Answer :
To determine which of the following proportions is false, we need to simplify each side of the proportions and see if they are equivalent. Let's break down each one:
1. Proportion 1: [tex]\(\frac{12}{15} = \frac{20}{25}\)[/tex]
- Simplify [tex]\(\frac{12}{15}\)[/tex]: Divide both the numerator and denominator by their greatest common divisor (GCD), which is 3.
[tex]\[
\frac{12 \div 3}{15 \div 3} = \frac{4}{5}
\][/tex]
- Simplify [tex]\(\frac{20}{25}\)[/tex]: Divide by the GCD, which is 5.
[tex]\[
\frac{20 \div 5}{25 \div 5} = \frac{4}{5}
\][/tex]
- Both simplify to [tex]\(\frac{4}{5}\)[/tex], so this proportion is true.
2. Proportion 2: [tex]\(\frac{18}{48} = \frac{30}{50}\)[/tex]
- Simplify [tex]\(\frac{18}{48}\)[/tex]: Divide by the GCD, which is 6.
[tex]\[
\frac{18 \div 6}{48 \div 6} = \frac{3}{8}
\][/tex]
- Simplify [tex]\(\frac{30}{50}\)[/tex]: Divide by the GCD, which is 10.
[tex]\[
\frac{30 \div 10}{50 \div 10} = \frac{3}{5}
\][/tex]
- [tex]\(\frac{3}{8} \neq \frac{3}{5}\)[/tex], so this proportion is false.
3. Proportion 3: [tex]\(\frac{20}{50} = \frac{40}{100}\)[/tex]
- Simplify [tex]\(\frac{20}{50}\)[/tex]: Divide by the GCD, which is 10.
[tex]\[
\frac{20 \div 10}{50 \div 10} = \frac{2}{5}
\][/tex]
- Simplify [tex]\(\frac{40}{100}\)[/tex]: Divide by the GCD, which is 20.
[tex]\[
\frac{40 \div 20}{100 \div 20} = \frac{2}{5}
\][/tex]
- Both simplify to [tex]\(\frac{2}{5}\)[/tex], so this proportion is true.
4. Proportion 4: [tex]\(\frac{25}{45} = \frac{50}{90}\)[/tex]
- Simplify [tex]\(\frac{25}{45}\)[/tex]: Divide by the GCD, which is 5.
[tex]\[
\frac{25 \div 5}{45 \div 5} = \frac{5}{9}
\][/tex]
- Simplify [tex]\(\frac{50}{90}\)[/tex]: Divide by the GCD, which is 10.
[tex]\[
\frac{50 \div 10}{90 \div 10} = \frac{5}{9}
\][/tex]
- Both simplify to [tex]\(\frac{5}{9}\)[/tex], so this proportion is true.
Based on these evaluations, the false proportion is from the second set: [tex]\(\frac{18}{48} = \frac{30}{50}\)[/tex].
1. Proportion 1: [tex]\(\frac{12}{15} = \frac{20}{25}\)[/tex]
- Simplify [tex]\(\frac{12}{15}\)[/tex]: Divide both the numerator and denominator by their greatest common divisor (GCD), which is 3.
[tex]\[
\frac{12 \div 3}{15 \div 3} = \frac{4}{5}
\][/tex]
- Simplify [tex]\(\frac{20}{25}\)[/tex]: Divide by the GCD, which is 5.
[tex]\[
\frac{20 \div 5}{25 \div 5} = \frac{4}{5}
\][/tex]
- Both simplify to [tex]\(\frac{4}{5}\)[/tex], so this proportion is true.
2. Proportion 2: [tex]\(\frac{18}{48} = \frac{30}{50}\)[/tex]
- Simplify [tex]\(\frac{18}{48}\)[/tex]: Divide by the GCD, which is 6.
[tex]\[
\frac{18 \div 6}{48 \div 6} = \frac{3}{8}
\][/tex]
- Simplify [tex]\(\frac{30}{50}\)[/tex]: Divide by the GCD, which is 10.
[tex]\[
\frac{30 \div 10}{50 \div 10} = \frac{3}{5}
\][/tex]
- [tex]\(\frac{3}{8} \neq \frac{3}{5}\)[/tex], so this proportion is false.
3. Proportion 3: [tex]\(\frac{20}{50} = \frac{40}{100}\)[/tex]
- Simplify [tex]\(\frac{20}{50}\)[/tex]: Divide by the GCD, which is 10.
[tex]\[
\frac{20 \div 10}{50 \div 10} = \frac{2}{5}
\][/tex]
- Simplify [tex]\(\frac{40}{100}\)[/tex]: Divide by the GCD, which is 20.
[tex]\[
\frac{40 \div 20}{100 \div 20} = \frac{2}{5}
\][/tex]
- Both simplify to [tex]\(\frac{2}{5}\)[/tex], so this proportion is true.
4. Proportion 4: [tex]\(\frac{25}{45} = \frac{50}{90}\)[/tex]
- Simplify [tex]\(\frac{25}{45}\)[/tex]: Divide by the GCD, which is 5.
[tex]\[
\frac{25 \div 5}{45 \div 5} = \frac{5}{9}
\][/tex]
- Simplify [tex]\(\frac{50}{90}\)[/tex]: Divide by the GCD, which is 10.
[tex]\[
\frac{50 \div 10}{90 \div 10} = \frac{5}{9}
\][/tex]
- Both simplify to [tex]\(\frac{5}{9}\)[/tex], so this proportion is true.
Based on these evaluations, the false proportion is from the second set: [tex]\(\frac{18}{48} = \frac{30}{50}\)[/tex].
