Equivalent Ratios with Variables (A) - Free Printable

Let's solve each of these equivalent ratio problems step by step. The goal is to find the unknown variable in each proportion.

We use the cross-multiplication method for solving proportions:

If $ a : b = c : d $, then $ \frac{a}{b} = \frac{c}{d} $, and so $ a \cdot d = b \cdot c $.

---

1. $ 80 : s = 10 : 7 $

$$
\frac{80}{s} = \frac{10}{7}
\Rightarrow 80 \cdot 7 = 10 \cdot s
\Rightarrow 560 = 10s
\Rightarrow s = \frac{560}{10} = 56
$$

✔ Answer: $ s = 56 $

---

2. $ 10 : 30 = 1 : p $

$$
\frac{10}{30} = \frac{1}{p}
\Rightarrow \frac{1}{3} = \frac{1}{p}
\Rightarrow p = 3
$$

✔ Answer: $ p = 3 $

---

3. $ v : 33 = 7 : 11 $

$$
\frac{v}{33} = \frac{7}{11}
\Rightarrow v \cdot 11 = 33 \cdot 7
\Rightarrow 11v = 231
\Rightarrow v = \frac{231}{11} = 21
$$

✔ Answer: $ v = 21 $

---

4. $ 3 : 2 = x : 4 $

$$
\frac{3}{2} = \frac{x}{4}
\Rightarrow 3 \cdot 4 = 2 \cdot x
\Rightarrow 12 = 2x
\Rightarrow x = 6
$$

✔ Answer: $ x = 6 $

---

5. $ q : 9 = 40 : 90 $

$$
\frac{q}{9} = \frac{40}{90} = \frac{4}{9}
\Rightarrow q = 9 \cdot \frac{4}{9} = 4
$$

✔ Answer: $ q = 4 $

---

6. $ 8 : h = 80 : 70 $

$$
\frac{8}{h} = \frac{80}{70} = \frac{8}{7}
\Rightarrow \frac{8}{h} = \frac{8}{7}
\Rightarrow h = 7
$$

✔ Answer: $ h = 7 $

---

7. $ 20 : 25 = t : 5 $

$$
\frac{20}{25} = \frac{t}{5}
\Rightarrow \frac{4}{5} = \frac{t}{5}
\Rightarrow t = 4
$$

✔ Answer: $ t = 4 $

---

8. $ m : 24 = 7 : 3 $

$$
\frac{m}{24} = \frac{7}{3}
\Rightarrow m \cdot 3 = 24 \cdot 7
\Rightarrow 3m = 168
\Rightarrow m = 56
$$

✔ Answer: $ m = 56 $

---

9. $ 1 : k = 6 : 72 $

$$
\frac{1}{k} = \frac{6}{72} = \frac{1}{12}
\Rightarrow k = 12
$$

✔ Answer: $ k = 12 $

---

10. $ 5 : 8 = 30 : d $

$$
\frac{5}{8} = \frac{30}{d}
\Rightarrow 5d = 8 \cdot 30 = 240
\Rightarrow d = \frac{240}{5} = 48
$$

✔ Answer: $ d = 48 $

---

11. $ 7 : 2 = 35 : y $

$$
\frac{7}{2} = \frac{35}{y}
\Rightarrow 7y = 2 \cdot 35 = 70
\Rightarrow y = 10
$$

✔ Answer: $ y = 10 $

---

12. $ 108 : 99 = n : 11 $

$$
\frac{108}{99} = \frac{n}{11}
\Rightarrow \frac{12}{11} = \frac{n}{11}
\Rightarrow n = 12
$$

✔ Answer: $ n = 12 $

---

13. $ j : 10 = 30 : 100 $

$$
\frac{j}{10} = \frac{30}{100} = \frac{3}{10}
\Rightarrow j = 10 \cdot \frac{3}{10} = 3
$$

✔ Answer: $ j = 3 $

---

14. $ 8 : 72 = 1 : y $

$$
\frac{8}{72} = \frac{1}{y}
\Rightarrow \frac{1}{9} = \frac{1}{y}
\Rightarrow y = 9
$$

✔ Answer: $ y = 9 $

---

15. $ 63 : 36 = c : 4 $

$$
\frac{63}{36} = \frac{c}{4}
\Rightarrow \frac{7}{4} = \frac{c}{4}
\Rightarrow c = 7
$$

✔ Answer: $ c = 7 $

---

16. $ 18 : 4 = 9 : r $

$$
\frac{18}{4} = \frac{9}{r}
\Rightarrow \frac{9}{2} = \frac{9}{r}
\Rightarrow r = 2
$$

✔ Answer: $ r = 2 $

---

17. $ w : 8 = 12 : 32 $

$$
\frac{w}{8} = \frac{12}{32} = \frac{3}{8}
\Rightarrow w = 8 \cdot \frac{3}{8} = 3
$$

✔ Answer: $ w = 3 $

---

18. $ 2 : 1 = 6 : b $

$$
\frac{2}{1} = \frac{6}{b}
\Rightarrow 2b = 6
\Rightarrow b = 3
$$

✔ Answer: $ b = 3 $

---

19. $ 10 : 50 = a : 5 $

$$
\frac{10}{50} = \frac{a}{5}
\Rightarrow \frac{1}{5} = \frac{a}{5}
\Rightarrow a = 1
$$

✔ Answer: $ a = 1 $

---

20. $ 9 : 7 = f : 35 $

$$
\frac{9}{7} = \frac{f}{35}
\Rightarrow 9 \cdot 35 = 7f
\Rightarrow 315 = 7f
\Rightarrow f = \frac{315}{7} = 45
$$

✔ Answer: $ f = 45 $

---

✔ Final Answers Summary:

| Problem | Unknown | Value |
|--------|--------|-------|
| 1 | $ s $ | 56 |
| 2 | $ p $ | 3 |
| 3 | $ v $ | 21 |
| 4 | $ x $ | 6 |
| 5 | $ q $ | 4 |
| 6 | $ h $ | 7 |
| 7 | $ t $ | 4 |
| 8 | $ m $ | 56 |
| 9 | $ k $ | 12 |
| 10 | $ d $ | 48 |
| 11 | $ y $ | 10 |
| 12 | $ n $ | 12 |
| 13 | $ j $ | 3 |
| 14 | $ y $ | 9 |
| 15 | $ c $ | 7 |
| 16 | $ r $ | 2 |
| 17 | $ w $ | 3 |
| 18 | $ b $ | 3 |
| 19 | $ a $ | 1 |
| 20 | $ f $ | 45 |

All solved using cross multiplication and simplification.

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