To determine if Dia's calculation of her average running speed was correct, we need to find her average speed for the entire distance she ran on that Saturday, rather than averaging the two speeds separately.
Step 1: Calculate the total distance ran
Distance to the beach: \( 9 \frac{3}{4} \) miles
Distance to her friend's house: \( 6 \frac{1}{4} \) miles
First, convert these mixed numbers into improper fractions:
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For \( 9 \frac{3}{4} \): \[ 9 \frac{3}{4} = \frac{39}{4} \text{ miles} \]
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For \( 6 \frac{1}{4} \): \[ 6 \frac{1}{4} = \frac{25}{4} \text{ miles} \]
Now add these distances together: \[ \text{Total distance} = \frac{39}{4} + \frac{25}{4} = \frac{39 + 25}{4} = \frac{64}{4} = 16 \text{ miles} \]
Step 2: Calculate the total time spent running
Time to the beach: \( 4 \frac{1}{3} \) hours
Time to her friend's house: \( 1 \frac{2}{3} \) hours
Convert these mixed numbers into improper fractions:
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For \( 4 \frac{1}{3} \): \[ 4 \frac{1}{3} = \frac{13}{3} \text{ hours} \]
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For \( 1 \frac{2}{3} \): \[ 1 \frac{2}{3} = \frac{5}{3} \text{ hours} \]
Now add these times together: \[ \text{Total time} = \frac{13}{3} + \frac{5}{3} = \frac{13 + 5}{3} = \frac{18}{3} = 6 \text{ hours} \]
Step 3: Calculate the average speed
Average speed is given by the formula: \[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} \]
Substituting in the total distance and total time: \[ \text{Average speed} = \frac{16 \text{ miles}}{6 \text{ hours}} = \frac{8}{3} \text{ miles per hour} \]
This can also be expressed as approximately \( 2.67 \) mph.
Dia's calculation of averaging the two speeds incorrectly led her to conclude that her average speed was 3 mph. However, the correct average speed she achieved was \( \frac{8}{3} \) mph or approximately 2.67 mph. Therefore, she was wrong.