How to subtract variables - ISEE Upper Level (grades 9-12) Quantitative Reasoning
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Simplify:
Simplify:
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Simplify:
Simplify:
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is negative. Which of these quantities is the greater?
(A) 
(B) 
(A)
(B)
,
So by the multiplication property of inequality, when each is multiplied by the negative number 
,
.
Also,
,
so by the addition property of inequality,
or
This makes (B) greater.
So by the multiplication property of inequality, when each is multiplied by the negative number
Also,
so by the addition property of inequality,
or
This makes (B) greater.
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Assume you know the values of all four variables in the expression
In which order do you perform the operations in order to evaluate the expression?
Assume you know the values of all four variables in the expression
In which order do you perform the operations in order to evaluate the expression?
Multiplication and division take precedence over subtraction in the order of operations, so these two operations are performed first. The two must be performed from left to right, so the division is worked first, followed by the multiplication. The subtraction is last.
Multiplication and division take precedence over subtraction in the order of operations, so these two operations are performed first. The two must be performed from left to right, so the division is worked first, followed by the multiplication. The subtraction is last.
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Consider the expression:
Which of the following expressions must be equal in value to the above expression?
I) 
II) 
III) 
Consider the expression:
Which of the following expressions must be equal in value to the above expression?
I)
II)
III)
The order of operations is as follows:
Exponents
Multiplication and division (left to right)
Addition and subtraction (left to right)
The expression
is therefore evaluated by multiplying, then dividing, then adding. The net result is that the product 
is added to the quotient 
.
If we examine (I), we see that, since the multiplication is in parentheses, it is worked first. The division is worked second, then the addition. The order of operations has not changed, so the expressions are equivalent.
If we examine (II), we see that the order of operations has changed so that the addition is worked first. We see through example that the expressions can have different values:
If we examine (III), we see that, since the division is in parentheses, it is worked first. The multiplication is worked second, then the addition. The upshot is the same as in the main expression, however - the product 
is added to the quotient 
. Therefore, the expressions are equivalent.
The correct response is (I) and (III)
The order of operations is as follows:
Exponents
Multiplication and division (left to right)
Addition and subtraction (left to right)
The expression
is therefore evaluated by multiplying, then dividing, then adding. The net result is that the product
If we examine (I), we see that, since the multiplication is in parentheses, it is worked first. The division is worked second, then the addition. The order of operations has not changed, so the expressions are equivalent.
If we examine (II), we see that the order of operations has changed so that the addition is worked first. We see through example that the expressions can have different values:
If we examine (III), we see that, since the division is in parentheses, it is worked first. The multiplication is worked second, then the addition. The upshot is the same as in the main expression, however - the product
The correct response is (I) and (III)
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When evaluating the expression
,
assuming you know the values of all five variables, what is the third operation that must be performed?
When evaluating the expression
assuming you know the values of all five variables, what is the third operation that must be performed?
In the order of operations, any operations in parentheses must be performed first - there are two, the leftmost addition and the middle subtraction. What remains are the leftmost subtraction, the rightmost subtraction, and the rightmost addition. Since additions and subtractions are performed from left to right, the next, or third, operation performed is the leftmost subtraction.
In the order of operations, any operations in parentheses must be performed first - there are two, the leftmost addition and the middle subtraction. What remains are the leftmost subtraction, the rightmost subtraction, and the rightmost addition. Since additions and subtractions are performed from left to right, the next, or third, operation performed is the leftmost subtraction.
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