Answer :
To determine which of the given proportions is false, we can compare the fractions by simplifying them and checking if they are equal. Let's go through each proportion:
1. [tex]\( \frac{20}{50} = \frac{40}{100} \)[/tex]:
- Simplify [tex]\( \frac{20}{50} \)[/tex] by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 10:
[tex]\[
\frac{20 \div 10}{50 \div 10} = \frac{2}{5}
\][/tex]
- Simplify [tex]\( \frac{40}{100} \)[/tex] by dividing the numerator and the denominator by their GCD, which is 20:
[tex]\[
\frac{40 \div 20}{100 \div 20} = \frac{2}{5}
\][/tex]
- Since both simplified fractions are equal, this proportion is true.
2. [tex]\( \frac{18}{48} = \frac{30}{50} \)[/tex]:
- Simplify [tex]\( \frac{18}{48} \)[/tex] by dividing the numerator and the denominator by their GCD, which is 6:
[tex]\[
\frac{18 \div 6}{48 \div 6} = \frac{3}{8}
\][/tex]
- Simplify [tex]\( \frac{30}{50} \)[/tex] by dividing the numerator and the denominator by their GCD, which is 10:
[tex]\[
\frac{30 \div 10}{50 \div 10} = \frac{3}{5}
\][/tex]
- Since [tex]\( \frac{3}{8} \)[/tex] is not equal to [tex]\( \frac{3}{5} \)[/tex], this proportion is false.
3. [tex]\( \frac{12}{15} = \frac{20}{25} \)[/tex]:
- Simplify [tex]\( \frac{12}{15} \)[/tex] by dividing the numerator and the denominator by their GCD, which is 3:
[tex]\[
\frac{12 \div 3}{15 \div 3} = \frac{4}{5}
\][/tex]
- Simplify [tex]\( \frac{20}{25} \)[/tex] by dividing the numerator and the denominator by their GCD, which is 5:
[tex]\[
\frac{20 \div 5}{25 \div 5} = \frac{4}{5}
\][/tex]
- Since both simplified fractions are equal, this proportion is true.
4. [tex]\( \frac{25}{45} = \frac{50}{90} \)[/tex]:
- Simplify [tex]\( \frac{25}{45} \)[/tex] by dividing the numerator and the denominator by their GCD, which is 5:
[tex]\[
\frac{25 \div 5}{45 \div 5} = \frac{5}{9}
\][/tex]
- Simplify [tex]\( \frac{50}{90} \)[/tex] by dividing the numerator and the denominator by their GCD, which is 10:
[tex]\[
\frac{50 \div 10}{90 \div 10} = \frac{5}{9}
\][/tex]
- Since both simplified fractions are equal, this proportion is true.
After checking all proportions, the false one is:
- [tex]\( \frac{18}{48} = \frac{30}{50} \)[/tex]
So, the answer is the second proportion.
1. [tex]\( \frac{20}{50} = \frac{40}{100} \)[/tex]:
- Simplify [tex]\( \frac{20}{50} \)[/tex] by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 10:
[tex]\[
\frac{20 \div 10}{50 \div 10} = \frac{2}{5}
\][/tex]
- Simplify [tex]\( \frac{40}{100} \)[/tex] by dividing the numerator and the denominator by their GCD, which is 20:
[tex]\[
\frac{40 \div 20}{100 \div 20} = \frac{2}{5}
\][/tex]
- Since both simplified fractions are equal, this proportion is true.
2. [tex]\( \frac{18}{48} = \frac{30}{50} \)[/tex]:
- Simplify [tex]\( \frac{18}{48} \)[/tex] by dividing the numerator and the denominator by their GCD, which is 6:
[tex]\[
\frac{18 \div 6}{48 \div 6} = \frac{3}{8}
\][/tex]
- Simplify [tex]\( \frac{30}{50} \)[/tex] by dividing the numerator and the denominator by their GCD, which is 10:
[tex]\[
\frac{30 \div 10}{50 \div 10} = \frac{3}{5}
\][/tex]
- Since [tex]\( \frac{3}{8} \)[/tex] is not equal to [tex]\( \frac{3}{5} \)[/tex], this proportion is false.
3. [tex]\( \frac{12}{15} = \frac{20}{25} \)[/tex]:
- Simplify [tex]\( \frac{12}{15} \)[/tex] by dividing the numerator and the denominator by their GCD, which is 3:
[tex]\[
\frac{12 \div 3}{15 \div 3} = \frac{4}{5}
\][/tex]
- Simplify [tex]\( \frac{20}{25} \)[/tex] by dividing the numerator and the denominator by their GCD, which is 5:
[tex]\[
\frac{20 \div 5}{25 \div 5} = \frac{4}{5}
\][/tex]
- Since both simplified fractions are equal, this proportion is true.
4. [tex]\( \frac{25}{45} = \frac{50}{90} \)[/tex]:
- Simplify [tex]\( \frac{25}{45} \)[/tex] by dividing the numerator and the denominator by their GCD, which is 5:
[tex]\[
\frac{25 \div 5}{45 \div 5} = \frac{5}{9}
\][/tex]
- Simplify [tex]\( \frac{50}{90} \)[/tex] by dividing the numerator and the denominator by their GCD, which is 10:
[tex]\[
\frac{50 \div 10}{90 \div 10} = \frac{5}{9}
\][/tex]
- Since both simplified fractions are equal, this proportion is true.
After checking all proportions, the false one is:
- [tex]\( \frac{18}{48} = \frac{30}{50} \)[/tex]
So, the answer is the second proportion.
