Inverse Proportion Lesson Plans & Worksheets Reviewed by Teachers - Free Printable

Let’s solve each problem one by one, carefully and step by step.

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Problem 1:
> Suppose P varies directly as Q, and Q = 8 when P = 20. Find P when Q = 5.

When two things vary directly, it means:
> P = k × Q
(where k is a constant)

We’re told that when Q = 8, P = 20. So plug those in to find k:

20 = k × 8
→ k = 20 ÷ 8 = 2.5

Now use this k to find P when Q = 5:

P = 2.5 × 5 = 12.5

✔ Final Answer for #1: 12.5

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Problem 2:
> If z varies inversely as w, and z = 3 when w = 8, find z when w = 12.

Inverse variation means:
> z = k / w

Plug in z = 3, w = 8:

3 = k / 8 → k = 3 × 8 = 24

Now find z when w = 12:

z = 24 / 12 = 2

✔ Final Answer for #2: 2

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Problem 3:
> If x varies jointly as y and z, and x = 6 when y = 9 and z = ½, what is x when y = 6 and z = 2?

Joint variation means:
> x = k × y × z

Plug in known values:

6 = k × 9 × (1/2)
→ 6 = k × 4.5
→ k = 6 ÷ 4.5 = 60 ÷ 45 = 4/3 ≈ 1.333...

Now find x when y = 6, z = 2:

x = (4/3) × 6 × 2
= (4/3) × 12
= 48/3 = 16

✔ Final Answer for #3: 16

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Problem 4:
> If G = 10 when R = 4, and G varies directly with R, then find G when R = 14.

Direct variation: G = k × R

10 = k × 4 → k = 10 ÷ 4 = 2.5

G = 2.5 × 14 = 35

✔ Final Answer for #4: 35

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Problem 5:
> If A varies directly as B², and A = 18 when B = 3, find A when B = 7.

A = k × B²

18 = k × (3)² = k × 9 → k = 18 ÷ 9 = 2

Now A = 2 × (7)² = 2 × 49 = 98

✔ Final Answer for #5: 98

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Problem 6:
> B varies inversely with C, and B = 15 when C = 18. What is B when C = 24?

B = k / C

15 = k / 18 → k = 15 × 18 = 270

B = 270 / 24 = let’s simplify:

Divide numerator and denominator by 6:
270 ÷ 6 = 45, 24 ÷ 6 = 4 → 45/4 = 11.25

✔ Final Answer for #6: 11.25

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Problem 7:
> H varies directly as J and inversely as M. If H = 12 when J = 4 and M = 4, find H when J = 6 and M = 8.

This is combined variation:
H = k × (J / M)

Plug in:
12 = k × (4 / 4) = k × 1 → k = 12

Now find H when J = 6, M = 8:

H = 12 × (6 / 8) = 12 × (3/4) = 36/4 = 9

✔ Final Answer for #7: 9

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Problem 8:
> If E varies directly as D, and E = 15 when D = 5, find E when D = 10.

E = k × D

15 = k × 5 → k = 3

E = 3 × 10 = 30

✔ Final Answer for #8: 30

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Problem 9:
> The number of hours h it takes to assemble computers varies inversely as the number of workers w. If 4 people can complete the job in 9 hours, how long will it take 6 people?

h = k / w

9 = k / 4 → k = 36

Now h = 36 / 6 = 6 hours

✔ Final Answer for #9: 6

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Problem 10:
> The weight of a body varies inversely as the square of its distance from the center of the earth. If the radius of the earth is about 4000 miles, how much would a 200-pound man weigh 1000 miles above the surface?

Weight W ∝ 1 / d² → W = k / d²

At surface: d = 4000 miles, W = 200 lbs

200 = k / (4000)² → k = 200 × 16,000,000 = 3,200,000,000

Now, 1000 miles above surface → d = 4000 + 1000 = 5000 miles

W = 3,200,000,000 / (5000)²
= 3,200,000,000 / 25,000,000
= 3200 / 25 = 128 pounds

✔ Final Answer for #10: 128

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Problems 11–14: Table Problems

We are given a table:

| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|----|----|----|---|---|---|---|
| y | 15% | 10% | 5% | 0% | 5% | 10% | 15% |

Wait — these look like percentages, but probably they mean actual numbers: 15, 10, 5, 0, 5, 10, 15? Because percentages don’t make sense here unless specified otherwise. Let’s assume it's just values: y = 15, 10, 5, 0, 5, 10, 15 for x = -3 to 3.

Looking at the pattern:

x: -3 → y=15
x: -2 → y=10
x: -1 → y=5
x: 0 → y=0
x: 1 → y=5
x: 2 → y=10
x: 3 → y=15

So y increases by 5 every time x increases by 1? But from x=-3 to x=-2, y goes from 15 to 10 → decrease of 5. Then from x=-2 to -1: 10 to 5 → decrease of 5. Then x=-1 to 0: 5 to 0 → decrease of 5. Then x=0 to 1: 0 to 5 → increase of 5. Hmm, not linear.

Actually, notice:
y = 5 × |x| ? Let’s check:

|x|: 3 → 5×3=15 ✔
|x|: 2 → 10 ✔
|x|: 1 → 5 ✔
|x|: 0 → 0 ✔
Yes! So y = 5 × |x|

But absolute value functions are not direct or inverse variations — they are piecewise.

However, looking at the questions:

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Problem 11: Does y vary directly with x?

Direct variation means y = kx (straight line through origin).

Check: When x = 1, y = 5 → k = 5
When x = 2, y = 10 → k = 5
When x = 3, y = 15 → k = 5
But when x = -1, y = 5 → if y = kx, then 5 = k(-1) → k = -5 → contradiction.

Also, when x = -2, y = 10 → k = -5 again? But earlier k was 5.

So no single k works for all x. Also, graph is V-shaped, not straight line through origin.

✘ No, y does NOT vary directly with x.

✔ Final Answer for #11: No

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Problem 12: Does y vary inversely with x?

Inverse variation: y = k/x

Try x=1, y=5 → k=5
x=2, y=10 → k=20 → not same
x=3, y=15 → k=45 → nope

Also, when x=0, y=0 — but inverse variation undefined at x=0.

✘ No.

✔ Final Answer for #12: No

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Problem 13: Does y vary directly with x²?

Check: y = k × x²

x=1 → x²=1 → y=5 → k=5
x=2 → x²=4 → y=10 → k=10/4=2.5 → not same
x=3 → x²=9 → y=15 → k=15/9≈1.666 → no

Not constant k.

Wait — what if we try y = 5 × |x|? That’s not x².

Alternatively, maybe y = 5 × |x|, which is not proportional to x².

For example, x=1 → x²=1 → y=5
x=2 → x²=4 → y=10 → ratio y/x² = 10/4 = 2.5 ≠ 5

So no.

But wait — let’s compute y / x² for each:

x=1: 5 / 1 = 5
x=2: 10 / 4 = 2.5
x=3: 15 / 9 ≈ 1.666
x=-1: 5 / 1 = 5
x=-2: 10 / 4 = 2.5
x=-3: 15 / 9 ≈ 1.666

Not constant → so not direct variation with x².

✔ Final Answer for #13: No

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Problem 14: Does y vary inversely with x²?

y = k / x²

x=1 → y=5 → k=5
x=2 → y=10 → k=10×4=40 → not same
x=3 → y=15 → k=15×9=135 → no

Also, at x=0, y=0 — inverse variation undefined.

✘ No.

✔ Final Answer for #14: No

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## ✔ FINAL ANSWERS SUMMARY:

1. 12.5
2. 2
3. 16
4. 35
5. 98
6. 11.25
7. 9
8. 30
9. 6
10. 128
11. No
12. No
13. No
14. No

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Final Answer:
1. 12.5
2. 2
3. 16
4. 35
5. 98
6. 11.25
7. 9
8. 30
9. 6
10. 128
11. No
12. No
13. No
14. No