ISEE Upper Level Quantitative Reasoning - How to divide variables

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

0.71

0.24

0.57

0.04

Explanation

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

; .

Which is the greater quantity?

(a) The reciprocal of

(b) The reciprocal of

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

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

Explanation

We show that the given information is insufficient by examining two cases.

Case 1:

$x = -\frac{1}{2}$

The reciprocal of is $- \frac{2} {1}$ , or .

Also, $x + 2 = -\frac{1}{2} + 2 = 1 \frac{1}{2} = \frac{3}{2}$ , the reciprocal of which is $\frac{2}{3}$ .

$\frac{2}{3} > -\frac{1}{2}$ , so (b) is the greater quantity.

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

The reciprocal of is $\frac{2} {1}$ , or 2.

Also, $x + 2 = \frac{1}{2} + 2 =2 \frac{1}{2} = \frac{5}{2}$ , the reciprocal of which is $\frac{2}{5}$ .

$2 > \frac{2}{5}$ , so (a) is the greater quantity.

in both cases, but in one case, (a) is greater and in the other, (b) is greater.

Let be negative. Which of the following is the greater quantity?

(A) $y \div \left ( -8\right )$

(B) $y \div \left ( -7\right )$

(B) is greater

(A) is greater

(A) and (B) are equal

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

Explanation

The quotient of two negative numbers is positive. The expressions can be rewritten as follows:

$y \div \left ( -8\right ) = \left | y\right | \div 8$

$y \div \left ( -7\right ) = \left | y\right | \div 7$

Both expressions have the same dividend; the second has the lesser divisor so it has the greater quotient. This makes (B) greater.

is a negative number.

Which is the greater quantity?

(a) The reciprocal of

(b) The reciprocal of

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

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

Explanation

Since is negative, its reciprocal $\frac{1}{x}$ is also negative. Since

$1 > \frac{1}{2}$ ,

by the Multiplication Property of Inequality,

$1 \cdot \frac{1}{x}< \frac{1}{2}\cdot \frac{1}{x}$

$\frac{1}{x}< \frac{1}{2 \cdot x}$

$\frac{1}{x}< \frac{1}{2 x}$

That is, the reciprocal of is greater than that of .

is a negative integer. Which is the greater quantity?

(A) $A \div (- \sqrt{5})$

(B) $A \div (- \sqrt{7})$

(A) is greater

(B) is greater

(A) and (B) are equal

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

Explanation

Since the quotient of negative numbers is positive, both results will be positive.

We can rewrite both of these as quotients of positive numbers, as follows:

$A \div (- \sqrt{5}) =\left | A \right | \div \sqrt{5}$

$A \div (- \sqrt{7}) =\left | A \right | \div \sqrt{7}$

Since the expressions have the same dividend and the second has the greater divisor, the first has the greater quotient.

Therefore, (A) is greater.

Divide:

$\frac{6x^3-3x^2+9x}{3x}$

Explanation

$\frac{6x^3-3x^2+9x}{3x}=\frac{6x^3}{3x}-\frac{-3x^2}{3x}+\frac{9x}{3}=2x^2-x+3$

Divide:

$\frac{4x^{3}+8x^{2}-12x}{4x}$

$x^{2}+2x-3$

$x^{3}+2x^{2}-3x$

$4x^{2}+8x-12$

The expression cannot be simplified further.

$x^{4}+2x^{3}-3x^{2}$

Explanation

To divide this problem we simplify it first. In this problem we can separate the big fraction into three smaller fractions.

$\frac{4x^{3}+8x^{2}-12x}{4x}=\frac{4x^{3}}{4x}+\frac{8x^{2}}{4x}-\frac{12x}{4x}$

Then from here, we can pull out a from both the numerator and denominator of each smaller fraction.

$\frac{4x(x^2)}{4x}+\frac{4x(2x)}{4x} - \frac{4x(3)}{4x}$

Now we cancel terms and get the following result:

Simplify the following expression:

$\frac{343w^7}{49w^3}$

$\frac{343w^4}{49}$

$\frac{49w^7}{w^3}$

Explanation

Simplify the following expression:

$\frac{343w^7}{49w^3}$

Let's begin by simplifying the coefficients

$\frac{343w^7}{49w^3}=\frac{49*7w^7}{49w^3}=\frac{7w^7}{w^3}$

Next, complete the question by subtracting the exponents:

$\frac{7w^7}{w^3}=7w^{7-3}=7w^4$

So, our answer is:

Which of the following is a factor of $(8-2)^{2}$ ?

Explanation

The first step to solving this question is to reduce $(8-2)^{2}$ .

$(8-2)^{2}$

$6^{2}$

The only number listed that is a factor of 36 is 18, given that 2 times 18 is 36. Therefore, 18 is the correct answer.

$\dpi{100} \frac{1}{3}\div \frac{1}{9}$

$\dpi{100} 3$

$\dpi{100} 2$

$\dpi{100} \frac{1}{27}$

$\dpi{100} \frac{1}{3}$

Explanation

Never divide fractions! Simply flip the fraction that follows the division symbol, and then multiply it by the first fraction. So this expression becomes:

$\dpi{100} \frac{1}{3}\times \frac{9}{1}=\frac{9}{3}=3$

How to divide variables

Help Questions

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation