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ISEE Middle Level Quantitative : ISEE Middle Level (grades 7-8) Quantitative Reasoning

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

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All ISEE Middle Level Quantitative Resources

6 Diagnostic Tests 110 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #286 : Measurement & Data

Joe has a piece of wallpaper that is 5ft by 3ft . How much of a wall can be covered by this piece of wallpaper?

Possible Answers:

15ft^2

13ft^2

16ft^2

17ft^2

14ft^2

Correct answer:

15ft^2

Explanation:

This problem asks us to calculate the amount of space that the wallpaper will cover. The amount of space that something covers can be described as its area. In this case area is calculated by using the formula $A=l \times w$

$A=5\times3$

A=15ft^2

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Example Question #221 : Quadrilaterals

Joe has a piece of wallpaper that is 9ft by 7ft . How much of a wall can be covered by this piece of wallpaper?

Possible Answers:

65ft^2

64ft^2

67ft^2

63ft^2

66ft^2

Correct answer:

63ft^2

Explanation:

$A=9\times7$

A=63ft^2

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Example Question #1 : Trapezoid

Square c

Note: Figure NOT drawn to scale

The above figure shows Square SQUA .

SX = UY + 1

Which is the greater quantity?

(a) The area of Trapezoid SXYA

(b) The area of Trapezoid XQUY

Possible Answers:

(a) is the greater quantity

(b) is the greater quantity

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

(a) and (b) are equal

Correct answer:

(a) is the greater quantity

Explanation:

The easiest way to answer the question is to locate on $\overline{SQ}$ such that SZ = UY :

Square c

Trapezoids SZYA and ZQUY have the same height, which is SA = QU . Their bases, by construction, have the same lengths - AY = ZQ and SZ = YU . Therefore, Trapezoids SZYA and ZQUY have the same area.

Since SX = UY + 1 , it follows that SX = SZ+ 1 , and SX > SZ . It follows that Trapezoid SXYA is greater in area than Trapezoids SZYA and ZQUY , and Trapezoid XQUY is less in area.

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Example Question #1 : Triangles

Which is the greater quantity?

(a) The perimeter of a right triangle with legs of length 5 feet and 12 feet

(b) 8 yards

Possible Answers:

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

(b) is greater

Correct answer:

(a) is greater

Explanation:

The length of the hypotenuse of a triangle with legs 5 feet and 12 feet is calculated using the Pythagorean Theorem, setting a = 5, b=12 :

$c = \sqrt{a^{2}+b^{2}} = \sqrt{5^{2}+12^{2}} = \sqrt{25+144} = \sqrt{169} =13$

The hypotenuse is 13 feet long. The perimeter is 5 + 12 + 13 = 30 feet, which is equal to 10 yards.

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Example Question #1 : Triangles

Which is the greater quantity?

(a) The perimeter of a right triangle with hypotenuse of length 25 centimeters and one leg of length 7 centimeters

(b) One-half of a meter

Possible Answers:

(a) and (b) are equal

(b) is greater

(a) is greater

It is impossible to tell from the information given

Correct answer:

(a) is greater

Explanation:

The length of the second leg of the triangle can be calculated using the Pythagorean Theorem, setting a = 7,c =25 :

$b = \sqrt{c^{2}-a^{2}} = \sqrt{25^{2}-7^{2}} = \sqrt{625-49}= \sqrt{576} = 24$

The second leg has length 24 centimeters, so the perimeter of the triangle is

7 + 24 + 25 = 56 centimeters.

One-half of a meter is one-half of 100 centimeters, or 50 centimeters, so (a) is greater.

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Example Question #3 : Triangles

Which is the greater quantity?

(a) The perimeter of an equilateral triangle with sidelength 30 inches

(b) The perimeter of a square with sidelength 2 feet

Possible Answers:

(a) is greater

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Correct answer:

(b) is greater

Explanation:

Each figure has sides that are congruent, so in each case, multiply the sidelength by the number of sides.

(a) The triangle has perimeter $30 \times 3 = 90$ inches

(b) 2 feet are equal to 24 inches, so the square has sidelength $24 \times 4 = 96$ inches.

The square has the greater perimeter.

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Example Question #4 : Triangles

$\Delta ABC$ is an equilateral triangle; AB = 25 .

Rectangle RECT ; RE = 35

Which is the greater quantity?

(a) The perimeter of $\Delta ABC$

(b) The perimeter of Rectangle RECT

Possible Answers:

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) is greater

Correct answer:

It is impossible to tell from the information given

Explanation:

(a) The perimeter of the equilateral triangle is $25 \times 3 = 75$ .

(b) CT = RE = 35 ; EC, RT are of unknown value, but they are equal, so we will call their common length .

Rectangle RECT has perimeter

35 + 35 + N + N = 70 + 2N .

Without knowing , it cannot be determined with certainty which figure has the longer perimeter. For example:

If N = 1 , then $70 + 2N = 70 + 2 \cdot 1 = 70 + 2 = 72 < 75$

If N = 3 , then $70 + 2N = 70 + 2 \cdot 3 = 70 + 6 = 76>75$

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Example Question #2 : Triangles

$\Delta ABC$ is an isosceles triangle; $\Delta DEF$ is an equilateral triangle

AB = 30,BC = 50, DE= 35

Which is the greater quantity?

(a) The perimeter of $\Delta ABC$

(b) The perimeter of $\Delta DEF$

Possible Answers:

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

(a) is greater

Correct answer:

(a) is greater

Explanation:

(a) As an isosceles triangle, $\Delta ABC$ , by definition, has two congruent sides. $AB \neq BC$ , so either :

AC = AB = 30

in which case the perimeter of $\Delta ABC$ is

AB + AC + BC = 30 + 30 + 50 = 110

AC =BC = 50

in which case the perimeter of $\Delta ABC$ is

AB + AC + BC = 30 + 50 + 50 = 130

(b) $\Delta DEF$ is an equilateral triangle, so, by definition, all of its sides are congruent; its perimeter is $35 \times 3 = 105$ .

Regardless of the length of $\overline{AC}$ , $\Delta ABC$ has the greater perimeter.

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Example Question #6 : Triangles

$\Delta ABC$ and $\Delta DEF$ are right triangles, with right angles $\angle B , \angle E$ , respectively.

AB = 6,BC = 8, DE = 24, DF = 26

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Correct answer:

(a) and (b) are equal

Explanation:

(a) $\overline{AC}$ is the hypotenuse of $\Delta ABC$ , so by the Pythagorean Theorem,

$\left (AC \right )^{2} =\left ( AB \right ) ^{2} + \left ( B C\right )^{2}$

$\left (AC \right )^{2} =6^{2} + 8^{2}$

$\left (AC \right )^{2} =36 + 64 = 100$

$AC = \sqrt{100} = 10$

(b) $\overline{EF}$ is a leg of $\Delta DEF$ , whose hypotenuse is $\overline{DF}$ , so by the Pythagorean Theorem,

$\left (EF \right )^{2} =\left ( DF \right ) ^{2} - \left ( DE\right )^{2}$

$\left (EF \right )^{2} =26^{2} - 24^{2}$

$\left (EF \right )^{2} =676 -576 = 100$

$EF= \sqrt{100} = 10$

AC = EF

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Example Question #3 : Triangles

Which is the greater quantity?

(a) The perimeter of a right triangle with legs of length 3 inches and 4 inches

(b) One foot

Possible Answers:

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

(a) is greater

Correct answer:

(a) and (b) are equal

Explanation:

The length of the hypotenuse of a triangle with legs 3 inches and 4 inches long is calculated using the Pythagorean Theorem, setting a = 3, b=4 :

$c = \sqrt{a^{2}+b^{2}} = \sqrt{3^{2}+4^{2}} = \sqrt{9+16} = \sqrt{25} =5$

The hypotenuse is 5 inches long. The perimeter is therefore 3 + 4 + 5 = 12 inches, which is equal to one foot.

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All ISEE Middle Level Quantitative Resources

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ISEE Middle Level Quantitative : ISEE Middle Level (grades 7-8) Quantitative Reasoning

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

All ISEE Middle Level Quantitative Resources

Example Questions

Example Question #286 : Measurement & Data

Example Question #221 : Quadrilaterals

Example Question #1 : Trapezoid

Example Question #1 : Triangles

Example Question #1 : Triangles

Example Question #3 : Triangles

Example Question #4 : Triangles

Example Question #2 : Triangles

Example Question #6 : Triangles

Example Question #3 : Triangles

All ISEE Middle Level Quantitative Resources

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