Equivalent Ratios worksheet for math practice, helping students understand ratios and fractions through fill-in-the-blank exercises.

Problem Explanation:

The task involves finding equivalent ratios for given ratios and expressing them as fractions. Equivalent ratios are ratios that represent the same relationship between two quantities, even though the numbers may be different. To solve this, we need to scale the given ratios up or down by multiplying or dividing both terms of the ratio by the same number.

Solution:

#### Step 1: Understand the structure
Each row has a given ratio (e.g., \(1 : 2\)), and we need to find two equivalent ratios:
1. One where the second term is a specific number (e.g., \(10\)).
2. Another where the first term is a specific number (e.g., \(9\)).

Additionally, we need to express each ratio as a fraction.

#### Step 2: Solve each part

##### Problem 1: \(1 : 2 = \_\_ : 10 = 9 : \_\_ \quad\) IN FRACTION
- First blank (\_\_ : 10):
- The original ratio is \(1 : 2\). To make the second term \(10\), we multiply both terms by \(5\) (since \(2 \times 5 = 10\)).
- \(1 \times 5 : 2 \times 5 = 5 : 10\).
- So, the first blank is \(5\).

- Second blank (9 : \_\_):
- The original ratio is \(1 : 2\). To make the first term \(9\), we multiply both terms by \(9\) (since \(1 \times 9 = 9\)).
- \(1 \times 9 : 2 \times 9 = 9 : 18\).
- So, the second blank is \(18\).

- Fraction:
- The ratio \(1 : 2\) can be written as the fraction \(\frac{1}{2}\).

##### Final Answer for Problem 1:
\[ 1 : 2 = 5 : 10 = 9 : 18 \quad \text{IN FRACTION: } \frac{1}{2} \]

##### Problem 2: \(1 : 3 = \_\_ : 15 = 2 : \_\_ \quad\) IN FRACTION
- First blank (\_\_ : 15):
- The original ratio is \(1 : 3\). To make the second term \(15\), we multiply both terms by \(5\) (since \(3 \times 5 = 15\)).
- \(1 \times 5 : 3 \times 5 = 5 : 15\).
- So, the first blank is \(5\).

- Second blank (2 : \_\_):
- The original ratio is \(1 : 3\). To make the first term \(2\), we multiply both terms by \(2\) (since \(1 \times 2 = 2\)).
- \(1 \times 2 : 3 \times 2 = 2 : 6\).
- So, the second blank is \(6\).

- Fraction:
- The ratio \(1 : 3\) can be written as the fraction \(\frac{1}{3}\).

##### Final Answer for Problem 2:
\[ 1 : 3 = 5 : 15 = 2 : 6 \quad \text{IN FRACTION: } \frac{1}{3} \]

##### Problem 3: \(1 : 4 = \_\_ : 24 = 5 : \_\_ \quad\) IN FRACTION
- First blank (\_\_ : 24):
- The original ratio is \(1 : 4\). To make the second term \(24\), we multiply both terms by \(6\) (since \(4 \times 6 = 24\)).
- \(1 \times 6 : 4 \times 6 = 6 : 24\).
- So, the first blank is \(6\).

- Second blank (5 : \_\_):
- The original ratio is \(1 : 4\). To make the first term \(5\), we multiply both terms by \(5\) (since \(1 \times 5 = 5\)).
- \(1 \times 5 : 4 \times 5 = 5 : 20\).
- So, the second blank is \(20\).

- Fraction:
- The ratio \(1 : 4\) can be written as the fraction \(\frac{1}{4}\).

##### Final Answer for Problem 3:
\[ 1 : 4 = 6 : 24 = 5 : 20 \quad \text{IN FRACTION: } \frac{1}{4} \]

##### Problem 4: \(1 : 5 = \_\_ : 35 = 3 : \_\_ \quad\) IN FRACTION
- First blank (\_\_ : 35):
- The original ratio is \(1 : 5\). To make the second term \(35\), we multiply both terms by \(7\) (since \(5 \times 7 = 35\)).
- \(1 \times 7 : 5 \times 7 = 7 : 35\).
- So, the first blank is \(7\).

- Second blank (3 : \_\_):
- The original ratio is \(1 : 5\). To make the first term \(3\), we multiply both terms by \(3\) (since \(1 \times 3 = 3\)).
- \(1 \times 3 : 5 \times 3 = 3 : 15\).
- So, the second blank is \(15\).

- Fraction:
- The ratio \(1 : 5\) can be written as the fraction \(\frac{1}{5}\).

##### Final Answer for Problem 4:
\[ 1 : 5 = 7 : 35 = 3 : 15 \quad \text{IN FRACTION: } \frac{1}{5} \]

##### Problem 5: \(1 : 6 = \_\_ : 12 = 4 : \_\_ \quad\) IN FRACTION
- First blank (\_\_ : 12):
- The original ratio is \(1 : 6\). To make the second term \(12\), we multiply both terms by \(2\) (since \(6 \times 2 = 12\)).
- \(1 \times 2 : 6 \times 2 = 2 : 12\).
- So, the first blank is \(2\).

- Second blank (4 : \_\_):
- The original ratio is \(1 : 6\). To make the first term \(4\), we multiply both terms by \(4\) (since \(1 \times 4 = 4\)).
- \(1 \times 4 : 6 \times 4 = 4 : 24\).
- So, the second blank is \(24\).

- Fraction:
- The ratio \(1 : 6\) can be written as the fraction \(\frac{1}{6}\).

##### Final Answer for Problem 5:
\[ 1 : 6 = 2 : 12 = 4 : 24 \quad \text{IN FRACTION: } \frac{1}{6} \]

##### Problem 6: \(1 : 7 = \_\_ : 21 = 6 : \_\_ \quad\) IN FRACTION
- First blank (\_\_ : 21):
- The original ratio is \(1 : 7\). To make the second term \(21\), we multiply both terms by \(3\) (since \(7 \times 3 = 21\)).
- \(1 \times 3 : 7 \times 3 = 3 : 21\).
- So, the first blank is \(3\).

- Second blank (6 : \_\_):
- The original ratio is \(1 : 7\). To make the first term \(6\), we multiply both terms by \(6\) (since \(1 \times 6 = 6\)).
- \(1 \times 6 : 7 \times 6 = 6 : 42\).
- So, the second blank is \(42\).

- Fraction:
- The ratio \(1 : 7\) can be written as the fraction \(\frac{1}{7}\).

##### Final Answer for Problem 6:
\[ 1 : 7 = 3 : 21 = 6 : 42 \quad \text{IN FRACTION: } \frac{1}{7} \]

##### Problem 7: \(1 : 8 = \_\_ : 16 = 7 : \_\_ \quad\) IN FRACTION
- First blank (\_\_ : 16):
- The original ratio is \(1 : 8\). To make the second term \(16\), we multiply both terms by \(2\) (since \(8 \times 2 = 16\)).
- \(1 \times 2 : 8 \times 2 = 2 : 16\).
- So, the first blank is \(2\).

- Second blank (7 : \_\_):
- The original ratio is \(1 : 8\). To make the first term \(7\), we multiply both terms by \(7\) (since \(1 \times 7 = 7\)).
- \(1 \times 7 : 8 \times 7 = 7 : 56\).
- So, the second blank is \(56\).

- Fraction:
- The ratio \(1 : 8\) can be written as the fraction \(\frac{1}{8}\).

##### Final Answer for Problem 7:
\[ 1 : 8 = 2 : 16 = 7 : 56 \quad \text{IN FRACTION: } \frac{1}{8} \]

##### Problem 8: \(1 : 9 = \_\_ : 72 = 6 : \_\_ \quad\) IN FRACTION
- First blank (\_\_ : 72):
- The original ratio is \(1 : 9\). To make the second term \(72\), we multiply both terms by \(8\) (since \(9 \times 8 = 72\)).
- \(1 \times 8 : 9 \times 8 = 8 : 72\).
- So, the first blank is \(8\).

- Second blank (6 : \_\_):
- The original ratio is \(1 : 9\). To make the first term \(6\), we multiply both terms by \(6\) (since \(1 \times 6 = 6\)).
- \(1 \times 6 : 9 \times 6 = 6 : 54\).
- So, the second blank is \(54\).

- Fraction:
- The ratio \(1 : 9\) can be written as the fraction \(\frac{1}{9}\).

##### Final Answer for Problem 8:
\[ 1 : 9 = 8 : 72 = 6 : 54 \quad \text{IN FRACTION: } \frac{1}{9} \]

Final Answers:

\[
\boxed{
\begin{array}{l}
1. \ 1 : 2 = 5 : 10 = 9 : 18 \quad \text{IN FRACTION: } \frac{1}{2} \\
2. \ 1 : 3 = 5 : 15 = 2 : 6 \quad \text{IN FRACTION: } \frac{1}{3} \\
3. \ 1 : 4 = 6 : 24 = 5 : 20 \quad \text{IN FRACTION: } \frac{1}{4} \\
4. \ 1 : 5 = 7 : 35 = 3 : 15 \quad \text{IN FRACTION: } \frac{1}{5} \\
5. \ 1 : 6 = 2 : 12 = 4 : 24 \quad \text{IN FRACTION: } \frac{1}{6} \\
6. \ 1 : 7 = 3 : 21 = 6 : 42 \quad \text{IN FRACTION: } \frac{1}{7} \\
7. \ 1 : 8 = 2 : 16 = 7 : 56 \quad \text{IN FRACTION: } \frac{1}{8} \\
8. \ 1 : 9 = 8 : 72 = 6 : 54 \quad \text{IN FRACTION: } \frac{1}{9} \\
\end{array}
}
\]