5/4 : 2/17 64/125 : 4 9 : 3/4 9 : 4 1 3/7 : 1 1/14 3 8/19 : 5 15/38

To solve the ratios given in your question, we will simplify each ratio.

1. \( \frac{5}{4} : \frac{2}{17} \)

To express this ratio as a single fraction, we can rewrite it as:

\[ \frac{5}{4} \div \frac{2}{17} = \frac{5}{4} \times \frac{17}{2} = \frac{5 \times 17}{4 \times 2} = \frac{85}{8} \]

This simplifies to \( 85 : 8 \).

2. \( \frac{64}{125} : 4 \)

We can express this as:

\[ \frac{64}{125} \div 4 = \frac{64}{125} \div \frac{4}{1} = \frac{64}{125} \times \frac{1}{4} = \frac{64}{500} \]

This fraction simplifies to \( \frac{16}{125} \) or \( 16 : 125 \).

3. \( 9 : \frac{3}{4} \)

Expressed as a single fraction:

\[ 9 \div \frac{3}{4} = 9 \times \frac{4}{3} = \frac{36}{3} = 12 \]

Thus, the ratio simplifies to \( 12 : 1 \).

4. \( 9 : 4 \)

This is already in simplest form, and we can write it as:

\[ 9 : 4 \]

5. \( 1 \frac{3}{7} : 1 \frac{1}{14} \)

We convert the mixed numbers to improper fractions:

\[ 1 \frac{3}{7} = \frac{10}{7}, \quad 1 \frac{1}{14} = \frac{15}{14} \]

Thus, we can express the ratio as:

\[ \frac{10}{7} \div \frac{15}{14} = \frac{10}{7} \times \frac{14}{15} = \frac{10 \times 14}{7 \times 15} = \frac{140}{105} \]

This fraction simplifies to \( \frac{4}{3} \), or \( 4 : 3 \).

6. \( 3 \frac{8}{19} : 5 \frac{15}{38} \)

Again, converting mixed numbers to improper fractions:

\[ 3 \frac{8}{19} = \frac{65}{19}, \quad 5 \frac{15}{38} = \frac{155}{38} \]

Thus, the ratio becomes:

\[ \frac{65}{19} \div \frac{155}{38} = \frac{65}{19} \times \frac{38}{155} = \frac{65 \times 38}{19 \times 155} \]

Now, simplifying:

1. \( 65 = 5 \times 13 \)
2. \( 155 = 5 \times 31 \)

This leads to cancellation of 5 in the numerator and denominator:

\[ \frac{13 \times 38}{19 \times 31} = \frac{494}{589} \]

Thus, the original ratio can be expressed as \( \frac{494}{589} \) or approximately \( 1 : 1.2 \) after simplifying.

To summarize the simple ratios:

1. \( 85 : 8 \)
2. \( 16 : 125 \)
3. \( 12 : 1 \)
4. \( 9 : 4 \)
5. \( 4 : 3 \)
6. \( \frac{494}{589} \) (approximately \( 1 : 1.2 \))

Make sure to check each simplification for accuracy!