Full-Length ISEE Upper-Level Math Practice Test-Answers and Explanations

Full-Length ISEE Upper-Level Math Practice Test-Answers and Explanations

Did you take the ISEE Upper-Level Math Practice Test? If so, then it’s time to review your results to see where you went wrong and what areas you need to improve.

ISEE Upper Level Math Practice Test Answers and Explanations

1- Choice C is correct
\(|15-(18÷|2-8|)|=|15-(18÷6)=|15-3|=12\)

2- Choice D is correct
Use FOIL (First, Out, In, Last) method.
\((x-4)(2x+2)=2x^2+2x-8x-8=2x^2-6x-8\)

3- Choice B is correct
Solve for \(x\). \(-4≤4x-5<7\)⇒ (add 5 all sides) \(5-4≤4x-5+5<7+5\) ⇒
\(1≤4x<12\) ⇒ (divide all sides by \(4\)) \(\frac{1}{4}≤x<3\), x is between \(\frac{1}{4}\) and \(3\). Choice B represent this inequality.

4- Choice D is correct
\(x_{1,2} =\frac{-b ± \sqrt{b^2-4ac}}{2a}, ax^2 + bx + c = 0, 3x^2 +8x + 4 = 0\)⇒then: \(a = 3, b = 8\) and \(c = 4\)
\(x = \frac{-8 + \sqrt{8^2 – 4 .3 .4}}{2 .3}= –\frac{2}{3}, x = \frac{-8 – \sqrt{8^2 – 4 .3 .4}(}{2 .3} = – 2\)

5- Choice B is correct
\(\frac{8}{23} = 0.347\) and \(45\% = 0.45\) therefore x should be between \(0.0.347\) and \(0.45\)
Choice B. \(\frac{8}{19} = 0.421\) is between \(0.347\) and \(0.45\)

6- Choice A is correct
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.

7- Choice D is correct
Translated 6 units down and 5 units to the left means: \((x.y) ⇒ (x-5,y-6)\)

8- Choice B is correct
Write the proportion and solve for missing side.
\(\frac{Smaller \space triangle \space height}{Smaller \space triangle \space base}=\frac{Bigger \space triangle \space height}{Bigger \space triangle \space base}
= ⇒ \frac{100 \space cm}{150 \space cm} =\frac{100+280 \space cm}{x} ⇒ x=570 cm\)

9- Choice B is correct
\(0.00000625= \frac{6.25}{1,000,000}⇒6.25 × 10^{–6}\)

10- Choice B is correct
The ratio of boy to girls is 7:5. Therefore, there are 7 boys out of 12 students. To find the answer, first divide the total number of students by 12, then multiply the result by 7.
\(540 ÷ 12 = 45 ⇒ 120 × 7 = 315\)

11- Choice C is correct
Probability = (number of desired outcomes)/ (number of total outcomes)
In this case, a desired outcome is selecting either a red or a yellow marble. Combine the number of red and yellow marbles: \(7 + 6 = 13\) and divide this by the total number of marbles: \(5 + 7 + 6 = 18.\) The probability is \(\frac{13}{18}\).

12- Choice A is correct
Emily = Daniel, Emily = 9 Claire, Daniel \(= 24 +\) Claire, Emily = Daniel →          Emily\( = 20 +\) Claire
Emily = 9 Claire →            9 Claire \(= 24 +\) Claire→9 Claire \(–\) Claire = 24, 8 Claire = 24, Claire = 3

13- Choice B is correct
\(\frac{1}{4}\) of the distance is \(4 \frac{2}{5}\) miles. Then: \(\frac{1}{4}×4 \frac{2}{5}=\frac{1}{4}×\frac{22}{5}=\frac{22}{20}\)
Converting \(\frac{22}{20}\) to a mixed number gives: \(\frac{22}{20}=1 \frac{2}{10}/20=1 \frac{1}{10}\)

14- Choice D is correct
Use Pythagorean theorem: \(a^2+b^2=c^2→s^2+h^2=(10s)^2→(6s)^2+h^2=100s^2\)
Subtracting \(s^2\) from both sides gives: \(h^2=64s^2\),
Square roots of both sides: \(h=\sqrt{64s^2}=8s\)

15- Choice C is correct
\(5 – 12 ÷ (2^4 ÷ 4) = 5 – 12 ÷ (16 ÷ 4) = 5 – 12 ÷ (4) = 5-3 =2\)

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16- Choice C is correct
The area of the trapezoid is: Area\(=\frac{1}{2} h(b_1+b_2 )=\frac{1}{2}(x)(14+10)=36\)
\(→12x=36→x=3, y=\sqrt{4^2+3^2}=\sqrt{16+9}=\sqrt{25}=5\)
The perimeter of the trapezoid is: \(10+4++10+3+5=32\)

17- Choice D is correct
average\(=\frac{sum}{total}=\frac{40+30+38}{3}=\frac{108}{3}=36\)

18- Choice A is correct
Area = w \(×\) h, Area \(= 212 × 58 = 12,296\)

19- Choice A is correct
\((6.2+6.3+6.5) x = x+2, 19x =x+2, Then x = \frac{1}{9}\)

20- Choice C is correct
\(16\%\) of \(x = 15.4, x = 16/100 x = 15.4, x =\frac{16 × 15.4}{100} = 96.25\)

21- Choice D is correct
\(\frac{9}{30} = 0.3\)

22- Choice B is correct
\(\frac{240}{8} < x < \frac{320}{8}, 30 < x < 40\), Then:Only choice b is correct

23- Choice D is correct
\(75 ÷ \frac{1}{6} =\frac{\frac{75}{1}}{\frac{1}{6}}= 75 × 6 = 450\)

24- Choice D is correct
For sum of 5: (1 & 4) and (4 & 1), (2 & 3) and (3 & 2), therefore we have 4 options.
For sum of 7: (1 & 6) and (6 & 1), (2 & 5) and (5 & 2), (3 & 4) and (4 & 3) we have 6 options.
To get a sum of 6 or 9 for two dice: 6 + 4 = 10
Since, we have 6 × 6 = 36 total number of options, the probability of getting a sum of 4 and 6 is 10 out of 36 or \(\frac{10}{36}=\frac{5}{18}\).

25- Choice D is correct
Simplify: \(\frac{\frac{1}{4}-\frac{x+7}{8}}{\frac{x^2}{4}-\frac{1}{4}}=\frac{\frac{1}{4}(1-\frac{x+7}{2})}{\frac{x^2-1}{4}}=\frac{1-\frac{x+7}{2}}{x^2-1}⇒Simplify:1-\frac{x+7}{2}=\frac{2-x-7}{2} =\frac{-x-5}{2}

Then:\frac{1-\frac{x+7}{2}}{x^2-1}=\frac{\frac{-x -5}{2}}{x^2-1}=\frac{-x – 5}{2(x^2 – 1)}=\frac{-x – 5}{2x^2- 2}\)

26- Choice C is correct
The distance of A to B on the coordinate plane is: \(\sqrt{(x_1-x_2 )^2+(y_1-y_2 )^2 }= \sqrt{(9-1)^2+(8-2)^2 }=\sqrt{8^2+6^2}, =\sqrt{64+36}=\sqrt{100}=10\)
The diameter of the circle is 10 and the radius of the circle is 5. Then: the circumference of the circle is: \(2πr=2π(5)=10π\)

27- Choice B is correct
\(102.5 ÷ 0.55 = 186.36\)

28- Choice C is correct
Diameter = 14, then: Radius = 7, Area of a circle \(= πr^2 ⇒ A = 3.14(7)^2 = 153.86\)

29- Choice D is correct
\(\frac{15}{45} = \frac{X}{120} → x =\frac{15 ×120}{45} = 40\)

30- Choice B is correct
\(4x^2 – 34 = 66, 4x^2 = 100, x^2 = 25, x = ± 5\)

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31- Choice B is correct
\(\frac{x+7+15+23+7}{5} = 12 → x + 52 = 60 → x = 62 – 52 = 8\)

32- Choice A is correct
The difference of the file added, and the file deleted is: \(793,352 – 652,159 + 599,986 = 741,179, 856,036 -741,179 = 114,857\)

33- Choice B is correct
6 days 14 hours 42 minutes – 4 days 12 hours 35 minutes = 2 days 2 hours 7 minutes

34- Choice D is correct
\(x^{\frac{1}{2}}\) equals to the root of x. Then: \(15+x^{\frac{1}{2}}=24→15+\sqrt{X}=24→\sqrt{X}=9→x=81\)
\(x=81\) and \(8×x\) equals: \(8×81=648\)

35- Choice D is correct
\(0.35% of 60 = 21, 60 + 21= 81\)

36- Choice C is correct
\(2(8h – 4) = 42\)

37- Choice D is correct
The area of the circle is 64π \(cm^2\), then, its diameter is 8cm.
area of a circle \(=πr^2=64π→r^2=64→r=8\)
The radius of the circle is 8 and the diameter is twice it, 16.
One side of the square equals the diameter of the circle. Then:
Area of square\(=side×side=16×16=256\)

38- Choice C is correct
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 \(= \frac{1}{2} (base × height) = \frac{1}{2} (4 × 4) = 8\)

39- Choice D is correct
When a point is reflected over y axes, the \((x)\) coordinate of that point changes to \((-x)\) while its x coordinate remains the same. \(C (5, 4) → C’ (-5, 4)\)

40- Choice A is correct
A set of ordered pairs represents \(y\) as a function of \(x\) if: \(x_1=x_2→y_1=y_2\)
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 \(x\).
In choice D: (6, 1) and (6, 3) are ordered pairs with same \(x\)and different \(y\), therefore \(y\)isn’t a function of \(x\).

41- Choice D is correct
If the length of the box is 24, then the width of the box is one-fourth of it, 6, and the height of the box is 2 (one-third of the width). The volume of the box is: V=lwh = (24)(6) (2) = 288

42- Choice C is correct
The area of the square is 64 square inches. Area of square\(=side×side=8×8=64\)
The length of the square is increased by 4 inches and its width decreased by 2 inches. Then, its area equals: Area of rectangle = width × Length\(=12×6=72\)
The area of the square will be increased by 3 square inches. \(72-64=8\)

43- Choice B is correct
Write a proportion and solve. \(\frac{3}{5}=\frac{x}{120}\)
Use cross multiplication: \(5x=360→x=72\)

44- Choice A is correct
Number of squares equal to \(\frac{42×20}{5×5}=33.6\)

45- Choice C is correct
David’s weekly salary is \($250\) plus \(12\%\) of \($1,800\). Then: \(12\%\) of \(1,800=0.12×1,800=216, 250+216=466\)

46- Choice C is correct
\(5x^3 y^4+14x^2 y-(3x^3 y^4-3x^2 y)=2x^3 y^4+17x^2 y\)

47- Choice C is correct
Let P be circumference of circle A, then; \(2πr_A=20π→r_A=10\)
\(r_A=5r_B→r_B=\frac{10}{5}=2\)→ Area of circle B is; \(πr_B^2=4π\)

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