240 MCGRAW-HILL’S SAT
If You Can’t Find What You Want, Find
What You Can!If you can’t find what you want right away, just
look at the parts of the problem one at a time,
and find what you can. Often, going step by
step and noticing the relationships among the
parts will lead you eventually to the answer
you need.Lesson 2:Analyzing Problems
Break Complicated Problems into
Simple Ones
Analyzing is key to solving many SAT math
problems. Analyzing a problem means simply
looking at its parts and seeing how they relate.
Often, a complicated problem can be greatly
simplified by looking at its individual parts. If
you’re given a geometry diagram, mark up the
angles and the sides when you can find them.
If you’re given algebraic expressions, notice
how they relate to one another.6 feet6 feet 6 feet6 feetFor a certain fence, vertical posts must be placed
6 feet apart with supports in between, as shown
above. How many vertical posts are needed for a
fence 120 feet in length?
You may want to divide 120 by 6 and get 20, which
seems reasonable. But how can you check this with-
out drawing a fence with 20 posts? Just change the
question to a much simpler one to check the relation-
ship between length and posts. How many posts are
needed for a 12-foot fence? The figure above provides
the answer. Obviously, it’s 3. But 12 ÷ 6 isn’t 3; it’s 2.
What gives? If you think about it, you will see that di-
viding only gives the number of spacesbetween the
posts, but there is always one more post than spaces.
So a 120-foot fence requires 20 + 1 = 21 vertical posts.Look for Simple RelationshipsOnce you see the parts of a problem, look for
simple relationships between them. Simple
relationships usually lead to simple solutions.If 2x^2 + 5y= 15, then what is the value of 12x^2 + 30y?
Don’t worry about solving for xand y. You only
need to see the simple relationship between the ex-
pressions. The expression you’re looking for, 12x^2 +
30 y, is 6 times the expression you’re given, 2x^2 + 5y. So,
by substitution, 12x^2 + 30ymust equal 6 times 15, or 90.AB
D C
E F
G
In the figure above, ABCDis a rectangle with area
60, and AB= 10. If E, F, and Gare the midpoints of
their respective sides, what is the area of the shaded
region?
This looks complicated at first, but it becomes
much simpler when you analyze the diagram. You
probably know that the formula for the area of a rec-
tangle is a= bh, but the shaded region is not a rec-
tangle. So how do you find its area? Analyze the
diagram using the given information. First, write the
fact that AB= 10 into the diagram. Since the area of
the rectangle is 60 and its base is 10, its height must
be 6. Then, knowing that E, F, and Gare midpoints,
you can mark up the diagram like this:AB
D C
E F
G
10
3
3 3 3
3
5 5
Notice that the dotted lines divide the shaded re-
gion into three right triangles, which are easy to
work with. The two bottom triangles have base 5
and height 3 (flip them up if it helps you to see), and
the top triangle has base 10 and height 3. Since the
formula for the area of a triangle is a= bh, the
areas of the triangles are 7.5, 7.5, and 15, for a total
area of 30.1
2