# 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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