Solving Proportions | Interactive Worksheet - Free Printable

Let's solve each proportion step by step using cross products and inverse operations, as demonstrated in the example.

---

🔷 Example Given:

Solve:
$$
\frac{2}{3} = \frac{x}{15}
$$

Step 1: Use cross products:
Multiply across the corners:
$$
2 \times 15 = 3 \times x \Rightarrow 30 = 3x
$$

Step 2: Solve for $x$:
$$
x = \frac{30}{3} = 10
$$

✔ So, $x = 10$

---

Now let’s solve each problem on the worksheet:

---

✔ Problem 1:

$$
\frac{m}{30} = \frac{24}{40}
$$

Cross multiply:
$$
m \times 40 = 30 \times 24
\Rightarrow 40m = 720
$$

Divide both sides by 40:
$$
m = \frac{720}{40} = 18
$$

✔ $m = 18$

---

✔ Problem 2:

$$
\frac{14}{35} = \frac{p}{5}
$$

Cross multiply:
$$
14 \times 5 = 35 \times p \Rightarrow 70 = 35p
$$

Divide both sides by 35:
$$
p = \frac{70}{35} = 2
$$

✔ $p = 2$

---

✔ Problem 3:

$$
\frac{9}{12} = \frac{f}{4}
$$

Cross multiply:
$$
9 \times 4 = 12 \times f \Rightarrow 36 = 12f
$$

Divide both sides by 12:
$$
f = \frac{36}{12} = 3
$$

✔ $f = 3$

---

✔ Problem 4:

$$
\frac{c}{16} = \frac{8}{20}
$$

Cross multiply:
$$
c \times 20 = 16 \times 8 \Rightarrow 20c = 128
$$

Divide by 20:
$$
c = \frac{128}{20} = 6.4
$$

Or as a fraction:
$$
\frac{128}{20} = \frac{32}{5} = 6\frac{2}{5}
$$

✔ $c = 6.4$ or $\frac{32}{5}$

---

✔ Problem 5:

$$
\frac{36}{18} = \frac{g}{9}
$$

Cross multiply:
$$
36 \times 9 = 18 \times g \Rightarrow 324 = 18g
$$

Divide by 18:
$$
g = \frac{324}{18} = 18
$$

✔ $g = 18$

---

✔ Problem 6:

$$
\frac{12}{n} = \frac{16}{20}
$$

Cross multiply:
$$
12 \times 20 = 16 \times n \Rightarrow 240 = 16n
$$

Divide by 16:
$$
n = \frac{240}{16} = 15
$$

✔ $n = 15$

---

✔ Problem 7:

$$
\frac{43}{48} = \frac{24}{e}
$$

Cross multiply:
$$
43 \times e = 48 \times 24 \Rightarrow 43e = 1152
$$

Divide by 43:
$$
e = \frac{1152}{43} \approx 26.8
$$

Let’s calculate exactly:
$$
1152 ÷ 43 = 26.837... \quad \text{(no simplification)}
$$

So:
$$
e = \frac{1152}{43} \quad \text{(exact value)}
$$

✔ $e = \frac{1152}{43}$ or approximately $26.84$

---

✔ Problem 8:

$$
\frac{48}{36} = \frac{k}{15}
$$

Cross multiply:
$$
48 \times 15 = 36 \times k \Rightarrow 720 = 36k
$$

Divide by 36:
$$
k = \frac{720}{36} = 20
$$

✔ $k = 20$

---

✔ Problem 9:

$$
\frac{28}{169} = \frac{52}{z}
$$

Cross multiply:
$$
28 \times z = 169 \times 52
\Rightarrow 28z = 8808
$$

Now divide:
$$
z = \frac{8808}{28}
$$

Let’s simplify:
- $8808 ÷ 4 = 2202$, $28 ÷ 4 = 7$
- So: $z = \frac{2202}{7} = 314.571...$

Wait — better to do direct division:

$$
8808 ÷ 28 = ?
$$

Divide:
- $28 \times 300 = 8400$
- $8808 - 8400 = 408$
- $28 \times 14 = 392$
- $408 - 392 = 16$

So:
$$
z = 314 + \frac{16}{28} = 314 + \frac{4}{7} = 314\frac{4}{7}
$$

✔ $z = 314\frac{4}{7}$ or $\frac{2202}{7}$

---

🟩 Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | $m = 18$ |
| 2 | $p = 2$ |
| 3 | $f = 3$ |
| 4 | $c = 6.4$ or $\frac{32}{5}$ |
| 5 | $g = 18$ |
| 6 | $n = 15$ |
| 7 | $e = \frac{1152}{43} \approx 26.84$ |
| 8 | $k = 20$ |
| 9 | $z = \frac{2202}{7} = 314\frac{4}{7}$ |

---

💡 Summary of Steps:

1. Set up the proportion.
2. Use cross multiplication: Multiply diagonally.
3. Write an equation (e.g., $a \cdot d = b \cdot c$).
4. Solve for the variable using inverse operations (division).

Let me know if you’d like these answers formatted neatly for printing!