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Answer :
To solve the given problem step-by-step, we need to find the number [tex]\( x \)[/tex] that, when subtracted from both the numerator and the denominator of the fraction [tex]\(\frac{18}{20}\)[/tex], results in the fraction [tex]\(\frac{11}{13}\)[/tex].
Let's break down the process:
1. Define the original fraction:
The fraction given is [tex]\(\frac{18}{20}\)[/tex].
2. Define the resulting fraction:
We need to obtain [tex]\(\frac{11}{13}\)[/tex] after subtracting some number [tex]\( x \)[/tex] from both the numerator and the denominator of the original fraction.
3. Set up the equation:
According to the problem, subtract [tex]\( x \)[/tex] from both the numerator and the denominator:
[tex]\[
\frac{18 - x}{20 - x} = \frac{11}{13}
\][/tex]
4. Cross-multiply to form an equation:
To solve the equation, cross-multiply:
[tex]\[
13(18 - x) = 11(20 - x)
\][/tex]
5. Expand both sides:
Expanding the expressions, we get:
[tex]\[
13 \cdot 18 - 13 \cdot x = 11 \cdot 20 - 11 \cdot x
\][/tex]
Simplifying further:
[tex]\[
234 - 13x = 220 - 11x
\][/tex]
6. Combine like terms:
To isolate [tex]\( x \)[/tex], move the terms involving [tex]\( x \)[/tex] to one side and the constant terms to the other side:
[tex]\[
234 - 220 = 13x - 11x
\][/tex]
Simplifying:
[tex]\[
14 = 2x
\][/tex]
7. Solve for [tex]\( x \)[/tex]:
Divide both sides by 2:
[tex]\[
x = \frac{14}{2} = 7
\][/tex]
So, the number [tex]\( x \)[/tex] that needs to be subtracted from the numerator and the denominator of the fraction [tex]\(\frac{18}{20}\)[/tex] to get the fraction [tex]\(\frac{11}{13}\)[/tex] is [tex]\( \boxed{7} \)[/tex].
Let's break down the process:
1. Define the original fraction:
The fraction given is [tex]\(\frac{18}{20}\)[/tex].
2. Define the resulting fraction:
We need to obtain [tex]\(\frac{11}{13}\)[/tex] after subtracting some number [tex]\( x \)[/tex] from both the numerator and the denominator of the original fraction.
3. Set up the equation:
According to the problem, subtract [tex]\( x \)[/tex] from both the numerator and the denominator:
[tex]\[
\frac{18 - x}{20 - x} = \frac{11}{13}
\][/tex]
4. Cross-multiply to form an equation:
To solve the equation, cross-multiply:
[tex]\[
13(18 - x) = 11(20 - x)
\][/tex]
5. Expand both sides:
Expanding the expressions, we get:
[tex]\[
13 \cdot 18 - 13 \cdot x = 11 \cdot 20 - 11 \cdot x
\][/tex]
Simplifying further:
[tex]\[
234 - 13x = 220 - 11x
\][/tex]
6. Combine like terms:
To isolate [tex]\( x \)[/tex], move the terms involving [tex]\( x \)[/tex] to one side and the constant terms to the other side:
[tex]\[
234 - 220 = 13x - 11x
\][/tex]
Simplifying:
[tex]\[
14 = 2x
\][/tex]
7. Solve for [tex]\( x \)[/tex]:
Divide both sides by 2:
[tex]\[
x = \frac{14}{2} = 7
\][/tex]
So, the number [tex]\( x \)[/tex] that needs to be subtracted from the numerator and the denominator of the fraction [tex]\(\frac{18}{20}\)[/tex] to get the fraction [tex]\(\frac{11}{13}\)[/tex] is [tex]\( \boxed{7} \)[/tex].
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