Direct Variation and
Proportion
Section 1.4
• A ratio is the comparison of two
quantities by division.
a
• Ex.
is read as “a is to b”
b
• A proportion is a statement that two
ratios are equal.
a
c

• Ex.
as read as “a is to b as c
b
d
is to d”
• Cross Product Property of Proportions
• For b  0, a n d d  0 :
• If a c , then ad = bc
b

d
• Important to check answers, sometimes
the answer will not work because it
causes one of the denominators to
equal zero
• Solve each proportion for the
variable. Check your answers.
• Ex. x  2  3 x
3
6
• Ex. 2 x  5  3 x
10
20
• Ex. 3 y  y  1
10
6
Why learn about direct
variation?
• Many situations in real life when one
quantity varies directly as the result
of another.
• Such as
• Distance traveled
• Paychecks of hourly workers
• Tax paid on purchase
Direct Variation
• The variable y varies directly as x if
there is a nonzero constant k such
that y = kx.
• The equation y = kx is called a direct
variation equation.
• K is called the constant of variation
• Find the constant of variation, k if y
varies directly as x and y = -72 when
x = -18.
• y = kx
• -72 = k(-18)
• k=4
• Find the constant of variation, k if y
varies directly as x and y = 2/3 when
x = 1/3.
• Write an equation of direct variation
that relates the two variables.
• y = 7 for x = 3
• Y = 3.2 for x = 12.8
• Each day Jonathon rides his bicycle
for exercise. When traveling at a
constant rate, he rides 4 miles in
about 20 minutes. At this rate, how
ling would it take him to ride 7 miles?
• Distance = (rate)(time)
• Find his rate
• Use that rate to solve for his new
distance.
• That problem also could have been
solved using a proportion.
• Proportion property of direct
variation: For x 1  0 a n d x 2  0 :
• If  x 1, y 1  a n d  x 2 , y 2  satisfy y = kx,
then y 1  k  y 2
x1
x2
• The speed of sound in air is about
335 ft per second. At this rate, how
far would sound travel in 25 seconds?
• Julie works for an hourly wage. Last week
she worked 18 hours and earned before
taxes $150.30.
• How many hours must Julie work to earn
$208.75?
• Write the direct variation equation that
gives Julie’s income in terms of hours
worked. What does the constant of
variation represent?
Day 2
• a varies directly as b
• If a is 6.3 when b is 70, find b when a
is 5.4.
• If b is 3/5 when a is 9/10, find a
when b is 1/3.
• Determine whether the values in the
table represent a direct variation. If
so, write an equation for the
variation. If not, explain.
• Ex.
x
y
-2
-6
-1
-3
0
0
1
3
2
6
• Ex.
x
y
5
49
4
28
3
20
2
5
1
2
• Ex.
x
y
1
2
3
6
5
10
7
14
9
18