1. Types of Variation
Direct Variation: y varies directly as x. As x increases, y
also increases. As x decreases, y also decreases.
Equation for Direct Variation: y = kx
Inverse Variation: y varies inversely as x. As x increases, y
decreases. As x decreases, y increases. The product of x
and y is always constant.
Equation for Inverse Variation: xy = k
Joint Variation: z varies jointly as x and y. As x and y
increases, z also increases. As x and y decrease, z also
decreases.
Equation for Joint Variation: y = kxz
4. Determine whether the relation between the quantities involved is
direct or inverse variation.
1. The number of workers and the time required to finish an amount
of work.
2. The number of workers and the amount of worked finished.
3. The height of a person and his weight.
Inverse
Direct
Direct
5. Steps in solving problems involving variations
Write the
equation or
formula
Solve for
the value of
k
Give the
equation of
variation
Solve for the
unknown
value
6. EXAMPLE
1. a is directly proportional to b. if a = 10 and b = 2.
Find a if b = 4.
Formula: a = kb
Solve for k: k = 5
Equation of variation: a = 5b
Solve for the
unknown
value:
a = 5b a = 5(4) a = 20
7. EXAMPLE
2. If y varies inversely as x and y = 10 when x = 2,
find y when x = 10.
Formula:
Solve for k: k = 20
Equation of variation:
Solve for the
unknown
value:
y = 2
8. EXAMPLE
3. d varies jointly as h and g. if d = 5 when h = 14
and g = 5, find g when h = 3 and d = 84.
Formula: d = khg
Solve for k: k = 1/14
Equation of variation: d = 1/14hg
Solve for the
unknown
value:
d = 1/14hg
84 = 14(3)g
g = 2
84 = 42g
9. EXAMPLE
4. If z varies directly as x and inversely as y, and z = 9 when
x = 6 and y = 2, find z when x = 8 and y = 12.
Formula:
Solve for k: k = 3
Equation of variation:
Solve for the
unknown
value: z = 2
10. EXAMPLE
4. If z varies directly as x and inversely as y, and z = 9 when
x = 6 and y = 2, find z when x = 8 and y = 12.
Formula:
Solve for k: k = 3
Equation of variation:
Solve for the
unknown
value: z = 2
11. EXAMPLE
4. If z varies directly as x and inversely as y, and z = 9 when
x = 6 and y = 2, find z when x = 8 and y = 12.
Formula:
Solve for k: k = 3
Equation of variation:
Solve for the
unknown
value: z = 2
