Lesson 1: Using algebraic expressions
To solve tough SAT math problems, you must be fluent in defining, manipulating, and analyzing algebraic expressions.
Corrine drives to her office at an average speed of 50 miles per hour. When she returns home by the same route, the traffic is lighter and she averages 60 miles per hour. If her trip home is 10 minutes shorter than her trip to her office, what is the distance, in miles, from Corrine’s home to her office?
(Medium-hard) Why does everyone hate “word problems” like this one? For most of us, the problem is that the equations aren’t “set up” for us—we have to set them up ourselves, which can be a pain in the neck. But we can make these problems much easier by breaking them down into clear steps.
Key Steps to Solving Tough Algebraic Problems:
Solving tough problems in mathematics and science frequently involves four essential steps:
1. identify the relevant quantities in the situation
2. express those quantities with algebraic expressions
3. translate the facts of the problem situation into equations involving those expressions
4. analyze and solve those equations
Step 1. Identify: In this problem, there are six relevant quantities:
• the speed from home to work
• the distance from home to work
• the time it takes to get from home to work
• the speed from work to home
• the distance from work to home
• the time it takes to get from work to home
This may seem like a lot, but as we will see, keeping track of them is quite manageable.
Step 2. Express: The problem gives us enough information to express all six quantities in terms of only two “unknowns.” If d is the distance, in miles, from her home to her office, and t is the time, in hours, it takes her to get home from the office, then we can express our six quantities, respectively, as
Step 3. Translate: To translate the facts of this problem into equations, we must know the formula
distance = average speed × time. Applying this to each trip gives us
Step 4. Analyze and Solve: We have now reduced the problem to a “two by two system,” that is, two equations with two unknowns. Since the number of equations equals the number of unknowns, we should be able to solve for those unknowns. (In Lessons 12 and 13, we will review these concepts and techniques.) Since the unknown d is isolated in both equations, substitution is simple:
Since t represents the time it took Corrine to return home, in hours, this means it took her 5/6 hours
(or 50 minutes) to get from her office to her home, and 5/6 hour + 1/6 hour = 1 hour to get to her
office from home. But remember, the question asks for the distance from her home to her office,
which we can find by substituting into either of our equations:
50(5/6 + 1/6) or 60(5/6) = 50 miles
Lesson 2: The Laws of Arithmetic
When expressing or simplifying a quantity, you frequently have many options. For instance, the
expression 4x2 – 12x can also be expressed as 4x(x – 3). Similarly, 3.2 can be expressed as 16/5 or 3
⅕ or 32/10. Which way is better? It depends on what you want to do with the expression. Different
forms of an expression can reveal different characteristics of that quantity or the equation in which it
appears. To gain fluency in expressing quantities, you must understand the Laws of Arithmetic.
What is the value of ?
To simplify complex expressions, you must know the Order of Operations: PG-ER-MD-AS
Step 1: PG (parentheses and other grouping symbols, from innermost to outermost and left to right)
Since this expression contains no parentheses, we don’t have to worry about “grouped” operations, right? Wrong! Remember that fraction bars and radicals are “grouping symbols” just like parentheses are.
In other words, we can think of this expression as
If a set of parentheses contains only one operation, then we simply do that operation:
If it contains more than one operation, then we must move on to the next step.
Step 2: ER (exponents and roots, from innermost to outermost and left to right)
Do any of the parentheses contain exponents or roots? Yes, so we must perform that operation next:
Step 3: MD (multiplication and division, from left to right)
Next, we do any multiplication inside the parentheses:
Step 4: AS (addition and subtraction, from left to right)
Now we do any addition and subtraction left in the parentheses:
Once all the “grouped” operations are completed, we run through the order of operations once again to finish up. Exponents or roots? No. Multiplication or division? Yes:1.875 + 2
Addition or subtraction? Yes: 1.875 + 2 = 3.875