How to multiply variables - ISEE Upper Level Quantitative Reasoning
Card 0 of 172
Multiply:
Multiply:
This can be solved using the pattern for the sum times difference of two terms:
This can be solved using the pattern for the sum times difference of two terms:
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Multiply:
Multiply:
This can be solved using the pattern for the sum times difference of two terms:
This can be solved using the pattern for the sum times difference of two terms:
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Simplify:
Simplify:
This can be solved using the pattern for the square of a binomial:
This can be solved using the pattern for the square of a binomial:
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N is a positive integer; 
. Which is greater?
(a) 
(b) 0
N is a positive integer;
(a)
(b) 0
. Since 
, 
, and 
. This makes (a) greater.
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N is a positive integer; . Which is greater?
(a) 
(b) 
N is a positive integer;
(a)
(b)
If 
, then 
.
If 
, then 
.
Therefore, at least two possibilities can be demonstrated.
If
If
Therefore, at least two possibilities can be demonstrated.
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Which is the greater quantity?
(a) 
(b) 
Which is the greater quantity?
(a)
(b)
Since 
,
.
Since
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Assume 
is nonzero. Which of the following is equivalent to 
?
Assume
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 
.
Using the associative property of multiplication,
Using the distributive property,
Using the commutative and associative properties of multiplication,
and
The expression
The only expression that can be restated as
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Assume 
is nonzero. Which expression is not equivalent to 
?
Assume
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 
.
Each expression can be simplified using the properties of exponents, among others:
The correct choice is
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Let 
be negative. Which of the following is the greater quantity?
(A) 
(B) 
Let
(A)
(B)
,
So by the multiplication property of inequality, when each is multiplied by the negative number 
,
and (B) is greater.
So by the multiplication property of inequality, when each is multiplied by the negative number
and (B) is greater.
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is a positive number. Which is greater?
(A) One ninth of 117% of 
(B) One eleventh of 143% of 
(A) One ninth of 117% of
(B) One eleventh of 143% of
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.
117% of
One-ninth of this is
143% of
One-eleventh of this is
Regardless of the value of
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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
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
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
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
(B) must be greater regardess of
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A hat sells for the same price in Store A, where the sales tax is 8%, and Store B, where there is no sales tax. Store A then discounts the hat by 8%; Store B does not discount it.
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
A hat sells for the same price in Store A, where the sales tax is 8%, and Store B, where there is no sales tax. Store A then discounts the hat by 8%; Store B does not discount it.
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
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.
Let
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
The hat will cost more in Store B, so (B) is greater.
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and 
are positive integers. Which is the greater quantity?
(A) 
(B) 
(A)
(B)
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
.
We can compare these positive numbers by comparing their squares; the greater number will have the greater square.
Since
Therefore,
and
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Which of the following expressions is equivalent to
?
Which of the following expressions is equivalent to
Use the difference of squares pattern as follows:
Use the difference of squares pattern as follows:
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Factor:
Factor:
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
We can rewrite as follows:
so the original polynomial is equal to
This is the difference of squares, so it can be factored as
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If m is a negative integer, which of the following is an expression that also represents a positive integer?
If m is a negative integer, which of the following is an expression that also represents a positive integer?
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.
The easiest way to solve this problem is to take a negative integer to use for m.
For example,
Plugging in
This simplifies to
Given that 24 is a positive number,
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When evaluating the expression
,
assuming you know the values of all five variables, what is the last operation that must be performed?
When evaluating the expression
assuming you know the values of all five variables, what is the last operation that must be performed?
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.
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.
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is a positive number. Which is the greater quantity?
(a) The number of inches in 
feet
(b) The number of ounces in 
pounds
(a) The number of inches in
(b) The number of ounces in
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 foot comprises twelve inches, so multiply the number of feet by conversion factor 12:
One pound comprises sixteen ounces, so multiply the number of ounces by conversion factor 16:
The two quantities are both equal to
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Which is the greater quantity?
(a) The number of inches in 
yards
(b) The number of days in 
weeks
Which is the greater quantity?
(a) The number of inches in
(b) The number of days in
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, 
.
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
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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.
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.
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.
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.
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