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Amy Ferguson Moncure

H Algebra 2/Trig · Unit 4

Inverse Variation

Find the constant of inverse variation, then find the new value.

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Key Ideas Easy Medium Hard

y = kx

k = xy

What to Remember

The formula

Inverse variation means y = kx.

The product xy stays equal to k.

Find k

Write k = xy.

Substitute the given pair.

A square

Inverse square variation means y = kx².

Then k = x²y.

The new value

Put k back into y = kx.

Substitute the new x and simplify the fraction.

Level 1

Easy

y = kx

Example
y varies inversely with x. y = 12 when x = 2. Find y when x = 8.

y = kx

k = xy

k = 2×12

k = 24

y = 24x

y = 248

y = 3

Answer: y = 3

1
y varies inversely with x. y = 9 when x = 4. Find y when x = 6.
Show Answer

y = kx

k = xy

k = 4×9

k = 36

y = 36x

y = 366

y = 6

Answer: y = 6

2
y varies inversely with x. y = 10 when x = 3. Find y when x = 5.
Show Answer

y = kx

k = xy

k = 3×10

k = 30

y = 30x

y = 305

y = 6

Answer: y = 6

3
y varies inversely with x. y = 4 when x = 6. Find y when x = 3.
Show Answer

y = kx

k = xy

k = 6×4

k = 24

y = 24x

y = 243

y = 8

Answer: y = 8

4
y varies inversely with x. y = 15 when x = 2. Find y when x = 6.
Show Answer

y = kx

k = xy

k = 2×15

k = 30

y = 30x

y = 306

y = 5

Answer: y = 5

5
y varies inversely with x. y = 8 when x = 5. Find y when x = 4.
Show Answer

y = kx

k = xy

k = 5×8

k = 40

y = 40x

y = 404

y = 10

Answer: y = 10

Level 2

Medium

y varies inversely with the square of x

Example
y varies inversely with the square of x. y = 18 when x = 2. Find y when x = 3.

y = kx²

k = x²y

x² = 4

k = 4×18

k = 72

y = 72x²

x² = 9

y = 729

y = 8

Answer: y = 8

1
y varies inversely with the square of x. y = 32 when x = 2. Find y when x = 4.
Show Answer

y = kx²

k = x²y

x² = 4

k = 4×32

k = 128

y = 128x²

x² = 16

y = 12816

y = 8

Answer: y = 8

2
y varies inversely with the square of x. y = 36 when x = 3. Find y when x = 2.
Show Answer

y = kx²

k = x²y

x² = 9

k = 9×36

k = 324

y = 324x²

x² = 4

y = 3244

y = 81

Answer: y = 81

3
y varies inversely with the square of x. y = 16 when x = 2. Find y when x = 4.
Show Answer

y = kx²

k = x²y

x² = 4

k = 4×16

k = 64

y = 64x²

x² = 16

y = 6416

y = 4

Answer: y = 4

4
y varies inversely with the square of x. y = 48 when x = 4. Find y when x = 2.
Show Answer

y = kx²

k = x²y

x² = 16

k = 16×48

k = 768

y = 768x²

x² = 4

y = 7684

y = 192

Answer: y = 192

5
y varies inversely with the square of x. y = 12 when x = 2. Find y when x = 6.
Show Answer

y = kx²

k = x²y

x² = 4

k = 4×12

k = 48

y = 48x²

x² = 36

y = 4836

y = 43

Answer: y = 43

Level 3

Hard

An inverse-variation situation

Example
6 students can paint a fence in 4 hours. The time varies inversely with the number of students. How long will 8 students take?

t = kn

k = nt

k = 6×4

k = 24

t = 24n

t = 248

t = 3

8 students will take 3 hours.

Answer: 8 students will take 3 hours.

1
4 pipes fill a pool in 6 hours. The time varies inversely with the number of pipes. How long will 3 pipes take?
Show Answer

t = kp

k = pt

k = 4×6

k = 24

t = 24p

t = 243

t = 8

3 pipes will take 8 hours.

Answer: 3 pipes will take 8 hours.

2
10 workers finish a job in 6 days. The time varies inversely with the number of workers. How many days will 15 workers take?
Show Answer

t = kw

k = wt

k = 10×6

k = 60

t = 60w

t = 6015

t = 4

15 workers will take 4 days.

Answer: 15 workers will take 4 days.

3
The time for a trip varies inversely with the speed. At 40 miles per hour the trip takes 6 hours. How long does the trip take at 60 miles per hour?
Show Answer

t = kr

k = rt

k = 40×6

k = 240

t = 240r

t = 24060

t = 4

The trip takes 4 hours.

Answer: The trip takes 4 hours.

4
8 printers finish a stack in 9 minutes. The time varies inversely with the number of printers. How long will 6 printers take?
Show Answer

t = kn

k = nt

k = 8×9

k = 72

t = 72n

t = 726

t = 12

6 printers will take 12 minutes.

Answer: 6 printers will take 12 minutes.

5
The time to harvest a field varies inversely with the number of machines. 5 machines take 12 hours. How long will 4 machines take?
Show Answer

t = km

k = mt

k = 5×12

k = 60

t = 60m

t = 604

t = 15

4 machines will take 15 hours.

Answer: 4 machines will take 15 hours.

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