Variables - ISEE Upper Level: Quantitative Reasoning

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Question

Simplify:

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Answer

Group and collect like terms:

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

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Question

Simplify the expression:

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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

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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) $z^{2}$

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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 :

$6 +x^{2}$

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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)

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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)

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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)

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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) $F_{778}$

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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?

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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)

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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)

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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) $(10 + X )\div 5$

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

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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)

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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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Question

is a negative number.

Which is the greater quantity?

(a)

(b)

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Answer

The expressions and are each other's opposite, so they have the same absolute value. Therefore, regardless of the value of ,

.

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Question

; and are both positive.

Which is the greater quantity?

(a)

(b)

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Answer

If ,

then

.

The absolute value of a negative number is its (positive) opposite, so

Also, if and are both positive, then is positive; the absolute value of a positive number is the number itself, so . Since , it follows that . Therefore,

Since is given to be positive,

.

and

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Question

Define . The graph of is a line with slope $\frac{1}{2}$ .

$(g \circ f) (2) = 6$ .

Which is the greater quantity?

(a)

(b)

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Answer

, so $f (2) = 3 \cdot 2 = 6$ .

$(g \circ f) (2) = 6$ , so, by definition, , or .

The graph of is a line through the point with coordinates and with slope $\frac{1}{2}$ . The equation of the line can be determined by setting $g(x) = x = 6 , m = \frac{1}{2}$ in the slope-intercept form:

$6 = \frac{1}{2} \cdot 6+ b$

.

The equation of the line is $g(x) = \frac{1}{2} x+ 3$ , which makes this the definition of . By setting ,

$g(2) = \frac{1}{2} \cdot 2+ 3 = 1 + 3 = 4$ .

Therefore,

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Question

The ratio of 10 to 14 is closest to what value?

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Answer

Another way to express ratios is through division. 10 divided by 14 is approximate 0.71.

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Question

If is the quotient of and , which statement could be true?

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Answer

A quotient is the result of division. If is the quotient of and , that means that $p\div v=q$ could be true.

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