How to multiply variables - ISEE Upper Level (grades 9-12) Quantitative Reasoning

This can be solved using the pattern for the sum times difference of two terms:

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

. Since , , and . This makes (a) greater.

If , then .

If , then .

Therefore, at least two possibilities can be demonstrated.

Since ,

.

Using the associative property of multiplication,

.

Using the distributive property,

.

Using the commutative and associative properties of multiplication,

and

.

The expression is the sum of unlike terms and cannot be simplified further.

The only expression that can be restated as is .

Each expression can be simplified using the properties of exponents, among others:

The correct choice is , as it is the only choice not equivalent to .

,

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

and (B) is greater.

117% of is equal to .

One-ninth of this is

143% of is equal to

One-eleventh of this is

Regardless of the value of , the quanitites are equal.

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

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

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

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

and must be positive, so .

(B) must be greater regardess of .

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

Store A discounts the hat by 8%, meaning that its discounted price before sales tax is

.

The amount paid after sales tax is this price plus 8% of it, or

.

Since Store B neither discounts the price nor charges sales tax, the amount paid will be the original .

, so .

The hat will cost more in Store B, so (B) is greater.

We can compare these positive numbers by comparing their squares; the greater number will have the greater square.

Since and are positive, and

.

Therefore,

and

.

Use the difference of squares pattern as follows:

We can rewrite as follows:

, and

is a perfect square polynomial, as seen here:

so the original polynomial is equal to

This is the difference of squares, so it can be factored as

The easiest way to solve this problem is to take a negative integer to use for m.

For example, can be used.

Plugging in into the expression, , we get:

This simplifies to

Given that 24 is a positive number, is the correct answer.

By the order of operations, all operations within grouping symbols must be performed first, with the innermost symbols taking precedence. Therefore, the three operations within the brackets - the subtraction, the division, and the cubing - must be performed before the remaining two.

Once these three operations are completed, there remain two more, a division and an addition. Division has precendence in the order of operations, so the last operation performed is the addition.

One foot comprises twelve inches, so multiply the number of feet by conversion factor 12:

inches.

One pound comprises sixteen ounces, so multiply the number of ounces by conversion factor 16:

ounces.

The two quantities are both equal to .

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

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

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

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,

,

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