Natasha and her two dogs were walking on a perfectly straight road when her two dogs ran away from her in opposite directions. Her beagle is now [tex]\frac{25}{4}[/tex] meters directly to her right, and her labrador is [tex]\frac{51}{20}[/tex] meters directly to her left.

Which two of the following expressions represent how far apart the two dogs are?

A. [tex]\frac{25}{4} + \frac{51}{20}[/tex]
B. [tex]\left|\frac{25}{4} - \frac{51}{20}\right|[/tex]
C. [tex]\frac{25}{4} \times \frac{51}{20}[/tex]
D. [tex]\left|\frac{25}{4} + \frac{51}{20}\right|[/tex]
E. [tex]\left|\frac{25}{4}\right| + \left|\frac{51}{20}\right|[/tex]



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.

Answer:

A

Step-by-step explanation:

The distance is the sum of the 2 distances

So A, 25/4 + 51/20 is 1 answer.