SAT Mc Graw Hill 2011

CHAPTER 7 / ESSENTIAL PRE-ALGEBRA SKILLS 283

Working with Ratios

When you see a ratio—such as 5:6—don’t let it

confuse you. If it is nota part-to-part ratio,

then just think of it as a fraction. For instance,

5:6 = 5/6. If it isa part-to-part ratio, just divide

each number by the sum to find the fraction of

each part to the whole. For instance, if the

ratio of boys to girls in a class is 5:6, then the

sum is 5 + 6 = 11, so the boys make up 5/11 of

the whole class, and the girls make up 6/11 of

the whole class. (Notice that these fractions

must add up to 1!)

Example:

If a $200 prize is divided up among three people

in a 1:4:5 ratio, then how much does each person

receive? The total of the parts is 1 + 4 + 5 = 10.

Therefore, the three people receive 1/10, 4/10, and

5/10 of the prize, respectively. So one person gets

(1/10) $200 = $20, another gets (4/10) $200 =

$80, and the other gets (5/10) $200 = $100.

Working with Proportions

A proportionis just an equation that says that

two fractions are equal, as in 3/5 = 9/15. Two

ways to simplify proportions are with the law

of cross-multiplicationand with the law of cross-

swapping. The law of cross-multiplication says

that if two fractions are equal, then their

“cross-products” also must be equal. The law

of cross-swapping says that if two fractions are

equal, then “cross-swapping” terms will create

another true proportion.

Example:

If we know that then by the law of

cross-multiplication, we know that 7x=12, and

by the law of cross-swapping, that.

In a word problem, the phrase “at this rate”

means that you can set up a proportion to

solve the problem. A rateis just a ratioof some

quantity to time. For instance, your reading

rateis in words per minute; that is, it is the ratio

x

3

4

7

=

x

4

3

7

= ,

of the number of words you read divided by

the number of minutes it takes you to read

them. (The word peracts like the : in the ratio.)

IMPORTANT: When setting up the propor-

tion, check that the units “match up”—that the

numerators share the same unit and the de-

nominators share the same unit.

Example:

A bird can fly 420 miles in one day if it flies con-

tinuously. At this rate, how many miles can the

bird fly in 14 hours?

To solve this, we can set up a proportion that says that

the two rates are the same.

Notice that the units “match up”—miles in the nu-

merator and hours in the denominator. Now we can

cross-multiply to get 420 14 = 24xand divide by 24

to get x= 245 miles.

Similarity

Two triangles are similar(have the same shape)

if their corresponding angles all havethe same

measure. If two triangles are similar, then their

corresponding sides are proportional.

Example:

In the figure above,

When setting up proportions of sides in simi-

lar figures, double-check that the correspond-

ing sides “match up” in the proportion. For

instance, notice how the terms “match up” in

the proportions above.

m

k

n

l

r

m

n

k

l

==o

m n kl

420 miles

24 hours

miles

hours

=

x

14

Lesson 4: Ratios and Proportions