ISEE Upper Level Quantitative Reasoning - How to multiply variables

N is a positive integer; . Which is greater?

(a) $N^{2} - 4N + 4$

(b)

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

(a) and (b) are equal.

Explanation

If , then $N^{2} - 4N + 4 = 2 ^{2} - 4 \cdot 2 + 4= 4 - 8 + 4 = 0$ .

If , then $N^{2} - 4N + 4 = 3 ^{2} - 4 \cdot 3 + 4= 9 - 12 + 4 = 1$ .

Therefore, at least two possibilities can be demonstrated.

Multiply:

$15 x ^{2}+31xy -24 y^{2}$

$15 x ^{2}-49xy +24 y^{2}$

$15x^{2} +49xy - 24y^{2}$

$15x^{2} - 24y^{2}$

$15x^{2} -31xy+ 24y^{2}$

Explanation

$= 5x \cdot 3x+5x \cdot 8y - 3y \cdot 3x - 3y \cdot 8y$

$= 5\cdot3 \cdot x \cdot x+5\cdot 8 \cdot x \cdot y - 3\cdot 3\cdot y\cdot x - 3\cdot 8 \cdot y \cdot y$

$= 15 x ^{2}+40xy - 9xy -24 y^{2}$

$= 15 x ^{2}+31xy -24 y^{2}$

Simplify:

$\left (\frac{4}{3}x - \frac{5}{3} \right )^{2}$

$\frac{16}{9}x^{2} - \frac{40}{9}x +\frac{25}{9}$

$\frac{16}{9}x^{2} - \frac{25}{9}$

$\frac{16}{9}x^{2} - \frac{40}{9}x -\frac{25}{9}$

$\frac{16}{9}x^{2} - \frac{20}{9}x +\frac{25}{9}$

$\frac{16}{9}x^{2} - \frac{20}{9}x -\frac{25}{9}$

Explanation

This can be solved using the pattern for the square of a binomial:

$\left (\frac{4}{3}x - \frac{5}{3} \right )^{2}$

$=\left (\frac{4}{3}x \right ) ^{2} -2 \cdot \frac{4}{3}x \cdot \frac{5}{3} + \left ( \frac{5}{3} \right ) ^{2}$

$=\frac{4^{2}}{3^{2}}\cdot x^{2} -2 \cdot \frac{4}{3}\cdot \frac{5}{3} \cdot x +\frac{5^{2}}{3^{2}}$

$=\frac{16}{9}x^{2} - \frac{40}{9}x +\frac{25}{9}$

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

(A)

(B)

(B) is greater

(A) and (B) are equal

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

(A) is greater

Explanation

So by the multiplication property of inequality, when each is multiplied by the negative number ,

and (B) is greater.

A hat sells for the same pre-tax price in Store A and Store B. Store A then discounts the hat 30%, and Store B discounts it 25%. After the discount, the hat is taxed at a 10% rate in Store A. The hat is taxed at a 5% rate in store B.

Which is greater?

(A) The amount that will be paid for the hat in Store A after discount and tax

(B) The amount that will be paid for the hat in Store B after discount and tax

(B) is greater

(A) is greater

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

(A) and (B) are equal

Explanation

Let be the original price of the hat before discount or tax.

Store A discounts the hat by 30%, meaning that its discounted price before sales tax is

$P - 0.30P = \left ( 1-0.30\right )P = 0.70P$

The amount paid after sales tax is this price plus 10% of it, or

$0.70P + 0.10 \cdot 0.70P = 0.70P + 0.07P= 0.77P$

Store B discounts the hat by 25%, meaning that its discounted price before sales tax is

$P - 0.25P = \left ( 1-0.25\right )P = 0.75P$

The amount paid after sales tax is this price plus 5% of it, or

$0.75P + 0.05 \cdot 0.75P = 0.75P + 0.0375P= 0.7875P$

and must be positive, so .

(B) must be greater regardess of .

Simplify the following:

$54x^{10}$

$54x^{30}$

$15x^{10}$

Explanation

Simplify the following:

To multiple this out, we need to recall how to multiple variables with exponents, and how to multiply coefficients.

To multiple the coefficients (numbers in front) simply treat them like regular multiplications.

So far so good.

To combine our x's, we will add the exponents. Recall that multiplying variables means you add the exponents.

$x^7*x^3=x^{7+3}=x^{10}$

Now, put those two together to get:

$54x^{10}$

Solve for $\dpi{100} x$ :

$\dpi{100} \frac{1}{3}x-14=7$

$\dpi{100} 63$

$\dpi{100} 21$

$\dpi{100} 3$

$\dpi{100} 7$

Explanation

$\dpi{100} \frac{1}{3}x-14=7$

$\dpi{100} \frac{1}{3}x-14+14=7+14$

$\dpi{100} \frac{1}{3}x=21$

$\dpi{100} 3\cdot \frac{1}{3}x=21\cdot 3$

$\dpi{100} x=63$

Which is the greater quantity?

(a) The number of inches in $\frac{7}{36} x$ yards

(b) The number of days in $\frac{36}{7}x$ weeks

(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

One yard comprises thirty-six inches, so multiply the number of yards by conversion factor 36:

$\frac{7}{36}x \cdot 36 = 7x$

One week comprises seven days, so multiply the number of weeks by conversion factor 7:

$\frac{36} {7} x \cdot 7 = 36x$

(b) is the greater quantity, since, if is positive, .

Which of the following expressions is equivalent to

$\left ( 11x - \sqrt {11x}\right )\left ( 11x +\sqrt {11x}\right )$ ?

Explanation

Use the difference of squares pattern as follows:

$\left ( 11x - \sqrt {11x}\right )\left ( 11x +\sqrt {11x}\right )$

$=\left ( 11x \right) ^{2} - \left ( \sqrt {11x}\right ) ^{2}$

$=11^{2}x^{2} - 11x$

$=121x^{2} - 11x$

Which is the greater quantity?

(a) The number of miles in 100 kilometers

(b) The number of kilometers in 100 miles

Note: You may use the conversion factor 1 mile = 1.609 kilometers.

(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 1 mile is equivalent to 1.609 kilometers, the number of kilometers equivalent to 100 miles can be found by multiplying 100 by 1.609. Conversely, the number of miles equivalent to 100 kilometers can be found by dividing 100 by 1.609.

You do not have to do the actual math to answer the question. Since the conversion factor is greater than one, multiplying any positive number by this factor yields a result greater than dividing that same number by it. Therefore,

$100 \times 1.609 > 100 \div 1.609$ ,

and the number of kilometers equivalent to 100 miles, (b), is the greater quantity.

How to multiply variables

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