Answer :
To determine how far apart Natasha's two dogs are, we need to add their distances from Natasha, as they ran in opposite directions.
Let's break this down step-by-step.
1. Determine the distance of the beagle from Natasha:
- The beagle is [tex]\(\frac{25}{4}\)[/tex] meters to her right.
2. Determine the distance of the labrador from Natasha:
- The labrador is [tex]\(\frac{51}{20}\)[/tex] meters to her left.
3. Calculate the distance between the two dogs:
- Since they ran in opposite directions, the total distance between them is the sum of their distances from Natasha.
- We can write this as:
[tex]\[ \text{Distance apart} = \left| \frac{25}{4} \right| + \left| -\frac{51}{20} \right| \][/tex]
4. Perform the addition using the given distances:
- [tex]\(\left| \frac{25}{4} \right| = \frac{25}{4} \)[/tex]
- [tex]\(\left| -\frac{51}{20} \right| = \frac{51}{20} \)[/tex]
5. Add the two fractions directly:
- Common denominator approach: To add the fractions [tex]\(\frac{25}{4}\)[/tex] and [tex]\(\frac{51}{20}\)[/tex], we need a common denominator.
- The least common multiple (LCM) of 4 and 20 is 20.
- Rewriting [tex]\(\frac{25}{4}\)[/tex] with denominator 20:
[tex]\[ \frac{25}{4} = \frac{25 \times 5}{4 \times 5} = \frac{125}{20} \][/tex]
- Now, add the fractions:
[tex]\[ \frac{125}{20} + \frac{51}{20} = \frac{125 + 51}{20} = \frac{176}{20} \][/tex]
- Simplify [tex]\(\frac{176}{20}\)[/tex]:
[tex]\[ \frac{176}{20} = 8.8 \][/tex]
6. Verify the result using another method (combined numerator approach):
- We use the formula:
[tex]\[ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \][/tex]
- For [tex]\(\frac{25}{4} + \frac{51}{20}\)[/tex]:
[tex]\[ \frac{25 \times 5 + 51 \times 4}{4 \times 20} = \frac{125 + 204}{80} = \frac{329}{80} = 8.8 \][/tex]
Therefore, the two different expressions that represent the distance between the dogs are:
[tex]\[ \frac{25}{4} + \frac{51}{20} \quad \text{and} \quad \frac{25 \times 5 + 51 \times 4}{4 \times 5} \][/tex]
The distance between the two dogs is [tex]\(8.8\)[/tex] meters, matching the numerical results from both expressions.
Let's break this down step-by-step.
1. Determine the distance of the beagle from Natasha:
- The beagle is [tex]\(\frac{25}{4}\)[/tex] meters to her right.
2. Determine the distance of the labrador from Natasha:
- The labrador is [tex]\(\frac{51}{20}\)[/tex] meters to her left.
3. Calculate the distance between the two dogs:
- Since they ran in opposite directions, the total distance between them is the sum of their distances from Natasha.
- We can write this as:
[tex]\[ \text{Distance apart} = \left| \frac{25}{4} \right| + \left| -\frac{51}{20} \right| \][/tex]
4. Perform the addition using the given distances:
- [tex]\(\left| \frac{25}{4} \right| = \frac{25}{4} \)[/tex]
- [tex]\(\left| -\frac{51}{20} \right| = \frac{51}{20} \)[/tex]
5. Add the two fractions directly:
- Common denominator approach: To add the fractions [tex]\(\frac{25}{4}\)[/tex] and [tex]\(\frac{51}{20}\)[/tex], we need a common denominator.
- The least common multiple (LCM) of 4 and 20 is 20.
- Rewriting [tex]\(\frac{25}{4}\)[/tex] with denominator 20:
[tex]\[ \frac{25}{4} = \frac{25 \times 5}{4 \times 5} = \frac{125}{20} \][/tex]
- Now, add the fractions:
[tex]\[ \frac{125}{20} + \frac{51}{20} = \frac{125 + 51}{20} = \frac{176}{20} \][/tex]
- Simplify [tex]\(\frac{176}{20}\)[/tex]:
[tex]\[ \frac{176}{20} = 8.8 \][/tex]
6. Verify the result using another method (combined numerator approach):
- We use the formula:
[tex]\[ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \][/tex]
- For [tex]\(\frac{25}{4} + \frac{51}{20}\)[/tex]:
[tex]\[ \frac{25 \times 5 + 51 \times 4}{4 \times 20} = \frac{125 + 204}{80} = \frac{329}{80} = 8.8 \][/tex]
Therefore, the two different expressions that represent the distance between the dogs are:
[tex]\[ \frac{25}{4} + \frac{51}{20} \quad \text{and} \quad \frac{25 \times 5 + 51 \times 4}{4 \times 5} \][/tex]
The distance between the two dogs is [tex]\(8.8\)[/tex] meters, matching the numerical results from both expressions.
Answer:
A
Step-by-step explanation:
The distance is the sum of the 2 distances
So A, 25/4 + 51/20 is 1 answer.
