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A jet flew from New York to Los Angeles, a distance of 4,200 kilometers. Then it completed the return trip. The speed for the return trip was 100 kilometers/hour faster than the outbound speed. This expression, where [tex]\( x \)[/tex] is the speed for the outbound trip, represents the situation.
[tex]\[
\frac{4,200}{x} + \frac{4,200}{x + 100}
\][/tex]

Which expression could be a step in rewriting this sum?

A. [tex]\(\frac{4,200}{x + (x + 100)} + \frac{4,200}{x + (x + 100)}\)[/tex]

B. [tex]\(\frac{8,400(x + 100)}{x(x + 100)}\)[/tex]

C. [tex]\(\frac{4,200(x + 100)}{x(x + 100)} + \frac{4,200 x}{x(x + 100)}\)[/tex]

D. [tex]\(\frac{1,200(x + 100) + 1,200}{x + 100}\)[/tex]



Answer :

GinnyAnswer
To solve this problem, we need to manipulate and simplify the given expression:
[tex]\[ \frac{4200}{x} + \frac{4200}{x+100} \][/tex]

The goal is to rewrite and possibly simplify this expression correctly. Let's go through the options provided.

### Option A
[tex]\[ \frac{4200}{x+(x+100)}+\frac{4200}{x+(x+100)} \][/tex]

This option is incorrect because the denominators do not correctly represent the time for each leg of the trip.

### Option B
[tex]\[ \frac{8400(x+100)}{x(x+100)} \][/tex]

This is a single combined fraction, and we see it uses a common denominator. We will compare this against our work later.

### Option C
[tex]\[ \frac{4200(x+100)}{x(x+100)}+\frac{4200 x}{x(x+100)} \][/tex]

This expression is structured correctly since it incorporates [tex]\( x \)[/tex] and [tex]\( x + 100 \)[/tex] into the numerators and utilizes the common denominator [tex]\( x(x+100) \)[/tex].

### Option D
[tex]\[ \frac{1200(x+100)+1200(x)}{(x+100)} \][/tex]

It appears incorrect with a common denominator that wasn't derived from the given expressions.

### Simplification Process
To validate and identify the correct simplification, let's properly rewrite the original expression using steps that might relate to the given options.

#### Original Expression
[tex]\[ \frac{4200}{x} + \frac{4200}{x+100} \][/tex]

#### Rewriting with a Common Denominator
To add these two fractions, we need a common denominator [tex]\( x(x+100) \)[/tex]:

[tex]\[ \frac{4200(x+100)}{x(x+100)} + \frac{4200x}{x(x+100)} \][/tex]

Combining fractions, we obtain:
[tex]\[ \frac{4200(x+100) + 4200x}{x(x+100)} \][/tex]

Upon combining like terms inside the numerator:
[tex]\[ \frac{4200x + 4200*100 + 4200x}{x(x+100)} \][/tex]
[tex]\[ \frac{4200x + 420000 + 4200x}{x(x+100)} \][/tex]
[tex]\[ \frac{8400(x+100)}{x(x+100)} \][/tex]

The outcome confirms that:

Option B,
[tex]\[ \frac{8400(x+100)}{x(x+100)} \][/tex]
is indeed the correct form of the expression in the given context.

In conclusion, the correct answer is:
[tex]\[ \boxed{2 \text { (Option B)}} \][/tex]

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