Do the ratios 15 cm to 2 m and 24 seconds to 3 minutes form a proportion ?

Answer

To determine if the ratios 15 cm to 2 m and 24 seconds to 3 minutes form a proportion, we need to compare the two ratios. Here's a step-by-step solution:

### Step 1: Convert the units to the same measurement
First, we need to convert 2 meters into centimeters since the first ratio is in centimeters.

- **Conversion**: 1 meter = 100 centimeters
- Therefore, 2 meters = 2 × 100 cm = 200 cm

Now, we can express the first ratio as:
\[ \text{First Ratio} = \frac{15 \text{ cm}}{200 \text{ cm}} \]

### Step 2: Simplify the first ratio
Next, we simplify the first ratio:
\[ \frac{15 \text{ cm}}{200 \text{ cm}} = \frac{15}{200} \]
To simplify, we can divide both the numerator and the denominator by 5:
\[ \frac{15 \div 5}{200 \div 5} = \frac{3}{40} \]

### Step 3: Convert the second ratio to the same unit
Now, we convert 3 minutes into seconds for the second ratio.

- **Conversion**: 1 minute = 60 seconds
- Therefore, 3 minutes = 3 × 60 seconds = 180 seconds

Now, we can express the second ratio as:
\[ \text{Second Ratio} = \frac{24 \text{ seconds}}{180 \text{ seconds}} \]

### Step 4: Simplify the second ratio
Next, we simplify the second ratio:
\[ \frac{24 \text{ seconds}}{180 \text{ seconds}} = \frac{24}{180} \]
To simplify, we can divide both the numerator and the denominator by 12:
\[ \frac{24 \div 12}{180 \div 12} = \frac{2}{15} \]

### Step 5: Compare the two simplified ratios
Now we have:
- First Ratio: \(\frac{3}{40}\)
- Second Ratio: \(\frac{2}{15}\)

To check if these ratios form a proportion, we need to see if:
\[ \frac{3}{40} = \frac{2}{15} \]
This can be done by cross-multiplying:
\[ 3 \times 15 \quad \text{and} \quad 2 \times 40 \]
Calculating these gives:
- \(3 \times 15 = 45\)
- \(2 \times 40 = 80\)

Since \(45 \neq 80\), the ratios do not form a proportion.

### Conclusion
The ratios \(15 \text{ cm} : 2 \text{ m}\) and \(24 \text{ seconds} : 3 \text{ minutes}\) do not form a proportion.

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