^2 \ - \ 6 \ x \ - \ 27\)
Use FOIL (First, Out, In, Last) method.
\((x \ - \ 9) \ (x \ + \ 3) = x^2 \ + \ 3 \ x \ - \ 9 \ x \ - \ 27 =x^2 \ - \ 6 \ x \ - \ 27\)

The correct answer is \(\frac{7}{48}\)
\(\frac{2}{25} = 0.08\) and \(25\% = 0.25\) therefore \(x \) should be between \(0.08\) and \(0.25\)
Choice B, \(\frac{7}{48} = 0.145\) is between \(0.08\) and \(0.25\)

The correct answr is \(s \ \sqrt{15}\)
Use Pythagorean theorem: \(a^2 \ + \ b^2 = c^2→\)
\(s^2 \ + \ ℎ^2=(4 \ s)^2→\)
\(s^2 \ + \ ℎ^2=16 \ s^2\)
Subtracting \(s^2\) from both sides gives: \(ℎ^2=15 \ s^2\)
Square roots of both sides:
\(ℎ=\sqrt{15 \ s^2}=\sqrt{s^2} \times \sqrt{15}  =s \ \sqrt{15}\)

The area of the trapezoid is \(40 \ + \ 4 \ \sqrt{ 10 } \)
Area \(=\frac{1}{2} \ ℎ \ (𝑏1 \ + \ 𝑏2)=\frac{1}{2} \ (x) \ (16 \ + \ 12)=168 →\)
\(14 \ x=168→x=12\)
\(y=\sqrt{4^2 \ + \ 12^2}=\sqrt{16 \ + \ 144}=\sqrt{160}=\sqrt{16 \times 10 } =4 \ \sqrt{ 10 }\)
The perimeter of the trapezoid is:
\(12 \ + \ 16 \ + \ 12 \ + \ 4 \ \sqrt{ 10 } =40 \ + \ 4 \ \sqrt{ 10 } \)

The correct answer is \(9,860\) sq. ft.
Area \(=\) w \(×\) h
Area \(= 145 \ × \ 68 = 9,860\) sq. ft.

The correct answer is \(6\)
Emily \(=\) Daniel
Emily \(= 3\) Claire
Daniel \(= 12 \ +\) Claire
Emily \(=\) Daniel \(→\) Emily \(= 12 \ +\) Claire
Emily \(= 3\) Claire \(→ 3\) Claire \(= 12 \ +\) Claire \(→\)
\(3\) Claire \(–\) Claire \(= 12\)
\(2\) Claire \(= 12\)
Claire \(= 6\)

The correct answer is \(465\)
\(93 \ ÷ \ \frac{1}{5} = \frac{\frac{93}{1}}{\frac{1}{5}} = 93 \ × \ 5=465\)

The correct answer is \(- \ 3\)
\(3 \ – \ 12 \ ÷ \ (5^2 \ ÷ \ 5) =\)
\(3 \ – \ 12 \ ÷ \ (25 \ ÷ \ 5) =\)
\(3 \ – \ 12 \ ÷ \ (3) = - \ 9 \ ÷ \ 3= - \ 3\)

The correct answer is \(2 \ \frac{1}{10}\) Miles
\(\frac{1}{2}\) of the distance is \(4 \ \frac{1}{5}\) miles.
Then: \(\frac{1}{2} \ × \ 4 \ \frac{1}{5}=\frac{1}{2} \ × \ \frac{21}{5}=\frac{21}{10}\)
Converting \(\frac{21}{10}\) to a mixed number gives:
\(\frac{21}{10}=2 \ \frac{1}{10}\)

The correct answer is \(0.25\)
\(\frac{5}{32}=0.25\)

The correct answer is \(\frac{7}{15}\)
Let \(x\) be the number of purple marbles.
Let’s review the choices provided:
A. \(\frac{1}{10}\)
if the probability of choosing a purple marble is one out of ten, then:
Probability \(=\frac{number \ of \ desired \ outcomes}{number \ of \ total \ outcomes}=\frac{x}{30 \ + \ 20 \ + \ 40 \ + \ x}=\frac{1}{10}\)
Use cross multiplication and solve for \(x\).
\(10 \ x=90 \ + \ x→9 \ x=90→x=9\)
Since, number of purple marbles can be \(9\), then, choice be the probability of randomly selecting a purple marble from the bag.
Use same method for other choices.
B. \(\frac{1}{4}\)
\(\frac{x}{20 \ + \ 30 \ + \ 40 \ + \ x}=\frac{1}{4}→4 \ x=90 \ + \ x→3 \ x=90→x=30\)
C. \(\frac{2}{5}\)
\(\frac{x}{20 \ + \ 30 \ + \ 40 \ + \ x}=\frac{2}{5}→5 \ x=180 \ + \ 2 \ x→3 \ x=180→x=60\)
D. \(\frac{7}{15}\)
\(\frac{x}{20 \ + \ 30 \ + \ 40 \ + \ x}=\frac{7}{15}→15 \ x=630 \ + \ 7 \ x→8 \ x=630→x=78.75\)
Number of purple marbles cannot be a decimal.

The correct answer is \(0\)
\((3.2 \ + \ 3.6 \ + \ 3.8) \ x = x\)
\(10.6 \ x = x\)
Then \(x = 0\)

The correct answer is \(\frac{1}{4}\)
For sum of \(6: \ (1 \) & \(5)\) and \((5\) & \(1), \ (2\) & \(4)\) and \((4\) & \(2), \ (3\) & \(3)\), therefore we have \(5\) options.
For sum of \(9: \ (3\) & \(6)\) and \((6\) & \(3), \ (4\) & \(5)\) and \((5\) & \(4)\), we have \(4\) options.
To get a sum of \(6\) or \(9\) for two dice: \(5 \ + \ 4 = 9\)
Since, we have \(6 \ × \ 6 = 36\) total number of options, the probability of getting a sum of \(6\) and \(9\) is \(9\) out of \(36\) or \(\frac{9}{36}=\frac{1}{4}\).

The correct answer is \(29\) Miles
average \(=\frac{𝑠𝑢𝑚}{total}=\frac{25 \ + \ 40 \ + \ 22}{3}=\frac{87}{3}=29\) Miles

The correct answer is \(10 \ 𝜋\)
The distance of A to B on the coordinate plane is:
\(\sqrt{(x_{1} \ − \ x_{2})^2 \ + \ (y_{1} \ − \ y_{2} )^2}= \sqrt{(12 \ − \ 4)^2 \ + \ (8 \ − \ 2)^2}=\sqrt{8^2 \ + \ 6^2}
=\sqrt{64 \ + \ 36}=\sqrt{100} = 10\)
The diameter of the circle is \(10\) and the radius of the circle is \(5\).
Then: the circumference of the circle is:
\(2 \ \pi \ 𝑟=2 \ \pi (5)=10 \ 𝜋\)

The correct answer is \(966.4\)
\(48.32 \ ÷ \ 0.05 =966.4\)

The correct answer is \(± \ 2\)
\(4 \ x^2 \ + \ 8 = 24\)
\(4 \ x^2 = 16\)
\(x^2 = 4\)
\(x =± \ 2\)

The correct answer is \($ 113.28\)
\(25\%\) of \(x = 28.32\)
\(\frac{25}{100 } \ x =28.32\)
\(x = \frac{100 \ × \ 28.32}{25} = 113.28\)

The correct answer is \(5\)
\(\frac{x \ + \ 10 \ + \ 12 \ + \ 15 \ + \ 8}{5} = 10 →\)
\(x \ + \ 45 = 50 →\)
\(x = 50 \ – \ 45 = 5\)

The correct answer is \( 254.34\) in\(^2\)
Diameter \(= 18\)
then: Radius \(= 9\)
Area of a circle \(= π \ r^2 ⇒\)
A \(= 3.14 \ (9)^2 = 254.34\) in\(^2\)

The correct answer is \(55\)
\(\frac{240}{6 } \ < \ x \ < \ \frac{390}{6}\)
\(40 \ < \ x \ < \ 65\)
Then:
Only choice C is correct

The correct answer is \(\frac{ - \ 6 \ x \ - \ 20}{6 \ x^2 \ - \ 9} \)
Simplify:
\(\frac{\frac{1}{3} \ - \ \frac{x \ + \ 4}{2}}{\frac{x^2}{2} \ - \ \frac{3}{4}} = \frac{\frac{1}{3} \ - \ \frac{x \ + \ 4}{2}}{\frac{2 \ x^2 \ - \ 3}{4}} = \frac{4 \ (\frac{1}{3} \ - \ \frac{x \ + \ 4}{2})}{2 \ x^2 \ - \ 3}⇒\)
Simplify: \(\frac{1}{3} \ - \ \frac{x \ + \ 4}{2}= \frac{2 \ - \ 3 \ (x \ + \ 4)}{6}= \frac{2 \ - \ 3 \ x \ - \ 12}{6}= \frac{ - \ 3 \ x \ - \ 10}{6}\)
Then:
\( \frac{4 \ (\frac{- \ 3 \ x \ - \ 10}{6})}{2 \ x^2 \ - \ 3} =\frac{2 \ (\frac{- \ 3 \ x \ - \ 10}{3})}{2 \ x^2 \ - \ 3} =\frac{ \frac{- \ 6 \ x \ - \ 20}{3}}{2 \ x^2 \ - \ 3} =\frac{- \ 6 \ x \ - \ 20}{3\ (2 \ x^2 \ - \ 3)}=\frac{ - \ 6 \ x \ - \ 20}{6 \ x^2 \ - \ 9} \)

The correct answer is \(8\) square feet
Formula of triangle area \(= \frac{1}{2}\) (base \(×\) height)
Since the angles are \(45-45-90\), then this is an isosceles triangle, meaning that the base and height of the triangle are equal.
Triangle area \(= \frac{1}{2}\) (base \(×\) height) \(= \frac{1}{2} \  (4 × 4) = 8\)

The correct answer is \(40\)
\(\frac{5}{25}= \frac{x}{200} → x = \frac{5 \ × \ 200}{25} = 40\)

The correct answer is \(2 \ (4 \ H \ - \ 2) = 25\)

The correct answer is \(952,691\)
The difference of the file added, and the file deleted is:
\(552,129 \ – \ 439,566 = 112,563\)
\(840,128 \ + \ 112,563 = 952,691\)

The correct answer is \(3,840\)
\(x^{\frac{1}{2}}\) equals to the root of \(x\). Then: \(5 \ + \ x^{\frac{1}{2}}=20→4 \ + \ \sqrt{x}=20→\sqrt{x}=16→x=256\)
\(x=256\) and \(15 \ × \ x\) equals: \(15 \ × \ 256=3,840\)

The correct answer is \(1,728\) cm\(^3\)
If the length of the box is \(36\), then the width of the box is one third of it, \(12\), and the height of the box is \(4\) (one third of the width).
The volume of the box is:
\(𝑉 = 𝑙𝑤ℎ = (36) \ (12) \ (4) = 1,728\)

The correct answer is \((7, − \ 9)\)
When a point is reflected over \(x \) axes, the \((y)\) coordinate of that point changes to \((− \ y)\) while its \(x \) coordinate remains the same.
C \((7, 9)\) → C’ \((7, − \ 9)\)

The correct answer is \(3\) days \(7\) hours \(7\) minutes
\(7\) days \(21\) hours \(21\) minutes \(– \ 4\) days \(14\) hours \(14\) minutes \(= 3\) days \(7\) hours \(7\) minutes

The correct answer is \(3\) sq decreased
The area of the square is \(36\) square inches.
Area of square \(=\) side \(×\) side \(=6 \ × \ 6=36\)
The length of the square is increased by \(5\) inches and its width decreased by \(3\) inches.
Then, its area equals:
Area of rectangle \(=\) width \(×\) length \(=11 \ × \ 3=33\)
The area of the square will be decreased by \(3\) square inches.
\(36 \ − \ 33=3\)

The correct answer is \(100\) cm\(^2\)
The area of the circle is \(25 \ π\) cm\(^2\), then, its diameter is \(10\) cm.
Area of circle \(=\pi \ r^2=12 \ π→𝑟^2=25→r=5\)
Radius of the circle is \(5\) and diameter is twice of it, \(10\).
One side of the square equals to the diameter of the circle. Then:
Area of square \(=\) side \(×\) side \(=10 \ × \ 10=100\)

The correct answer is \(- \ 2 \ x^2 \ y^3 \ + \ 5 \ x^3 \ y^5\)
\(6 \ x^2 \ y^3 \ + \ 2 \ x^3 \ y^5 \ – \ (8 \ x^2 \ y^3 \ – \ 3 \ x^3 \ y^5) =\)
\(6 \ x^2 \ y^3 \ + \ 2 \ x^3 \ y^5 \ – \ 8 \ x^2 \ y^3 \ + \ 3 \ x^3 \ y^5 =\)
\(- \ 2 \ x^2 \ y^3 \ + \ 5 \ x^3 \ y^5\)

The correct answer is \(131.95\)
\(45\%\) of \(91 = 40.95\)
\(91 \ + \ 40.95= 131.95\)

The correct answer is \(70\)
Number of squares equal to: \(\frac{35 \ ×\ 18}{3 \ × \ 3}=35 \ × \ 2=70\)

The correct answer is \(4 \ 𝜋\)
Let P be circumference of circle A, then; \(2 \ \pi \ r_{A} = 12 \ 𝜋→r_{𝐴}=6\)
\(r_{𝐴}=3 \ r_{B}→r_{B}=\frac{6}{3}=2→\)
Area of circle B is; \(𝜋 \ r_{B}^2=4 \ 𝜋\)

The correct answer is \( \left\{(6, 1), (3, 1), (0, 5), (4, 5)\right\}\)
A set of ordered pairs represents \(y\) as a function of \(x\) if: \(x_{1}=x_{2}→y_{1}=y_{2}\)
In choice A: \((3, - \ 2)\) and \((3, 7)\) are ordered pairs with same \(x\) and different \(y\), therefore \(y\) isn’t a function of \( x\).
In choice B: \((4, 2)\) and \((4, 7)\) are ordered pairs with same \(x\) and different \(y\), therefore \(y\) isn’t a function of \(x\).
In choice C: \((5, 7)\) and \((5, 18)\) are ordered pairs with same \(x\) and different \(y\), therefore \(y\) isn’t a function of \(x\).

The correct answer is \($500\)
David’s weekly salary is \($350\) plus \(10\%\) of \($1500\).
Then: \(10\%\) of \(1,500=0.1 \ × \ 1,500=150\)
\(350 \ + \ 150=500\)

The correct answer is \(40\)
Write a proportion and solve. \(\frac{4}{5}=\frac{x}{50}\)
Use cross multiplication: \(5 \ x=200→x=40\)