High School

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The denominator of a rational number is greater than its numerator by 8. If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is [tex] \frac{3}{2} [/tex]. Find the rational number.

1) [tex] \frac{13}{21} [/tex]
2) [tex] \frac{15}{21} [/tex]
3) [tex] \frac{18}{20} [/tex]
4) [tex] \frac{12}{22} [/tex]

Answer :

To solve this problem, let's use the given conditions to create an equation and find the rational number step by step.

### Step 1: Define the Variables
Let's denote the numerator by [tex]\( x \)[/tex]. According to the problem, the denominator is greater than the numerator by 8, so we can write the denominator as [tex]\( x + 8 \)[/tex].

### Step 2: Create the First Condition
The problem states that if the numerator is increased by 17 and the denominator is decreased by 1, the resulting fraction is [tex]\( \frac{3}{2} \)[/tex]. This can be written as an equation:

[tex]\[
\frac{x + 17}{(x + 8) - 1} = \frac{3}{2}
\][/tex]

This simplifies to:

[tex]\[
\frac{x + 17}{x + 7} = \frac{3}{2}
\][/tex]

### Step 3: Cross-multiply to Solve for [tex]\( x \)[/tex]
To solve this equation, we can cross-multiply:

[tex]\[
2(x + 17) = 3(x + 7)
\][/tex]

Expanding both sides gives:

[tex]\[
2x + 34 = 3x + 21
\][/tex]

### Step 4: Rearrange the Equation
To isolate [tex]\( x \)[/tex], we rearrange the terms:

[tex]\[
34 - 21 = 3x - 2x
\][/tex]

Simplifying this gives:

[tex]\[
13 = x
\][/tex]

### Step 5: Find the Denominator
Now that we have [tex]\( x = 13 \)[/tex], we can find the denominator [tex]\( x + 8 \)[/tex]:

[tex]\[
13 + 8 = 21
\][/tex]

### Step 6: Form the Rational Number
The rational number is therefore:

[tex]\[
\frac{13}{21}
\][/tex]

### Conclusion
So, the rational number, where the denominator is greater than the numerator by 8, and when adjusted as described in the problem, results in [tex]\( \frac{3}{2} \)[/tex], is:

[tex]\[
\boxed{\frac{13}{21}}
\][/tex]

Thus, the correct answer is:

1) [tex]\(\frac{13}{21}\)[/tex]

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