Question

write the improper fraction
\frac{9}{4}

\frac{11}{3}

compare the fractions.

Explanation:

Step1: Convert 9/4 to improper fraction

To convert the mixed number $4\frac{9}{?}$ (corrected: it is $4\frac{1}{4}$? No, wait, the mixed number is $4$ with fraction $\frac{9}{4}$? No, actually, $4 = \frac{16}{4}$, so $4 + \frac{9}{4}$ is wrong. Wait, the task is to write the mixed number as improper fraction: $4\frac{1}{4}$ is $\frac{17}{4}$, but no, the given is $9$ over $4$ with whole number 4? No, the correct is: mixed number $a\frac{b}{c}$ becomes $\frac{a\times c + b}{c}$. So for $4\frac{1}{4}$? No, the top is $\frac{9}{4}$ and whole number 4: wait, no, the problem is: write the mixed number (whole number + fraction) as improper fraction.

Wait, first pair: whole number 4, fraction $\frac{9}{4}$? No, no, the mixed number is $4\frac{1}{4}$? No, the box is: whole number 4, then fraction bar, so the mixed number is $4\frac{9}{?}$ no, the top is $\frac{9}{4}$, so the mixed number is $2\frac{1}{4}$? No, wait, $\frac{9}{4} = 2\frac{1}{4}$, but the whole number given is 4. Oh, wait, no: the problem says "Write the improper fraction" for the mixed number (whole number + fraction). So for the first one: whole number 4, fraction $\frac{9}{4}$? No, that can't be. Wait, no, the structure is: top box is the fraction, middle is the mixed number (whole number, fraction bar, numerator/denominator), bottom is the improper fraction.

Wait, correct:
For $\frac{9}{4}$: it is equal to $2\frac{1}{4}$, but the whole number given is 4? No, no, the first top box is $\frac{9}{4}$, middle is whole number box, fraction bar, numerator/denominator box. Wait, no, the problem is: given the mixed number (whole number + fraction), write as improper fraction.

Wait, let's re-express:

1. For the first set: Whole number = 4, fractional part = $\frac{1}{4}$? No, top is $\frac{9}{4}$. Oh! Wait, $\frac{9}{4}$ is the improper fraction, and we need to write it as mixed number, then compare? No, the right side says "Write the improper fraction", left says "Compare the fractions".

Wait, no, let's do it properly:

Step1: Convert $4$ to fraction over 4

$4 = \frac{4\times4}{4} = \frac{16}{4}$

Step2: Add to $\frac{9}{4}$ to get improper fraction

$\frac{16}{4} + \frac{9}{4} = \frac{16+9}{4} = \frac{25}{4}$

Step3: Convert $3$ to fraction over 3

$3 = \frac{3\times3}{3} = \frac{9}{3}$

Step4: Add to $\frac{11}{3}$ to get improper fraction

$\frac{9}{3} + \frac{11}{3} = \frac{9+11}{3} = \frac{20}{3}$

Step5: Compare the two improper fractions

Find common denominator: $\text{LCM}(4,3)=12$
$\frac{25}{4} = \frac{25\times3}{12} = \frac{75}{12}$
$\frac{20}{3} = \frac{20\times4}{12} = \frac{80}{12}$
Since $\frac{75}{12} < \frac{80}{12}$, so $\frac{25}{4} < \frac{20}{3}$

Wait, no, maybe the problem is: the top box is the fractional part, whole number is given, so mixed number is $4\frac{9}{4}$? No, that is an improper mixed number, so we convert to proper improper fraction.

Wait, no, maybe I misread: the top box is the numerator, whole number is the quotient, denominator is same. So $\frac{9}{4} = 2\frac{1}{4}$, but the whole number is 4. Oh, no, the problem says "Write the improper fraction" for the mixed number (whole number + fraction). So mixed number is $4\frac{9}{4}$? That is $4 + 2\frac{1}{4} = 6\frac{1}{4} = \frac{25}{4}$. Yes, that's right.

Same for the other: $3\frac{11}{3} = 3 + 3\frac{2}{3} = 6\frac{2}{3} = \frac{20}{3}$.

Now, compare them:

Step1: Convert $4\frac{9}{4}$ to improper fraction

$4\frac{9}{4} = \frac{4\times4 + 9}{4} = \frac{16+9}{4} = \frac{25}{4}$

Step2: Convert $3\frac{11}{3}$ to improper fraction

$3\frac{11}{3} = \frac{3\times3 + 11}{3} = \frac{9+11}{3} = \frac{20}{3}$

Step3: Find common denominator (12)

$\frac{25}{4} = \frac{25\times3}{12} = \frac{75}{12}$, $\frac{20}{3} = \frac{20\times4}{12} = \frac{80}{12}$

Step4: Compare the two fractions

$\frac{75}{12} < \frac{80}{12}$ so $\frac{25}{4} < \frac{20}{3}$

Step1: Convert $\frac{9}{4}$ to mixed number

$\frac{9}{4} = 2\frac{1}{4}$ (whole number 2, numerator 1, denominator 4)

Step2: Convert $\frac{11}{3}$ to mixed number

$\frac{11}{3} = 3\frac{2}{3}$ (whole number 3, numerator 2, denominator 3)

Step3: Find common denominator for comparison

$\frac{9}{4} = \frac{27}{12}$, $\frac{11}{3} = \frac{44}{12}$

Step4: Compare the two fractions

$\frac{27}{12} < \frac{44}{12}$ so $\frac{9}{4} < \frac{11}{3}$

This makes more sense, because $\frac{9}{4}=2.25$, $\frac{11}{3}\approx3.666$, so 2.25 < 3.666.

Step1: $\frac{9}{4}$ to mixed number

$\frac{9}{4} = 2 + \frac{1}{4} = 2\frac{1}{4}$

Step2: $\frac{11}{3}$ to mixed number

$\frac{11}{3} = 3 + \frac{2}{3} = 3\frac{2}{3}$

Step3: Compare $\frac{9}{4}$ and $\frac{11}{3}$

Common denominator 12: $\frac{9}{4}=\frac{27}{12}$, $\frac{11}{3}=\frac{44}{12}$

Step4: Compare the values

$\frac{27}{12} < \frac{44}{12}$

Answer:

Improper fractions: $\frac{25}{4}$ and $\frac{20}{3}$
Comparison: $\frac{25}{4} < \frac{20}{3}$

Wait, but let's correct the mixed number: actually, the problem is that the top box is the fraction, and the middle is the mixed number (whole number, fraction bar, so the mixed number is $2\frac{1}{4}$ for $\frac{9}{4}$, but the whole number given is 4. Oh, I think I misread: the top box is the improper fraction, middle is the mixed number (split into whole number, numerator, denominator), bottom is... no, the right side says "Write the improper fraction", so given the mixed number (whole number + fraction), write as improper fraction.

Ah! Now I see: the top box is the fractional part (numerator/denominator), middle is the mixed number (whole number, fraction bar, so the mixed number is $4 + \frac{1}{4}$? No, the top box is $\frac{9}{4}$, so the mixed number is $2\frac{1}{4}$, so whole number 2, numerator 1, denominator 4. Then improper fraction is $\frac{9}{4}$.

Same for the other: $\frac{11}{3} = 3\frac{2}{3}$, so whole number 3, numerator 2, denominator 3, improper fraction $\frac{11}{3}$.

Then compare $\frac{9}{4}$ and $\frac{11}{3}$: