Question

$$\frac{3}{4}, \frac{9}{12}, \frac{27}{36}, \frac{81}{108}, \dots$$ which statement is true about the pattern above? each fraction in the pattern is equal to the previous fraction multiplied by 2. the next fraction in the pattern is $\frac{243}{324}$. each fraction in the pattern is less than the previous fraction. the fraction $\frac{12}{16}$ is also part of the pattern.

Explanation:

Step1: Analyze the pattern of fractions

First, simplify each fraction or find the ratio between consecutive fractions. Let's check the ratio of the second fraction to the first: $\frac{9/12}{3/4}=\frac{9}{12}\times\frac{4}{3}=\frac{36}{36} = 1$? Wait, no, wait, let's compute $\frac{9}{12}\div\frac{3}{4}=\frac{9}{12}\times\frac{4}{3}=\frac{36}{36} = 1$? Wait, that's not right. Wait, $\frac{3}{4}$, then $\frac{9}{12}=\frac{3}{4}$, $\frac{27}{36}=\frac{3}{4}$, $\frac{81}{108}=\frac{3}{4}$. Oh! Wait, all these fractions simplify to $\frac{3}{4}$. Wait, no, $\frac{9}{12}=\frac{3}{4}$, $\frac{27}{36}=\frac{3}{4}$, $\frac{81}{108}=\frac{3}{4}$. Wait, so actually, each fraction is equal to $\frac{3}{4}$. Wait, but let's check the multiplication factor. Wait, $\frac{3}{4}\times3=\frac{9}{4}$, no, that's not. Wait, no, the numerators: 3, 9, 27, 81... that's a geometric sequence with common ratio 3. Denominators: 4, 12, 36, 108... also geometric sequence with common ratio 3. So each fraction is $\frac{3\times3^{n - 1}}{4\times3^{n - 1}}=\frac{3}{4}$ for $n = 1,2,3,4...$. Wait, so actually, each fraction is equal to $\frac{3}{4}$. Wait, but let's check the options.

Wait, let's re - evaluate. Let's take the first fraction $\frac{3}{4}$, second $\frac{9}{12}=\frac{3}{4}$, third $\frac{27}{36}=\frac{3}{4}$, fourth $\frac{81}{108}=\frac{3}{4}$. So all fractions are equal to $\frac{3}{4}$. Now let's check each option:

Option 1: Each fraction is previous multiplied by 2. Let's check $\frac{3}{4}\times2=\frac{6}{4}=\frac{3}{2}
eq\frac{9}{12}$. So this is false.

Option 2: Next fraction. Since numerator is 81×3 = 243, denominator is 108×3 = 324. So the next fraction is $\frac{243}{324}$, and $\frac{243}{324}=\frac{3}{4}$. So this is true. Let's check other options to be sure.

Option 3: Each fraction is less than previous. But all fractions are equal to $\frac{3}{4}$, so they are equal, not less. So this is false.

Option 4: $\frac{12}{16}=\frac{3}{4}$? Wait, $\frac{12}{16}=\frac{3}{4}$? Yes! Wait, $\frac{12\div4}{16\div4}=\frac{3}{4}$. Wait, but wait, the pattern of numerators: 3, 9, 27, 81... which is 3^1, 3^2, 3^3, 3^4... and denominators: 4, 12, 36, 108... which is 4×3^0, 4×3^1, 4×3^2, 4×3^3... So the general term is $\frac{3^n}{4\times3^{n - 1}}=\frac{3}{4}$? Wait, no, the numerator for n = 1 is 3 = 3^1, n = 2 is 9 = 3^2, n = 3 is 27 = 3^3, n = 4 is 81 = 3^4. Denominator for n = 1 is 4 = 4×3^0, n = 2 is 12 = 4×3^1, n = 3 is 36 = 4×3^2, n = 4 is 108 = 4×3^3. So the nth term is $\frac{3^n}{4\times3^{n - 1}}=\frac{3}{4}$? Wait, no, $\frac{3^n}{4\times3^{n - 1}}=\frac{3}{4}$. Wait, but $\frac{12}{16}$: numerator 12 is not a power of 3 (12 = 3×4), denominator 16 is not a multiple of 4×3^{k}. Wait, but $\frac{12}{16}=\frac{3}{4}$, but according to the numerator and denominator pattern (numerator: 3,9,27,81...; denominator:4,12,36,108...), the numerators are powers of 3, denominators are 4×3^{n - 1}. So 12 is not a power of 3 (3^1 = 3, 3^2 = 9, 3^3 = 27...), so $\frac{12}{16}$ is not part of the pattern. So option 4 is false.

Option 2: Next fraction. The next numerator is 81×3 = 243, next denominator is 108×3 = 324. So the next fraction is $\frac{243}{324}$, which simplifies to $\frac{3}{4}$, same as the previous ones. So this is true.

Step2: Confirm the correct option

After analyzing all options, we see that the second option (The next fraction in the pattern is $\frac{243}{324}$) is correct.

Answer:

The next fraction in the pattern is $\frac{243}{324}$ (the second option in the list of options provided in the question).