Answer :
To determine which proportion is false, we need to compare each pair of fractions to see if they are equal. Let's evaluate them one by one:
1. Proportion 1: [tex]\(\frac{24}{30}\)[/tex] and [tex]\(\frac{20}{25}\)[/tex]
- Simplify [tex]\(\frac{24}{30}\)[/tex]: Divide the numerator and denominator by their greatest common divisor (GCD), which is 6.
[tex]\[
\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}
\][/tex]
- Simplify [tex]\(\frac{20}{25}\)[/tex]: Divide the numerator and denominator by their GCD, which is 5.
[tex]\[
\frac{20}{25} = \frac{20 \div 5}{25 \div 5} = \frac{4}{5}
\][/tex]
Both fractions simplify to [tex]\(\frac{4}{5}\)[/tex], so Proportion 1 is true.
2. Proportion 2: [tex]\(\frac{25}{45}\)[/tex] and [tex]\(\frac{75}{135}\)[/tex]
- Simplify [tex]\(\frac{25}{45}\)[/tex]: Divide the numerator and denominator by their GCD, which is 5.
[tex]\[
\frac{25}{45} = \frac{25 \div 5}{45 \div 5} = \frac{5}{9}
\][/tex]
- Simplify [tex]\(\frac{75}{135}\)[/tex]: Divide the numerator and denominator by their GCD, which is 15.
[tex]\[
\frac{75}{135} = \frac{75 \div 15}{135 \div 15} = \frac{5}{9}
\][/tex]
Both fractions simplify to [tex]\(\frac{5}{9}\)[/tex], so Proportion 2 is true.
3. Proportion 3: [tex]\(\frac{10}{25}\)[/tex] and [tex]\(\frac{40}{100}\)[/tex]
- Simplify [tex]\(\frac{10}{25}\)[/tex]: Divide the numerator and denominator by their GCD, which is 5.
[tex]\[
\frac{10}{25} = \frac{10 \div 5}{25 \div 5} = \frac{2}{5}
\][/tex]
- Simplify [tex]\(\frac{40}{100}\)[/tex]: Divide the numerator and denominator by their GCD, which is 20.
[tex]\[
\frac{40}{100} = \frac{40 \div 20}{100 \div 20} = \frac{2}{5}
\][/tex]
Both fractions simplify to [tex]\(\frac{2}{5}\)[/tex], so Proportion 3 is true.
4. Proportion 4: [tex]\(\frac{18}{48}\)[/tex] and [tex]\(\frac{20}{50}\)[/tex]
- Simplify [tex]\(\frac{18}{48}\)[/tex]: Divide the numerator and denominator by their GCD, which is 6.
[tex]\[
\frac{18}{48} = \frac{18 \div 6}{48 \div 6} = \frac{3}{8}
\][/tex]
- Simplify [tex]\(\frac{20}{50}\)[/tex]: Divide the numerator and denominator by their GCD, which is 10.
[tex]\[
\frac{20}{50} = \frac{20 \div 10}{50 \div 10} = \frac{2}{5}
\][/tex]
[tex]\(\frac{18}{48}\)[/tex] simplifies to [tex]\(\frac{3}{8}\)[/tex], and [tex]\(\frac{20}{50}\)[/tex] simplifies to [tex]\(\frac{2}{5}\)[/tex]. These are not equal, so Proportion 4 is false.
Therefore, the false proportion is the fourth one: [tex]\(\frac{18}{48} \neq \frac{20}{50}\)[/tex].
1. Proportion 1: [tex]\(\frac{24}{30}\)[/tex] and [tex]\(\frac{20}{25}\)[/tex]
- Simplify [tex]\(\frac{24}{30}\)[/tex]: Divide the numerator and denominator by their greatest common divisor (GCD), which is 6.
[tex]\[
\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}
\][/tex]
- Simplify [tex]\(\frac{20}{25}\)[/tex]: Divide the numerator and denominator by their GCD, which is 5.
[tex]\[
\frac{20}{25} = \frac{20 \div 5}{25 \div 5} = \frac{4}{5}
\][/tex]
Both fractions simplify to [tex]\(\frac{4}{5}\)[/tex], so Proportion 1 is true.
2. Proportion 2: [tex]\(\frac{25}{45}\)[/tex] and [tex]\(\frac{75}{135}\)[/tex]
- Simplify [tex]\(\frac{25}{45}\)[/tex]: Divide the numerator and denominator by their GCD, which is 5.
[tex]\[
\frac{25}{45} = \frac{25 \div 5}{45 \div 5} = \frac{5}{9}
\][/tex]
- Simplify [tex]\(\frac{75}{135}\)[/tex]: Divide the numerator and denominator by their GCD, which is 15.
[tex]\[
\frac{75}{135} = \frac{75 \div 15}{135 \div 15} = \frac{5}{9}
\][/tex]
Both fractions simplify to [tex]\(\frac{5}{9}\)[/tex], so Proportion 2 is true.
3. Proportion 3: [tex]\(\frac{10}{25}\)[/tex] and [tex]\(\frac{40}{100}\)[/tex]
- Simplify [tex]\(\frac{10}{25}\)[/tex]: Divide the numerator and denominator by their GCD, which is 5.
[tex]\[
\frac{10}{25} = \frac{10 \div 5}{25 \div 5} = \frac{2}{5}
\][/tex]
- Simplify [tex]\(\frac{40}{100}\)[/tex]: Divide the numerator and denominator by their GCD, which is 20.
[tex]\[
\frac{40}{100} = \frac{40 \div 20}{100 \div 20} = \frac{2}{5}
\][/tex]
Both fractions simplify to [tex]\(\frac{2}{5}\)[/tex], so Proportion 3 is true.
4. Proportion 4: [tex]\(\frac{18}{48}\)[/tex] and [tex]\(\frac{20}{50}\)[/tex]
- Simplify [tex]\(\frac{18}{48}\)[/tex]: Divide the numerator and denominator by their GCD, which is 6.
[tex]\[
\frac{18}{48} = \frac{18 \div 6}{48 \div 6} = \frac{3}{8}
\][/tex]
- Simplify [tex]\(\frac{20}{50}\)[/tex]: Divide the numerator and denominator by their GCD, which is 10.
[tex]\[
\frac{20}{50} = \frac{20 \div 10}{50 \div 10} = \frac{2}{5}
\][/tex]
[tex]\(\frac{18}{48}\)[/tex] simplifies to [tex]\(\frac{3}{8}\)[/tex], and [tex]\(\frac{20}{50}\)[/tex] simplifies to [tex]\(\frac{2}{5}\)[/tex]. These are not equal, so Proportion 4 is false.
Therefore, the false proportion is the fourth one: [tex]\(\frac{18}{48} \neq \frac{20}{50}\)[/tex].
