Variables - ISEE Upper Level (grades 9-12) Quantitative Reasoning

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Question

Simplify:

$5x^{2} y ^{2}+7 + 6x^{2}y^{2} + 11 xy$

Answer

Group and combine like terms $5x^{2} y ^{2},6x^{2}y^{2}$ :

$5x^{2} y ^{2}+7 + 6x^{2}y^{2} + 11 xy$

$= 5x^{2} y ^{2}+ 6x^{2}y^{2}+ 11 xy +7$

$=\left ( 5+ 6 \right )x^{2}y^{2}+ 11 xy +7$

$=11x^{2}y^{2}+ 11 xy +7$

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Question

Assume that and are not both zero. Which is the greater quantity?

(a) $\frac{(x+y)^{2} + (x - y)^{2}}{x^{2}+y^{2}}$

(b)

Answer

Simplify the expression in (a):

$\frac{(x+y)^{2} + (x - y)^{2}}{x^{2}+y^{2}}$

$=\frac{(x^{2}+2xy + y^{2})+ (x^{2}-2xy + y^{2})}{x^{2}+y^{2}}$

$=\frac{ x^{2}+x^{2} +2xy -2xy+ y^{2}+ y^{2}}{x^{2}+y^{2}}$

$=\frac{ 2x^{2} +2 y^{2}}{x^{2}+y^{2}} =\frac{ 2(x^{2}+y^{2})}{x^{2}+y^{2}} = 2$

Therefore, whether (a) or (b) is greater depends on the values of and , neither of which are known.

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Question

Which is the greater quantity?

(a) $(x + y)^{2}$

(b) $x^{2}+ y^{2}$

Answer

$(x + y)^{2} = x^{2}+2xy + y^{2}$

Since and have different signs,

, and, subsequently,

Therefore,

$(x + y)^{2} = x^{2}+2xy + y^{2} < x^{2} + y^{2}$

This makes (b) the greater quantity.

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Question

Which is the greater quantity?

(a) $x^{2} + 3x$

(b)

Answer

We give at least one positive value of for which (a) is greater and at least one positive value of for which (b) is greater.

Case 1:

(a) $x^{2} + 3x = 2^{2} + 3 \cdot 2 = 4 + 6 = 10$

(b) $4x = 4 \cdot 2 = 8$

Case 2: $x = \frac{1}{2}$

(a) $x^{2} + 3x = \left ( \frac{1}{2} \right ) ^{2} + 3 \cdot \frac{1}{2} = \frac{1}{4} + \frac{3}{2} = \frac{7}{4}= 1 \frac{3}{4}$

(b) $4x = 4 \cdot \frac{1}{2} = 2$

Therefore, either (a) or (b) can be greater.

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Question

Assume all variables to be nonzero.

Simplify: $\left (12x^{5}y ^{4}z^{3} \right )^{0} + \left (3x^{5}y ^{4}z^{3} \right )^{0}$

Answer

Any nonzero expression raised to the power of 0 is equal to 1. Therefore,

$\left (12x^{5}y ^{4}z^{3} \right )^{0} + \left (3x^{5}y ^{4}z^{3} \right )^{0} = 1 + 1 = 2$ .

None of the given expressions are correct.

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Question

Simplify:

Answer

Group and collect like terms:

$=\left ( 5+ 8 \right )x +\left ( 7 - 9 \right ) y$

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Question

Simplify the expression:

Answer

There are no like terms among these three; one has only as a variable, one has only , one has both. Therefore, no terms can be collected, and the expression cannot be simplified further.

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Question

Which is the greater quantity?

(a) The sum of the integers from 1 to 1,000

(b) The sum of the integers from 1,001 to 1,500

Answer

(a) The sum of the integers from 1 to 1,000 is $\frac{(1+1,000) \times 1,000}{2} = 500,500$

(b) The sum of the integers from 1,001 to 1,500 is

$\frac{(1,001+1,500 ) \times 500}{2} = 625,250$

(b) is greater

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Question

Which of the following is the greater quantity?

(A) $x + y^{2}$

(B) $z^{2}$

Answer

(A) Substitute and evaluate, remembering that by order of operations, squaring precedes addition:

$x + y^{2} = 4 + 10^{2} = 4 + 100 = 104$

(B) Substitute and evaluate

$z^{2} = 14^{2} = 196$

(B) is greater

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Question

Evaluate for :

$6 +x^{2}$

Answer

Substitute and evaluate, remembering that in order of operations, squaring comes before adding:

$6 +x^{2}$

$= 6 + (-6)^{2}$

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Question

Which is the greater quantity?

(A)

(B)

Answer

Which quantity is the greater depends on the value of .

Case 1:

Then

and

$2x = 2 \cdot 0 = 0$

Subsequently, (A) is greater

Case 2:

Then

and

$2x = 2 \cdot 1,000,001 = 2,000,002$

Subsequently, (B) is greater

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Question

is negative. Which of these quantities is the greater?

(A)

(B)

Answer

Since and is negative, then by the multiplication property of inequality,

Also, , so by the addition property of inequality,

making (B) greater.

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Question

is positive. Which of these quantities is greater?

(A)

(B)

Answer

The values of both terms depend on the value of .

Case 1: $y = \frac{1}{2}$

$9 + 6y = 9 + 6 \cdot \frac{1}{2} = 9 + 3 = 12$

$11 + 4y = 11 + 4 \cdot \frac{1}{2} = 11+ 2 = 13$

, making (B) greater.

Case 2:

$9 + 6y = 9 + 6 \cdot 2= 9 + 12 = 21$

$11 + 4y = 11 + 4 \cdot 2 = 11+ 8 = 19$

, making (A) greater.

Therefore, the information given is insufficient.

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Question

The Fibonacci sequence is defined as follows:

$F_{1} = F_{2} = 1$

For integers , $F_{n} = F_{n-1}+ F_{n-2}$ .

Which is the greater quantity?

(a) The sum of $F_{776}$ and $F_{777}$

(b) $F_{778}$

Answer

Using the definition of the sequence for all terms beginning with the third, substitute 778 for :

$F_{778} = F_{778-1}+ F_{778-2}$

$F_{778} = F_{777}+ F_{776}$

The two quantities are equal.

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Question

When evaluating the expression

$a+ b \cdot \left (c \div d \right ) ^{2}- e$ ,

assuming you know the values of all five variables, what is the third operation that must be performed?

Answer

The operation that must be performed first is the operation in parentheses, which is the division.

After this, the order of operations is:

Exponents

Multiplication and division (left to right)

Addition and subtraction (left to right)

The second thing that must be done is the exponentiation - the squaring. Of the three remaining operations, the multiplication takes precedence. The correct response is multiplication.

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Question

is a negative integer. Which is the greater quantity?

(a)

(b)

Answer

$3(X+10) = 3 \cdot X + 3 \cdot 10 = 3X + 30$

Since

,

it follows that

and

regardless of the value of .

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Question

is a negative integer. Which is the greater quantity?

(A)

(B)

Answer

, so

and

is the greater quantity, regardless of .

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Question

is a negative integer. Which is the greater quantity?

(A) $10 + X \div 5$

(B) $(10 + X )\div 5$

Answer

$10 + X \div 5 = 10 + \frac{X}{5}$

$(10 + X )\div 5 = \frac{(10 + X )}{5} = 2 + \frac{X}{5}$

,

so

$10 + \frac{X}{5}> 2 + \frac{X}{5}$

and

$10 + X \div 5> (10 + X )\div 5$ regardless of the value of .

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Question

Which is the greater quantity?

(a) $\left | - 7 \right | \cdot \left | x\right | + \left | -4 \right |$

(b) $(-7) \cdot x+ (-4)$

Answer

Suppose is nonnegative.

Then .

Consequently,

$\left | - 7 \right | \cdot \left | x\right | + \left | -4 \right | = 7 x +4$ ,

which must be positive,

and

$(-7) \cdot x+ (-4) = -7x-4$ ,

which is the opposite of and consequently must be negative. Therefore, (a) is greater.

Suppose is negative.

Then .

Consequently,

$\left | - 7 \right | \cdot \left | x\right | + \left | -4 \right | = 7( -x) +4 = -7x+4$ ,

and

$(-7) \cdot x+ (-4) = -7x-4$ .

, so

,

and (a) is greater.

(a) is the greater quantity either way.

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Question

is a positive number.

Which is the greater quantity?

(a)

(b)

Answer

By examining two scenarios, we see that we cannot determine which is the greater quantity.

Case 1: $x = \frac{1}{2}$

Then $|7x - 9| =\left | 7 \cdot \frac{1}{2}- 9 \right |= \left | 3 \frac{1}{2}- 9 \right |= \left | -5 \frac{1}{2} \right |=5 \frac{1}{2}$

and $|9x - 7|=\left | 9 \cdot \frac{1}{2}- 7 \right |= \left | 4 \frac{1}{2}- 7 \right |= \left | -2 \frac{1}{2} \right |=2 \frac{1}{2}$

This makes (a) the greater quantity.

Case 2:

Then $|7x - 9| =\left | 7 \cdot 2- 9 \right |= \left | 14 - 9 \right |= \left | 5 \right |=5$

and $|9x - 7|=\left | 9 \cdot 2- 7 \right |= \left | 18 - 7 \right |= \left | 11 \right |=11$

This makes (b) the greater quantity.

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