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Amy Ferguson Moncure

H Algebra 2/Trig · Unit 4

Combined Variation

Find a value when a variable varies directly with one quantity and inversely with another.

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Key Ideas Easy Medium Hard

y = kxz

k = yzx

What to Remember

Both kinds at once

Combined variation has a direct part and an inverse part.

y = kxz varies directly with x and inversely with z.

Find k

Rearrange so k is alone.

For y = kxz, the rearranged formula is k = yzx.

A square

y = kxz² varies inversely with the square of z.

Square z before you multiply.

Put k back

Write the formula again with the number you found for k.

Then substitute the new values.

Level 1

Easy

Directly with x and inversely with z

Example
y varies directly with x and inversely with z. y = 10 when x = 4 and z = 2. Find y when x = 6 and z = 3.

y = kxz

k = yzx

k = 10×24

k = 204

k = 5

y = 5xz

y = 5×63

y = 303

y = 10

Answer: y = 10

1
y varies directly with x and inversely with z. y = 18 when x = 6 and z = 3. Find y when x = 4 and z = 2.
Show Answer

y = kxz

k = yzx

k = 18×36

k = 546

k = 9

y = 9xz

y = 9×42

y = 362

y = 18

Answer: y = 18

2
y varies directly with x and inversely with z. y = 12 when x = 3 and z = 2. Find y when x = 5 and z = 4.
Show Answer

y = kxz

k = yzx

k = 12×23

k = 243

k = 8

y = 8xz

y = 8×54

y = 404

y = 10

Answer: y = 10

3
y varies directly with x and inversely with z. y = 16 when x = 4 and z = 2. Find y when x = 3 and z = 6.
Show Answer

y = kxz

k = yzx

k = 16×24

k = 324

k = 8

y = 8xz

y = 8×36

y = 246

y = 4

Answer: y = 4

4
y varies directly with x and inversely with z. y = 20 when x = 5 and z = 4. Find y when x = 2 and z = 2.
Show Answer

y = kxz

k = yzx

k = 20×45

k = 805

k = 16

y = 16xz

y = 16×22

y = 322

y = 16

Answer: y = 16

5
y varies directly with x and inversely with z. y = 24 when x = 8 and z = 3. Find y when x = 4 and z = 6.
Show Answer

y = kxz

k = yzx

k = 24×38

k = 728

k = 9

y = 9xz

y = 9×46

y = 366

y = 6

Answer: y = 6

Level 2

Medium

Inversely with a square

Example
y varies directly with x and inversely with the square of z. y = 12 when x = 4 and z = 2. Find y when x = 3 and z = 1.

y = kxz²

k = yz²x

z² = 4

k = 12×44

k = 484

k = 12

y = 12xz²

z² = 1

y = 12×31

y = 36

Answer: y = 36

1
y varies directly with x and inversely with the square of z. y = 6 when x = 3 and z = 2. Find y when x = 2 and z = 1.
Show Answer

y = kxz²

k = yz²x

z² = 4

k = 6×43

k = 243

k = 8

y = 8xz²

z² = 1

y = 8×21

y = 16

Answer: y = 16

2
y varies directly with x and inversely with the square of z. y = 32 when x = 4 and z = 2. Find y when x = 3 and z = 2.
Show Answer

y = kxz²

k = yz²x

z² = 4

k = 32×44

k = 1284

k = 32

y = 32xz²

z² = 4

y = 32×34

y = 964

y = 24

Answer: y = 24

3
y varies directly with x and inversely with the square of z. y = 27 when x = 3 and z = 3. Find y when x = 2 and z = 1.
Show Answer

y = kxz²

k = yz²x

z² = 9

k = 27×93

k = 2433

k = 81

y = 81xz²

z² = 1

y = 81×21

y = 162

Answer: y = 162

4
y varies directly with x and inversely with the square of z. y = 16 when x = 2 and z = 4. Find y when x = 1 and z = 2.
Show Answer

y = kxz²

k = yz²x

z² = 16

k = 16×162

k = 2562

k = 128

y = 128xz²

z² = 4

y = 128×14

y = 1284

y = 32

Answer: y = 32

5
y varies directly with x and inversely with the square of z. y = 10 when x = 5 and z = 1. Find y when x = 4 and z = 2.
Show Answer

y = kxz²

k = yz²x

z² = 1

k = 10×15

k = 105

k = 2

y = 2xz²

z² = 4

y = 2×44

y = 84

y = 2

Answer: y = 2

Level 3

Hard

A combined-variation situation

Example
The time needed to mow lawns varies directly with the area and inversely with the number of people. 2 people take 6 hours to mow 4 acres. How long will 3 people take to mow 6 acres?

t = kap

k = tpa

k = 6×24

k = 124

k = 3

t = 3ap

t = 3×63

t = 183

t = 6

3 people will take 6 hours.

Answer: 3 people will take 6 hours.

1
The time to paint rooms varies directly with the number of rooms and inversely with the number of painters. 4 painters take 6 hours to paint 8 rooms. How long will 3 painters take to paint 6 rooms?
Show Answer

t = krn

k = tnr

k = 6×48

k = 248

k = 3

t = 3rn

t = 3×63

t = 183

t = 6

3 painters will take 6 hours.

Answer: 3 painters will take 6 hours.

2
The pressure of a gas varies directly with its temperature and inversely with its volume. The pressure is 12 when the temperature is 300 and the volume is 50. What is the pressure when the temperature is 280 and the volume is 40?
Show Answer

P = kTV

k = PVT

k = 12×50300

k = 600300

k = 2

P = 2TV

P = 2×28040

P = 56040

P = 14

The pressure is 14.

Answer: The pressure is 14.

3
The cost of a trip varies directly with the miles driven and inversely with the number of people sharing the cost. 4 people drive 120 miles and each pays 30 dollars. How much does each person pay when 5 people drive 200 miles?
Show Answer

c = kmp

k = cpm

k = 30×4120

k = 120120

k = 1

c = 1mp

c = 1×2005

c = 2005

c = 40

Each person pays 40 dollars.

Answer: Each person pays 40 dollars.

4
The time to type a paper varies directly with the number of pages and inversely with the typing speed in pages per hour. A 10-page paper takes 2 hours at 5 pages per hour. How long does a 12-page paper take at 4 pages per hour?
Show Answer

t = knr

k = trn

k = 2×510

k = 1010

k = 1

t = 1nr

t = 1×124

t = 124

t = 3

The paper takes 3 hours.

Answer: The paper takes 3 hours.

5
The brightness of a light varies directly with the wattage and inversely with the square of the distance. The brightness is 18 at 36 watts and a distance of 2 meters. What is the brightness at 24 watts and a distance of 4 meters?
Show Answer

B = kwd²

k = Bd²w

d² = 4

k = 18×436

k = 7236

k = 2

B = 2wd²

d² = 16

B = 2×2416

B = 4816

B = 3

The brightness is 3.

Answer: The brightness is 3.

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