ISEE Middle Level Quantitative : Operations

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

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Example Questions

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

Which quantity is greater if \(\displaystyle x\neq 0\)?

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

\(\displaystyle (b)\ x^2+2\)

Possible Answers:

\(\displaystyle (b)\) is greater

\(\displaystyle (a)\) is greater

It is not possible to tell from the information given

\(\displaystyle (a)\) and \(\displaystyle (b)\) are equal

Correct answer:

\(\displaystyle (b)\) is greater

Explanation:

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

\(\displaystyle -x^2< x^2\)

So we have:

\(\displaystyle x^2+2>-x^2+2\)

\(\displaystyle (b)>(a)\)

\(\displaystyle (b)\) is the greater quantity.

Example Question #1 : Variables

Which quantity is greater if \(\displaystyle x\neq0\)?

\(\displaystyle (a)-x^3+2\)

\(\displaystyle (b)\ x^3+2\)

Possible Answers:

It is not possible to tell from the information given

\(\displaystyle (b)\) is greater

\(\displaystyle (a)\) and \(\displaystyle (b)\) are equal

\(\displaystyle (a)\) is greater

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 \(\displaystyle x>0\), then \(\displaystyle x^3>0\) and \(\displaystyle -x^3< 0\).

If \(\displaystyle x< 0\), then \(\displaystyle x^3< 0\) and \(\displaystyle -x^3>0\).

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

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

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

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

Which quantity is greater if \(\displaystyle x>0\)?

\(\displaystyle (a) -x^3-x+1\)

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

Possible Answers:

\(\displaystyle (b)\) is greater

\(\displaystyle (a)\) is greater

It is not possible to tell from the information given

\(\displaystyle (a)\) and \(\displaystyle (b)\) are equal

Correct answer:

\(\displaystyle (b)\) is greater

Explanation:

When \(\displaystyle x>0\) we can write:

\(\displaystyle x^3>0\ \text{and}\ -x^3< 0\)

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

\(\displaystyle x^3+x+1>-x^3-x+1\)

\(\displaystyle (b)>(a)\)

\(\displaystyle (b)\) is the greater quantity.

 

Example Question #1 : Variables

Which quantity is greater if \(\displaystyle x\geq 3\)?

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

\(\displaystyle (b)\ 0\)

Possible Answers:

It is not possible to tell from the information given

\(\displaystyle (a)\) is greater

\(\displaystyle (b)\) is greater

\(\displaystyle (a)\) and \(\displaystyle (b)\) are equal

Correct answer:

\(\displaystyle (a)\) is greater

Explanation:

We know that \(\displaystyle x\) is greater than \(\displaystyle 3\). We can easily test a few values for \(\displaystyle x\) to determine if the values are increasing or decreasing.

If \(\displaystyle {x=3}\):

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

If \(\displaystyle x=4\):

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

If \(\displaystyle x=5\):

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

The value of \(\displaystyle (a)\) is increasing, with the smallest possible value being \(\displaystyle 1\). From this, we know that \(\displaystyle (a)>0\), so \(\displaystyle (a)>(b)\).

Example Question #3 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Which of the following is equivalent to \(\displaystyle 8t\) ?

Possible Answers:

\(\displaystyle 4 \cdot 4t\)

\(\displaystyle 6 + 2t\)

\(\displaystyle 3t + 5t\)

\(\displaystyle 16t \div 8\)

\(\displaystyle 80t - 10t\)

Correct answer:

\(\displaystyle 3t + 5t\)

Explanation:

Using the distributive property:

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

and

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

Using the associative property of multiplication:

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

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

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

\(\displaystyle 6 + 2t\) is the sum of unlike terms and cannot be simplified.

\(\displaystyle 3t + 5t\) is the correct choice.

 

Example Question #2 : Variables

\(\displaystyle t\) is a positive integer.

Which is the greater quantity?

(A) \(\displaystyle 5t + 8t + 7\)

(B) \(\displaystyle 5+ 8t + 7t\)

Possible Answers:

(A) is greater

(A) and (B) are equal

(B) is greater

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

Correct answer:

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

Explanation:

Depending on the value of \(\displaystyle t\), it is possible for either expression to be greater or for both to be equal.

Case 1: \(\displaystyle t = 1\)

\(\displaystyle 5t + 8t + 7 = 5 \cdot 1 + 8 \cdot 1 + 7 = 5 + 8 + 7 = 20\)

and 

\(\displaystyle 5+ 8t + 7t = 5+ 8 \cdot 1 + 7 \cdot 1 = 5 + 8 + 7 = 20\)

So the two are equal.

 

Case 2: \(\displaystyle t = 2\)

\(\displaystyle 5t + 8t + 7 = 5 \cdot 2 + 8 \cdot 2 + 7 = 10 + 16 + 7 = 33\)

and 

\(\displaystyle 5+ 8t + 7t = 5+ 8 \cdot 2 + 7 \cdot 2 = 5 + 16 + 14 = 35\)

So (B) is greater. 

 

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

Example Question #4 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

\(\displaystyle t\) is a positive integer.

Which is the greater quantity?

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

(B) \(\displaystyle 7t+5t+3t+t\)

Possible Answers:

(B) is greater

(A) is greater

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

(A) and (B) are equal

Correct answer:

(A) is greater

Explanation:

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

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

Since \(\displaystyle 21 > 16\), and \(\displaystyle t\) is positive,

then by the multiplication property of inequality,

\(\displaystyle 21 t > 16t\)

making (A) greater regardless of the value of \(\displaystyle t\).

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

\(\displaystyle t\) is a positive integer.

Which is the greater quantity?

(A) \(\displaystyle 7t + 11 + 4t\)

(B) \(\displaystyle 7 + 11t + 4\)

Possible Answers:

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

(A) and (B) are equal

(B) is greater

(A) is greater

Correct answer:

(A) and (B) are equal

Explanation:

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

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

Regardless of the value of \(\displaystyle t\), the expressions are equal.

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

Which of the following is equivalent to \(\displaystyle 2t + 14\) ?

Possible Answers:

\(\displaystyle 16t^{2}\)

\(\displaystyle 16t\)

\(\displaystyle 28t^{2}\)

None of the other responses is correct.

\(\displaystyle 28t\)

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.

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

\(\displaystyle t\) is a positive integer.

Which is the greater quantity?

(A) \(\displaystyle 8t + 7 + 5t + 9\)

(B) \(\displaystyle 10 + 4t + 9t +5\)

Possible Answers:

(B) is greater

(A) is greater

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

(A) and (B) are equal

Correct answer:

(A) is greater

Explanation:

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

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

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

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