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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Example Questions

Example Question #67 : Percentage

\displaystyle A is \displaystyle \frac{1}{100} \% of \displaystyle B, which is 10,000 % of \displaystyle C.

Which is a true statement?

Possible Answers:

\displaystyle A = 0.001 C

\displaystyle A = 0.1 C

\displaystyle A = 0.01 C

\displaystyle A=C

Correct answer:

\displaystyle A = 0.01 C

Explanation:

\displaystyle B is 10,000 % of \displaystyle C, so \displaystyle B = \frac{10,000}{100} \cdot C = 100 C.

 

\displaystyle A is \displaystyle \frac{1}{100} \%, or \displaystyle 0.01 \%, of \displaystyle B, so \displaystyle A = \frac{0.01}{100} \cdot B = 0.0001 B.

Therefore, 

\displaystyle A = 0.0001 B = 0.0001 \cdot 100 C = 0.01 C

Example Question #291 : Numbers And Operations

Scrabble

A popular word game uses one hundred tiles, each of which is marked with a letter or a blank. The distribution of the tiles is shown above, with each letter paired with the number of tiles marked with that letter. Notice that there are two blank tiles.

To the nearest whole percent, what percent of the vowel tiles are "E's"?

(Note: for this problem, "Y" is considered a consonant)

Possible Answers:

\displaystyle 24 \%

\displaystyle 14 \%

\displaystyle 19 \%

\displaystyle 29 \%

Correct answer:

\displaystyle 29 \%

Explanation:

There are nine "A" tiles, twelve "E" tiles, nine "I" tiles, eight "O" tiles, and four "U" tiles. This is a total of 

\displaystyle 9 + 12 + 9 + 8 + 4 = 42 vowel tiles.

12 of the tiles are "E's"; they therefore comprise

\displaystyle \frac{12}{42} \times 100 \% = \frac{1,200}{42} \% \approx 28.57 \%.

This rounds to 29%.

Example Question #21 : How To Find Percentage

The Ace of Spades and the King of Spades are both removed from a standard deck of 52 playing cards. What percent of the remaining cards are spades?

Possible Answers:

\displaystyle 24 \%

\displaystyle 25 \%

\displaystyle 23 \%

\displaystyle 22 \%

Correct answer:

\displaystyle 22 \%

Explanation:

13 of the 52 cards in a standard deck are spades. If two spades are removed, then there will remain 11 spades out of 50 cards, or

\displaystyle \frac{11}{50} \times 100 \% = \frac{1,100}{50} \% = 22 \% 

of the remaining cards.

Example Question #1 : How To Find The Distributive Property

Which is the greater quantity?

(a) \displaystyle -4 (5 - y)

(b) \displaystyle 4y - 20

Possible Answers:

(b) is greater

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

Apply the distributive and commutative properties to the expression in (a):

\displaystyle -4 (5 - y)

\displaystyle = -4 \cdot 5 - \left (-4 \right ) \cdot y

\displaystyle = -20 - \left (-4 y\right )

\displaystyle = -20+4 y

\displaystyle = 4 y +(-20)

\displaystyle = 4 y -20

The two expressions are equivalent.

Example Question #1 : Distributive Property

Which is the greater quantity?

(a) \displaystyle -5(x+6)

(b) \displaystyle -5x+30

Possible Answers:

(a) is greater

(b) is greater

(a) and (b) are equal

Correct answer:

(b) is greater

Explanation:

Apply the distributive property to the expression in (a):

\displaystyle -5(x+6) = -5 \cdot x+ (-5 ) \cdot 6 = -5x -30

\displaystyle -30 < 30, so \displaystyle 5x -30 < 5x+ 30 regardless of \displaystyle x.

Therefore, \displaystyle -5(x+6) < 5x+ 30

Example Question #951 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which is the greater quantity?

(a) \displaystyle 8(y-5)

(b) \displaystyle 8y-5

Possible Answers:

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

Correct answer:

(b) is greater

Explanation:

Apply the distributive property to the expression in (a):

\displaystyle 8(y-5) = 8 \cdot y - 8 \cdot 5= 8y - 40

Since \displaystyle -40 < -5\displaystyle 8y - 40 < 8y - 5, and therefore, regardless of \displaystyle y

\displaystyle 8\left ( y - 5 \right )< 8y - 5

Example Question #1 : How To Find The Distributive Property

Which is the greater quantity?

(a) \displaystyle -7 (-10+y)

(b) \displaystyle 7y + 70

Possible Answers:

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

(a) is greater

Correct answer:

It is impossible to tell from the information given

Explanation:

We show that there is at least one value of \displaystyle y that makes the (a) greater and at least one that makes (b) greater:

Case 1: \displaystyle y = 1

(a) \displaystyle -7 (-10+y) = -7 (-10+1) = -7 (-9) = 63

(b) \displaystyle 7y + 70 = 7 \cdot 1 + 70 = 7 + 70 = 77

(b) is greater here

Case 2: \displaystyle y =- 1

(a) \displaystyle -7 (-10-y) = -7 (-10-1) = -7 (-11) = 77

(b) \displaystyle 7y + 70 = 7 \cdot \left ( -1 \right )+ 70 = -7 + 70 = 63

(a) is greater here

Example Question #4 : How To Find The Distributive Property

Which of the following is equivalent to \displaystyle 8y + 4z ?

Possible Answers:

\displaystyle 4(2y+ 2z + 2)

\displaystyle 4 (4y + z)

\displaystyle 4 (y+2z)

\displaystyle 4 (2y+z)

\displaystyle 4 (2y + 4z)

Correct answer:

\displaystyle 4 (2y+z)

Explanation:

We can best solve this by factoring 4 from both terms, and distributing it out:

\displaystyle 8y + 4z

\displaystyle = 4 \cdot 2y + 4 \cdot z

\displaystyle = 4 (2y+z)

Example Question #2 : How To Find The Distributive Property

\displaystyle x and \displaystyle y are positive integers.

Which of the following is greater?

(A) \displaystyle 7x + 2y

(b) \displaystyle 2 (x+y) +5x

Possible Answers:

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

(B) is greater

(A) and (B) are equal

(A) is greater

Correct answer:

(A) and (B) are equal

Explanation:

\displaystyle 2 (x+y) +5x

\displaystyle = 2 x+2 y +5x

\displaystyle = 2 x+5x +2 y

\displaystyle =\left ( 2 +5 \right ) x +2 y

\displaystyle =7 x +2 y

(A) and (B) are equivalent variable expressions and are therefore equal regardless of the values of \displaystyle x and \displaystyle y.

Example Question #5 : How To Find The Distributive Property

Simplify the below: 

\displaystyle -2(x-4)+3x

Possible Answers:

\displaystyle x-8

\displaystyle 4x+8

\displaystyle x+8

\displaystyle -6x-8

Correct answer:

\displaystyle x+8

Explanation:

In order to simiplify we must first distribute the -2 only to what is inside the ( ): 

\displaystyle -2(x-4)+3x=-2x+8+3x

Now, we must combine like terms: 

\displaystyle -2x+8+3x =-2x+3x+8 =x+8

This gives us the final answer:

\displaystyle x+8

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