Answer :
To determine which proportion is false among the given options, we will evaluate each pair of fractions to see if they are equal by simplifying them.
1. Checking the first proportion:
[tex]\[
\frac{18}{48} \quad \text{and} \quad \frac{30}{50}
\][/tex]
- Simplify [tex]\(\frac{18}{48}\)[/tex]:
[tex]\[
\text{GCD of 18 and 48 is 6, so } \frac{18}{48} = \frac{18 \div 6}{48 \div 6} = \frac{3}{8}
\][/tex]
- Simplify [tex]\(\frac{30}{50}\)[/tex]:
[tex]\[
\text{GCD of 30 and 50 is 10, so } \frac{30}{50} = \frac{30 \div 10}{50 \div 10} = \frac{3}{5}
\][/tex]
Since [tex]\(\frac{3}{8} \neq \frac{3}{5}\)[/tex], this proportion is false.
2. Checking the second proportion:
[tex]\[
\frac{25}{45} \quad \text{and} \quad \frac{50}{90}
\][/tex]
- Simplify [tex]\(\frac{25}{45}\)[/tex]:
[tex]\[
\text{GCD of 25 and 45 is 5, so } \frac{25}{45} = \frac{25 \div 5}{45 \div 5} = \frac{5}{9}
\][/tex]
- Simplify [tex]\(\frac{50}{90}\)[/tex]:
[tex]\[
\text{GCD of 50 and 90 is 10, so } \frac{50}{90} = \frac{50 \div 10}{90 \div 10} = \frac{5}{9}
\][/tex]
Since [tex]\(\frac{5}{9} = \frac{5}{9}\)[/tex], this proportion is true.
3. Checking the third proportion:
[tex]\[
\frac{12}{15} \quad \text{and} \quad \frac{20}{25}
\][/tex]
- Simplify [tex]\(\frac{12}{15}\)[/tex]:
[tex]\[
\text{GCD of 12 and 15 is 3, so } \frac{12}{15} = \frac{12 \div 3}{15 \div 3} = \frac{4}{5}
\][/tex]
- Simplify [tex]\(\frac{20}{25}\)[/tex]:
[tex]\[
\text{GCD of 20 and 25 is 5, so } \frac{20}{25} = \frac{20 \div 5}{25 \div 5} = \frac{4}{5}
\][/tex]
Since [tex]\(\frac{4}{5} = \frac{4}{5}\)[/tex], this proportion is true.
4. Checking the fourth proportion:
[tex]\[
\frac{20}{50} \quad \text{and} \quad \frac{40}{100}
\][/tex]
- Simplify [tex]\(\frac{20}{50}\)[/tex]:
[tex]\[
\text{GCD of 20 and 50 is 10, so } \frac{20}{50} = \frac{20 \div 10}{50 \div 10} = \frac{2}{5}
\][/tex]
- Simplify [tex]\(\frac{40}{100}\)[/tex]:
[tex]\[
\text{GCD of 40 and 100 is 20, so } \frac{40}{100} = \frac{40 \div 20}{100 \div 20} = \frac{2}{5}
\][/tex]
Since [tex]\(\frac{2}{5} = \frac{2}{5}\)[/tex], this proportion is true.
Therefore, the false proportion is:
[tex]\[
\frac{18}{48} = \frac{30}{50}
\][/tex]
1. Checking the first proportion:
[tex]\[
\frac{18}{48} \quad \text{and} \quad \frac{30}{50}
\][/tex]
- Simplify [tex]\(\frac{18}{48}\)[/tex]:
[tex]\[
\text{GCD of 18 and 48 is 6, so } \frac{18}{48} = \frac{18 \div 6}{48 \div 6} = \frac{3}{8}
\][/tex]
- Simplify [tex]\(\frac{30}{50}\)[/tex]:
[tex]\[
\text{GCD of 30 and 50 is 10, so } \frac{30}{50} = \frac{30 \div 10}{50 \div 10} = \frac{3}{5}
\][/tex]
Since [tex]\(\frac{3}{8} \neq \frac{3}{5}\)[/tex], this proportion is false.
2. Checking the second proportion:
[tex]\[
\frac{25}{45} \quad \text{and} \quad \frac{50}{90}
\][/tex]
- Simplify [tex]\(\frac{25}{45}\)[/tex]:
[tex]\[
\text{GCD of 25 and 45 is 5, so } \frac{25}{45} = \frac{25 \div 5}{45 \div 5} = \frac{5}{9}
\][/tex]
- Simplify [tex]\(\frac{50}{90}\)[/tex]:
[tex]\[
\text{GCD of 50 and 90 is 10, so } \frac{50}{90} = \frac{50 \div 10}{90 \div 10} = \frac{5}{9}
\][/tex]
Since [tex]\(\frac{5}{9} = \frac{5}{9}\)[/tex], this proportion is true.
3. Checking the third proportion:
[tex]\[
\frac{12}{15} \quad \text{and} \quad \frac{20}{25}
\][/tex]
- Simplify [tex]\(\frac{12}{15}\)[/tex]:
[tex]\[
\text{GCD of 12 and 15 is 3, so } \frac{12}{15} = \frac{12 \div 3}{15 \div 3} = \frac{4}{5}
\][/tex]
- Simplify [tex]\(\frac{20}{25}\)[/tex]:
[tex]\[
\text{GCD of 20 and 25 is 5, so } \frac{20}{25} = \frac{20 \div 5}{25 \div 5} = \frac{4}{5}
\][/tex]
Since [tex]\(\frac{4}{5} = \frac{4}{5}\)[/tex], this proportion is true.
4. Checking the fourth proportion:
[tex]\[
\frac{20}{50} \quad \text{and} \quad \frac{40}{100}
\][/tex]
- Simplify [tex]\(\frac{20}{50}\)[/tex]:
[tex]\[
\text{GCD of 20 and 50 is 10, so } \frac{20}{50} = \frac{20 \div 10}{50 \div 10} = \frac{2}{5}
\][/tex]
- Simplify [tex]\(\frac{40}{100}\)[/tex]:
[tex]\[
\text{GCD of 40 and 100 is 20, so } \frac{40}{100} = \frac{40 \div 20}{100 \div 20} = \frac{2}{5}
\][/tex]
Since [tex]\(\frac{2}{5} = \frac{2}{5}\)[/tex], this proportion is true.
Therefore, the false proportion is:
[tex]\[
\frac{18}{48} = \frac{30}{50}
\][/tex]
