Solving Proportions Worksheet with 15 problems to solve for variables in given proportions.

Let's solve each proportion step by step. We'll use the cross-multiplication method for solving proportions. For a proportion:

$$
\frac{a}{b} = \frac{c}{d}
$$

We cross-multiply:
$$
a \cdot d = b \cdot c
$$

Then solve for the unknown variable.

---

1. $\frac{4}{9} = \frac{10}{x}$

Cross-multiply:
$$
4x = 9 \cdot 10 = 90
$$
$$
x = \frac{90}{4} = 22.5
$$

✔ Answer: $x = 22.5$

---

2. $\frac{5}{2} = \frac{6}{x}$

Cross-multiply:
$$
5x = 2 \cdot 6 = 12
$$
$$
x = \frac{12}{5} = 2.4
$$

✔ Answer: $x = 2.4$

---

3. $\frac{5}{2} = \frac{2}{x}$

Cross-multiply:
$$
5x = 2 \cdot 2 = 4
$$
$$
x = \frac{4}{5} = 0.8
$$

✔ Answer: $x = 0.8$

---

4. $\frac{21}{27} = \frac{x}{18}$

Cross-multiply:
$$
21 \cdot 18 = 27x
$$
$$
378 = 27x
$$
$$
x = \frac{378}{27} = 14
$$

✔ Answer: $x = 14$

---

5. $\frac{15}{21} = \frac{20}{y}$

Cross-multiply:
$$
15y = 21 \cdot 20 = 420
$$
$$
y = \frac{420}{15} = 28
$$

✔ Answer: $y = 28$

---

6. $\frac{26}{b} = \frac{39}{9}$

Cross-multiply:
$$
26 \cdot 9 = 39b
$$
$$
234 = 39b
$$
$$
b = \frac{234}{39} = 6
$$

✔ Answer: $b = 6$

---

7. $\frac{h}{108} = \frac{7}{18}$

Cross-multiply:
$$
18h = 108 \cdot 7 = 756
$$
$$
h = \frac{756}{18} = 42
$$

✔ Answer: $h = 42$

---

8. $\frac{45}{792} = \frac{70}{w}$

Cross-multiply:
$$
45w = 792 \cdot 70 = 55,440
$$
$$
w = \frac{55,440}{45} = 1,232
$$

✔ Answer: $w = 1,232$

---

9. $\frac{16}{120} = \frac{j}{15}$

Cross-multiply:
$$
16 \cdot 15 = 120j
$$
$$
240 = 120j
$$
$$
j = \frac{240}{120} = 2
$$

✔ Answer: $j = 2$

---

10. $\frac{350}{p} = \frac{1050}{60}$

Cross-multiply:
$$
350 \cdot 60 = 1050p
$$
$$
21,000 = 1050p
$$
$$
p = \frac{21,000}{1050} = 20
$$

✔ Answer: $p = 20$

---

11. $\frac{a}{1134} = \frac{27}{729}$

Cross-multiply:
$$
a \cdot 729 = 1134 \cdot 27
$$
First calculate:
$$
1134 \cdot 27 = (1100 + 34) \cdot 27 = 1100 \cdot 27 = 29,700; \quad 34 \cdot 27 = 918 \Rightarrow 29,700 + 918 = 30,618
$$
So:
$$
729a = 30,618
$$
$$
a = \frac{30,618}{729}
$$

Let’s simplify:
Divide numerator and denominator by 3:
- $30,618 ÷ 3 = 10,206$
- $729 ÷ 3 = 243$

Again:
- $10,206 ÷ 3 = 3,402$
- $243 ÷ 3 = 81$

Again:
- $3,402 ÷ 3 = 1,134$
- $81 ÷ 3 = 27$

Again:
- $1,134 ÷ 3 = 378$
- $27 ÷ 3 = 9$

Now:
- $378 ÷ 3 = 126$
- $9 ÷ 3 = 3$

Now:
- $126 ÷ 3 = 42$
- $3 ÷ 3 = 1$

So total division by $3^6 = 729$, so:
$$
a = \frac{30,618}{729} = 42
$$

✔ Answer: $a = 42$

---

12. $\frac{40}{65} = \frac{z}{104}$

Cross-multiply:
$$
40 \cdot 104 = 65z
$$
$$
4,160 = 65z
$$
$$
z = \frac{4,160}{65} = 64
$$

✔ Answer: $z = 64$

---

13. $\frac{15}{y} = \frac{40}{12}$

Simplify right side: $\frac{40}{12} = \frac{10}{3}$

So:
$$
\frac{15}{y} = \frac{10}{3}
$$

Cross-multiply:
$$
15 \cdot 3 = 10y \Rightarrow 45 = 10y
\Rightarrow y = \frac{45}{10} = 4.5
$$

✔ Answer: $y = 4.5$

---

14. $\frac{y}{32} = \frac{16}{10}$

Simplify right side: $\frac{16}{10} = \frac{8}{5}$

So:
$$
\frac{y}{32} = \frac{8}{5}
$$

Cross-multiply:
$$
5y = 32 \cdot 8 = 256
\Rightarrow y = \frac{256}{5} = 51.2
$$

✔ Answer: $y = 51.2$

---

15. $\frac{32.5}{25} = \frac{97.5}{q}$

Cross-multiply:
$$
32.5 \cdot q = 25 \cdot 97.5
$$

Calculate right side:
$$
25 \cdot 97.5 = 25 \cdot (100 - 2.5) = 2500 - 62.5 = 2437.5
$$

So:
$$
32.5q = 2437.5
\Rightarrow q = \frac{2437.5}{32.5}
$$

To make it easier, multiply numerator and denominator by 10 to eliminate decimals:
$$
q = \frac{24,375}{325}
$$

Now divide:
Divide both by 25:
- $24,375 ÷ 25 = 975$
- $325 ÷ 25 = 13$

So:
$$
q = \frac{975}{13} = 75
$$

✔ Answer: $q = 75$

---

✔ Final Answers:

1. $x = 22.5$
2. $x = 2.4$
3. $x = 0.8$
4. $x = 14$
5. $y = 28$
6. $b = 6$
7. $h = 42$
8. $w = 1,232$
9. $j = 2$
10. $p = 20$
11. $a = 42$
12. $z = 64$
13. $y = 4.5$
14. $y = 51.2$
15. $q = 75$

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