Answer :
We need to determine which fractions are equivalent to
[tex]$$\frac{12}{15}.$$[/tex]
First, simplify the fraction:
[tex]$$\gcd(12,15)=3,$$[/tex]
so
[tex]$$\frac{12}{15}=\frac{12\div 3}{15\div 3}=\frac{4}{5}.$$[/tex]
Now, we compare each option after simplifying:
1. Option A: [tex]$\frac{4}{20}$[/tex]
Find the greatest common divisor (GCD) of 4 and 20:
[tex]$$\gcd(4,20)=4.$$[/tex]
Simplify:
[tex]$$\frac{4}{20}=\frac{4\div 4}{20\div 4}=\frac{1}{5}.$$[/tex]
Since [tex]$\frac{1}{5}\neq \frac{4}{5}$[/tex], option A is not equivalent.
2. Option B: [tex]$\frac{4}{5}$[/tex]
This is already in simplest form and matches [tex]$\frac{4}{5}$[/tex] exactly. Hence, option B is equivalent.
3. Option C: [tex]$\frac{24}{30}$[/tex]
Find the GCD of 24 and 30:
[tex]$$\gcd(24,30)=6.$$[/tex]
Simplify:
[tex]$$\frac{24}{30}=\frac{24\div 6}{30\div 6}=\frac{4}{5}.$$[/tex]
This matches, so option C is equivalent.
4. Option D: [tex]$\frac{4}{16}$[/tex]
Find the GCD of 4 and 16:
[tex]$$\gcd(4,16)=4.$$[/tex]
Simplify:
[tex]$$\frac{4}{16}=\frac{4\div 4}{16\div 4}=\frac{1}{4}.$$[/tex]
Since [tex]$\frac{1}{4}\neq \frac{4}{5}$[/tex], option D is not equivalent.
Thus, the fractions equivalent to [tex]$\frac{12}{15}$[/tex] are:
[tex]$$\boxed{\text{B and C}}.$$[/tex]
[tex]$$\frac{12}{15}.$$[/tex]
First, simplify the fraction:
[tex]$$\gcd(12,15)=3,$$[/tex]
so
[tex]$$\frac{12}{15}=\frac{12\div 3}{15\div 3}=\frac{4}{5}.$$[/tex]
Now, we compare each option after simplifying:
1. Option A: [tex]$\frac{4}{20}$[/tex]
Find the greatest common divisor (GCD) of 4 and 20:
[tex]$$\gcd(4,20)=4.$$[/tex]
Simplify:
[tex]$$\frac{4}{20}=\frac{4\div 4}{20\div 4}=\frac{1}{5}.$$[/tex]
Since [tex]$\frac{1}{5}\neq \frac{4}{5}$[/tex], option A is not equivalent.
2. Option B: [tex]$\frac{4}{5}$[/tex]
This is already in simplest form and matches [tex]$\frac{4}{5}$[/tex] exactly. Hence, option B is equivalent.
3. Option C: [tex]$\frac{24}{30}$[/tex]
Find the GCD of 24 and 30:
[tex]$$\gcd(24,30)=6.$$[/tex]
Simplify:
[tex]$$\frac{24}{30}=\frac{24\div 6}{30\div 6}=\frac{4}{5}.$$[/tex]
This matches, so option C is equivalent.
4. Option D: [tex]$\frac{4}{16}$[/tex]
Find the GCD of 4 and 16:
[tex]$$\gcd(4,16)=4.$$[/tex]
Simplify:
[tex]$$\frac{4}{16}=\frac{4\div 4}{16\div 4}=\frac{1}{4}.$$[/tex]
Since [tex]$\frac{1}{4}\neq \frac{4}{5}$[/tex], option D is not equivalent.
Thus, the fractions equivalent to [tex]$\frac{12}{15}$[/tex] are:
[tex]$$\boxed{\text{B and C}}.$$[/tex]
