Combined variation worksheet featuring six math problems on direct and inverse variation.

You're working on a Combined Variation worksheet. The key phrase in every problem is:

> "z varies directly as x and inversely as y"

This means the relationship can be written as:

> z = k · (x / y)

where k is the constant of variation.

---

We’ll solve each problem step-by-step using this formula.

---

✔ Problem 1:

> z varies directly as x and inversely as y.
> z = 7/3 when x = 8 and y = 4.
> What is z when x = 4 and y = 6?

Step 1: Find k

Use the first set of values:

z = k · (x / y)
→ 7/3 = k · (8 / 4)
→ 7/3 = k · 2
→ k = (7/3) ÷ 2 = 7/6

Step 2: Use k to find new z

Now plug in x = 4, y = 6, k = 7/6:

z = (7/6) · (4 / 6) = (7/6) · (2/3) = 14/18 = 7/9

✔ Answer: z = 7/9

---

✔ Problem 2:

> z = 15/4 when x = 9, y = 4.
> Find z when x = 2, y = 9.

Step 1: Find k

15/4 = k · (9 / 4)
→ k = (15/4) ÷ (9/4) = (15/4) × (4/9) = 15/9 = 5/3

Step 2: Find new z

z = (5/3) · (2 / 9) = 10/27

✔ Answer: z = 10/27

---

✔ Problem 3:

> z = 20 when x = 8, y = 2.
> Find z when x = 5, y = 8.

Step 1: Find k

20 = k · (8 / 2) → 20 = k · 4 → k = 5

Step 2: Find new z

z = 5 · (5 / 8) = 25/8

✔ Answer: z = 25/8

---

✔ Problem 4:

> z = -5/4 when x = 5, y = 2.
> Find x when y = 5, z = -1/5.

Step 1: Find k

-5/4 = k · (5 / 2)
→ k = (-5/4) ÷ (5/2) = (-5/4) × (2/5) = -10/20 = -1/2

Step 2: Solve for x

z = k · (x / y)
→ -1/5 = (-1/2) · (x / 5)

Multiply both sides by 5:

→ -1 = (-1/2) · x
→ x = (-1) ÷ (-1/2) = 2

✔ Answer: x = 2

---

✔ Problem 5:

> z = 1/12 when x = 3, y = 6.
> Find x when y = 4, z = 1/3.

Step 1: Find k

1/12 = k · (3 / 6) → 1/12 = k · (1/2) → k = (1/12) ÷ (1/2) = 1/6

Step 2: Solve for x

z = k · (x / y)
→ 1/3 = (1/6) · (x / 4)

Multiply both sides by 6:

→ 2 = x / 4
→ x = 8

✔ Answer: x = 8

---

✔ Problem 6:

> z = 10/9 when x = 2, y = 3.
> Find x when y = 6, z = 25/18.

Step 1: Find k

10/9 = k · (2 / 3)
→ k = (10/9) ÷ (2/3) = (10/9) × (3/2) = 30/18 = 5/3

Step 2: Solve for x

z = k · (x / y)
→ 25/18 = (5/3) · (x / 6)

Multiply both sides by 6:

→ (25/18) × 6 = (5/3) · x
→ 150/18 = (5/3)x
→ 25/3 = (5/3)x

Divide both sides by 5/3:

→ x = (25/3) ÷ (5/3) = 25/3 × 3/5 = 5

✔ Answer: x = 5

---

## 📝 Final Answers:

1. 7/9
2. 10/27
3. 25/8
4. 2
5. 8
6. 5

Let me know if you’d like a summary sheet or to visualize any of these!