Math worksheet focusing on equivalent ratios with exercises for students to practice ratio equivalence and solving for unknowns.

Let's solve each part of this "Equivalent Ratios" worksheet step by step.

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Part 1: Write two equivalent ratios

We are given a ratio in the form of a fraction, and we need to write two equivalent ratios (i.e., fractions that represent the same value).

To find equivalent ratios, multiply or divide both numerator and denominator by the same number.

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#### 1) $ \frac{5}{11} $

Multiply numerator and denominator by 2:
- $ \frac{5 \times 2}{11 \times 2} = \frac{10}{22} $
- Multiply by 3: $ \frac{15}{33} $

✔ Equivalent ratios: $ \frac{10}{22}, \frac{15}{33} $

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#### 2) $ \frac{7}{8} $

Multiply by 2: $ \frac{14}{16} $
Multiply by 3: $ \frac{21}{24} $

✔ $ \frac{14}{16}, \frac{21}{24} $

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#### 3) $ \frac{7}{4} $

Multiply by 2: $ \frac{14}{8} $
Multiply by 3: $ \frac{21}{12} $

✔ $ \frac{14}{8}, \frac{21}{12} $

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#### 4) $ \frac{11}{5} $

Multiply by 2: $ \frac{22}{10} $
Multiply by 3: $ \frac{33}{15} $

✔ $ \frac{22}{10}, \frac{33}{15} $

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#### 5) $ \frac{4}{5} $

Multiply by 2: $ \frac{8}{10} $
Multiply by 3: $ \frac{12}{15} $

✔ $ \frac{8}{10}, \frac{12}{15} $

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#### 6) $ \frac{8}{5} $

Multiply by 2: $ \frac{16}{10} $
Multiply by 3: $ \frac{24}{15} $

✔ $ \frac{16}{10}, \frac{24}{15} $

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✔ Part 1 Answers:

| # | Equivalent Ratios |
|---|-------------------|
| 1) | $ \frac{10}{22}, \frac{15}{33} $ |
| 2) | $ \frac{14}{16}, \frac{21}{24} $ |
| 3) | $ \frac{14}{8}, \frac{21}{12} $ |
| 4) | $ \frac{22}{10}, \frac{33}{15} $ |
| 5) | $ \frac{8}{10}, \frac{12}{15} $ |
| 6) | $ \frac{16}{10}, \frac{24}{15} $ |

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Part 2: Determine whether the ratios are equivalent

Two ratios are equivalent if their cross products are equal.

That is:
$ \frac{a}{b} = \frac{c}{d} $ iff $ a \cdot d = b \cdot c $

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#### 7) $ \frac{8}{7} $ and $ \frac{9}{7} $

Cross multiply:
$ 8 \cdot 7 = 56 $, $ 7 \cdot 9 = 63 $ → Not equal

✘ Not equivalent

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#### 8) $ \frac{4}{7} $ and $ \frac{3}{4} $

$ 4 \cdot 4 = 16 $, $ 7 \cdot 3 = 21 $ → Not equal

✘ Not equivalent

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#### 9) $ \frac{7}{6} $ and $ \frac{11}{3} $

$ 7 \cdot 3 = 21 $, $ 6 \cdot 11 = 66 $ → Not equal

✘ Not equivalent

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#### 10) $ \frac{11}{10} $ and $ \frac{5}{6} $

$ 11 \cdot 6 = 66 $, $ 10 \cdot 5 = 50 $ → Not equal

✘ Not equivalent

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#### 11) $ \frac{5}{7} $ and $ \frac{9}{2} $

$ 5 \cdot 2 = 10 $, $ 7 \cdot 9 = 63 $ → Not equal

✘ Not equivalent

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#### 12) $ \frac{5}{12} $ and $ \frac{4}{3} $

$ 5 \cdot 3 = 15 $, $ 12 \cdot 4 = 48 $ → Not equal

✘ Not equivalent

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✔ Part 2 Answers:

| # | Equivalent? |
|---|-------------|
| 7) | No |
| 8) | No |
| 9) | No |
| 10) | No |
| 11) | No |
| 12) | No |

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Part 3: Use equivalent ratios to find the unknown value

We solve using cross multiplication.

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#### 13) $ \frac{2}{9} = \frac{r}{36} $

Cross multiply:
$ 2 \cdot 36 = 9 \cdot r $
$ 72 = 9r $
$ r = \frac{72}{9} = 8 $

✔ $ r = 8 $

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#### 14) $ \frac{k}{24} = \frac{11}{8} $

Cross multiply:
$ k \cdot 8 = 24 \cdot 11 $
$ 8k = 264 $
$ k = \frac{264}{8} = 33 $

✔ $ k = 33 $

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#### 15) $ \frac{2}{5} = \frac{8}{z} $

Cross multiply:
$ 2 \cdot z = 5 \cdot 8 $
$ 2z = 40 $
$ z = 20 $

✔ $ z = 20 $

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#### 16) $ \frac{3}{2} = \frac{15}{z} $

Cross multiply:
$ 3 \cdot z = 2 \cdot 15 $
$ 3z = 30 $
$ z = 10 $

✔ $ z = 10 $

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#### 17) $ \frac{9}{8} = \frac{r}{16} $

Cross multiply:
$ 9 \cdot 16 = 8 \cdot r $
$ 144 = 8r $
$ r = \frac{144}{8} = 18 $

✔ $ r = 18 $

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#### 18) $ \frac{7f}{7} = \frac{11}{7} $

Wait — this is written as $ \frac{7f}{7} = \frac{11}{7} $

Simplify left side: $ \frac{7f}{7} = f $

So:
$ f = \frac{11}{7} $

But let’s check if it's really $ \frac{7f}{7} $. That simplifies to $ f $, so:

$ f = \frac{11}{7} $

✔ $ f = \frac{11}{7} $ or $ 1\frac{4}{7} $

Alternatively, if it was meant to be $ \frac{7}{f} = \frac{11}{7} $, that would be different. But based on what’s written: $ \frac{7f}{7} = \frac{11}{7} $, then yes:

$ f = \frac{11}{7} $

✔ $ f = \frac{11}{7} $

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✔ Part 3 Answers:

| # | Unknown Value |
|---|---------------|
| 13) | $ r = 8 $ |
| 14) | $ k = 33 $ |
| 15) | $ z = 20 $ |
| 16) | $ z = 10 $ |
| 17) | $ r = 18 $ |
| 18) | $ f = \frac{11}{7} $ |

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✔ Final Answer Summary:

#### Part 1: Equivalent Ratios
1) $ \frac{10}{22}, \frac{15}{33} $
2) $ \frac{14}{16}, \frac{21}{24} $
3) $ \frac{14}{8}, \frac{21}{12} $
4) $ \frac{22}{10}, \frac{33}{15} $
5) $ \frac{8}{10}, \frac{12}{15} $
6) $ \frac{16}{10}, \frac{24}{15} $

#### Part 2: Are they equivalent?
7) No
8) No
9) No
10) No
11) No
12) No

#### Part 3: Solve for unknowns
13) $ r = 8 $
14) $ k = 33 $
15) $ z = 20 $
16) $ z = 10 $
17) $ r = 18 $
18) $ f = \frac{11}{7} $

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