Answer :
To determine which ratios are equivalent to [tex]\(6:8\)[/tex], we need to find ratios that have the same simplified form as [tex]\(6:8\)[/tex].
1. Simplifying the Ratio [tex]\(6:8\)[/tex]:
- The greatest common divisor (GCD) of 6 and 8 is 2.
- Simplifying the ratio by dividing both terms by 2, we get:
[tex]\[
\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}
\][/tex]
- So, the ratio [tex]\(6:8\)[/tex] simplifies to [tex]\(3:4\)[/tex].
2. Checking Each Option:
- 24:36
- Simplify [tex]\(24:36\)[/tex] to [tex]\(\frac{24 \div 12}{36 \div 12} = \frac{2}{3}\)[/tex] (using GCD of 12).
- [tex]\(\frac{2}{3}\)[/tex] is not equal to [tex]\(\frac{3}{4}\)[/tex], so this is not equivalent.
- [tex]\(\frac{36}{40}\)[/tex]
- Simplify [tex]\(\frac{36}{40}\)[/tex] to [tex]\(\frac{36 \div 4}{40 \div 4} = \frac{9}{10}\)[/tex] (using GCD of 4).
- [tex]\(\frac{9}{10}\)[/tex] is not equal to [tex]\(\frac{3}{4}\)[/tex], so this is not equivalent.
- [tex]\(\frac{15}{20}\)[/tex]
- Simplify [tex]\(\frac{15}{20}\)[/tex] to [tex]\(\frac{15 \div 5}{20 \div 5} = \frac{3}{4}\)[/tex] (using GCD of 5).
- [tex]\(\frac{3}{4}\)[/tex] is equal to [tex]\(\frac{3}{4}\)[/tex], so this is equivalent.
- 9:12
- Simplify [tex]\(9:12\)[/tex] to [tex]\(\frac{9 \div 3}{12 \div 3} = \frac{3}{4}\)[/tex] (using GCD of 3).
- [tex]\(\frac{3}{4}\)[/tex] is equal to [tex]\(\frac{3}{4}\)[/tex], so this is equivalent.
- 12 to 16
- Simplify [tex]\(12:16\)[/tex] to [tex]\(\frac{12 \div 4}{16 \div 4} = \frac{3}{4}\)[/tex] (using GCD of 4).
- [tex]\(\frac{3}{4}\)[/tex] is equal to [tex]\(\frac{3}{4}\)[/tex], so this is equivalent.
- [tex]\(\frac{1}{8}\)[/tex] to 24
- Interpreting this ratio as [tex]\(\frac{1/8}{24}\)[/tex] gives a very small number, which is clearly not [tex]\(\frac{3}{4}\)[/tex].
- Thus, this is not equivalent.
3. Equivalent Ratios:
- Based on the simplification, the ratios that are equivalent to [tex]\(6:8\)[/tex] are:
- [tex]\(\frac{15}{20}\)[/tex]
- 9:12
- 12 to 16
These are the ratios that simplify to the same value as [tex]\(6:8\)[/tex].
1. Simplifying the Ratio [tex]\(6:8\)[/tex]:
- The greatest common divisor (GCD) of 6 and 8 is 2.
- Simplifying the ratio by dividing both terms by 2, we get:
[tex]\[
\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}
\][/tex]
- So, the ratio [tex]\(6:8\)[/tex] simplifies to [tex]\(3:4\)[/tex].
2. Checking Each Option:
- 24:36
- Simplify [tex]\(24:36\)[/tex] to [tex]\(\frac{24 \div 12}{36 \div 12} = \frac{2}{3}\)[/tex] (using GCD of 12).
- [tex]\(\frac{2}{3}\)[/tex] is not equal to [tex]\(\frac{3}{4}\)[/tex], so this is not equivalent.
- [tex]\(\frac{36}{40}\)[/tex]
- Simplify [tex]\(\frac{36}{40}\)[/tex] to [tex]\(\frac{36 \div 4}{40 \div 4} = \frac{9}{10}\)[/tex] (using GCD of 4).
- [tex]\(\frac{9}{10}\)[/tex] is not equal to [tex]\(\frac{3}{4}\)[/tex], so this is not equivalent.
- [tex]\(\frac{15}{20}\)[/tex]
- Simplify [tex]\(\frac{15}{20}\)[/tex] to [tex]\(\frac{15 \div 5}{20 \div 5} = \frac{3}{4}\)[/tex] (using GCD of 5).
- [tex]\(\frac{3}{4}\)[/tex] is equal to [tex]\(\frac{3}{4}\)[/tex], so this is equivalent.
- 9:12
- Simplify [tex]\(9:12\)[/tex] to [tex]\(\frac{9 \div 3}{12 \div 3} = \frac{3}{4}\)[/tex] (using GCD of 3).
- [tex]\(\frac{3}{4}\)[/tex] is equal to [tex]\(\frac{3}{4}\)[/tex], so this is equivalent.
- 12 to 16
- Simplify [tex]\(12:16\)[/tex] to [tex]\(\frac{12 \div 4}{16 \div 4} = \frac{3}{4}\)[/tex] (using GCD of 4).
- [tex]\(\frac{3}{4}\)[/tex] is equal to [tex]\(\frac{3}{4}\)[/tex], so this is equivalent.
- [tex]\(\frac{1}{8}\)[/tex] to 24
- Interpreting this ratio as [tex]\(\frac{1/8}{24}\)[/tex] gives a very small number, which is clearly not [tex]\(\frac{3}{4}\)[/tex].
- Thus, this is not equivalent.
3. Equivalent Ratios:
- Based on the simplification, the ratios that are equivalent to [tex]\(6:8\)[/tex] are:
- [tex]\(\frac{15}{20}\)[/tex]
- 9:12
- 12 to 16
These are the ratios that simplify to the same value as [tex]\(6:8\)[/tex].
