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ISEE Middle Level Quantitative : Variables

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

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All ISEE Middle Level Quantitative Resources

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Example Questions

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Example Question #1 : How To Add Variables

Which quantity is greater if $x\neq 0$ ?

$(a)\ -x^2+2$

$(b)\ x^2+2$

Possible Answers:

(a) and (b) are equal

It is not possible to tell from the information given

(a) is greater

(b) is greater

Correct answer:

(b) is greater

Explanation:

We know that x^2 is always positive for all values of . Therefore (-x^2) would be negative for all values of . From this conclusion, we know:

-x^2<x^2

So we have:

x^2+2>-x^2+2

(b)>(a)

(b) is the greater quantity.

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Example Question #1 : How To Add Variables

Which quantity is greater if $x\neq0$ ?

(a)-x^3+2

$(b)\ x^3+2$

Possible Answers:

It is not possible to tell from the information given

(a) is greater

(b) is greater

(a) and (b) are equal

Correct answer:

It is not possible to tell from the information given

Explanation:

A positive number raised to the third power will be positive, while a negative number raised to the third power will remain negative.

If x>0 , then x^3>0 and -x^3<0 .

If x<0 , then x^3<0 and -x^3>0 .

Since we do not know if is positive or negative, we cannot draw a conclusion about which option is greater.

If x>0 , then (b) is greater.

If x<0 , then (a) is greater.

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Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which quantity is greater if x>0 ?

(a) -x^3-x+1

$(b)\ x^3+x+1$

Possible Answers:

(a) is greater

(a) and (b) are equal

It is not possible to tell from the information given

(b) is greater

Correct answer:

(b) is greater

Explanation:

When x>0 we can write:

$x^3>0\ \text{and}\ -x^3<0$

We know that x^3>-x^3 and x>-x . Based on this, we can compare the two given quantities.

x^3+x+1>-x^3-x+1

(b)>(a)

(b) is the greater quantity.

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Example Question #1 : How To Add Variables

Which quantity is greater if $x\geq 3$ ?

$(a)\ x^2-6x+10$

$(b)\ 0$

Possible Answers:

(a) and (b) are equal

(a) is greater

It is not possible to tell from the information given

(b) is greater

Correct answer:

(a) is greater

Explanation:

We know that is greater than . We can easily test a few values for to determine if the values are increasing or decreasing.

If {x=3} :

(a)=3^2-6(3)+10=9-18+10=1

If x=4 :

(a)=4^2-6(4)+10=16-24+10=2

If x=5 :

(a)=5^2-6(5)+10=25-30+10=5

The value of (a) is increasing, with the smallest possible value being . From this, we know that (a)>0 , so (a)>(b) .

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Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which of the following is equivalent to ?

Possible Answers:

3t + 5t

$16t \div 8$

80t - 10t

$4 \cdot 4t$

6 + 2t

Correct answer:

3t + 5t

Explanation:

Using the distributive property:

3t + 5t = (3 + 5)t = 8t

and

80t - 10t = (80-10)t = 70t

Using the associative property of multiplication:

$4 \cdot 4t =\left ( 4 \cdot 4 \right ) t= 16t$

We can rewrite $16t \div 8$ as $16t \cdot \frac{1}{8}$ ; using the commutative and associative properties of multiplication:

$16t \cdot \frac{1}{8} = \left ( 16 \cdot \frac{1}{8} \right ) \cdot t = 2t$

6 + 2t is the sum of unlike terms and cannot be simplified.

3t + 5t is the correct choice.

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Example Question #1 : How To Add Variables

is a positive integer.

Which is the greater quantity?

(A) 5t + 8t + 7

(B) 5+ 8t + 7t

Possible Answers:

(A) and (B) are equal

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

(B) is greater

(A) is greater

Correct answer:

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

Explanation:

Depending on the value of , it is possible for either expression to be greater or for both to be equal.

Case 1: t = 1

$5t + 8t + 7 = 5 \cdot 1 + 8 \cdot 1 + 7 = 5 + 8 + 7 = 20$

and

$5+ 8t + 7t = 5+ 8 \cdot 1 + 7 \cdot 1 = 5 + 8 + 7 = 20$

So the two are equal.

Case 2: t = 2

$5t + 8t + 7 = 5 \cdot 2 + 8 \cdot 2 + 7 = 10 + 16 + 7 = 33$

and

$5+ 8t + 7t = 5+ 8 \cdot 2 + 7 \cdot 2 = 5 + 16 + 14 = 35$

So (B) is greater.

The correct response is that it cannot be determined which is greater.

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Example Question #7 : How To Add Variables

is a positive integer.

Which is the greater quantity?

(A) 6t+ 5t + 4t + 3t + 2 t + t

(B) 7t+5t+3t+t

Possible Answers:

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

(A) and (B) are equal

(A) is greater

(B) is greater

Correct answer:

(A) is greater

Explanation:

6t+ 5t + 4t + 3t + 2 t + t = (6+5+4+3+2+1)t = 21t

7t+5t+3t+t = (7 + 5 + 3 + 1)t = 16t

Since 21 > 16 , and is positive,

then by the multiplication property of inequality,

21 t > 16t

making (A) greater regardless of the value of .

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Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

is a positive integer.

Which is the greater quantity?

(A) 7t + 11 + 4t

(B) 7 + 11t + 4

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:

7t + 11 + 4t = 7t + 4t + 11= (7+4)t + 11 = 11t + 11

7 + 11t + 4 = 11t + 7 + 4 = 11t + 11

Regardless of the value of , the expressions are equal.

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Example Question #2 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which of the following is equivalent to 2t + 14 ?

Possible Answers:

None of the other responses is correct.

16t

$28t^{2}$

28t

$16t^{2}$

Correct answer:

None of the other responses is correct.

Explanation:

The expression is the sum of two unlike terms, and therefore cannot be further simplified. None of these responses is correct.

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Example Question #1 : Variables

is a positive integer.

Which is the greater quantity?

(A) 8t + 7 + 5t + 9

(B) 10 + 4t + 9t +5

Possible Answers:

(B) is greater

(A) and (B) are equal

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

(A) is greater

Correct answer:

(A) is greater

Explanation:

$8t + 7 + 5t + 9 = \left ( 8 + 5 \right ) t + 9+ 7 = 13t + 16$

10 + 4t + 9t +5 = (4+9)t + 10 + 5 = 13t + 15

Since 16 > 15 , 13t+ 16 > 13t+ 15 , so (A) is greater regardless of the value of .

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All ISEE Middle Level Quantitative Resources

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ISEE Middle Level Quantitative : Variables

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

All ISEE Middle Level Quantitative Resources

Example Questions

Example Question #1 : How To Add Variables

Example Question #1 : How To Add Variables

Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Example Question #1 : How To Add Variables

Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Example Question #1 : How To Add Variables

Example Question #7 : How To Add Variables

Example Question #1 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Example Question #2 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Example Question #1 : Variables

All ISEE Middle Level Quantitative Resources

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