Express each of the following ratios in simplest form :
`6 (2)/(3)` hours : 1 day

Answer

To express the ratio of `6 (2)/(3)` hours to 1 day in simplest form, we can follow these steps:

### Step 1: Convert the mixed number to an improper fraction
The mixed number `6 (2)/(3)` can be converted to an improper fraction.

To do this:
- Multiply the whole number (6) by the denominator (3):
\( 6 \times 3 = 18 \)
- Add the numerator (2) to this result:
\( 18 + 2 = 20 \)

So, `6 (2)/(3)` hours can be expressed as:
\[ \frac{20}{3} \text{ hours} \]

### Step 2: Convert 1 day to hours
Next, we need to convert 1 day into hours.
1 day is equal to:
\[ 24 \text{ hours} \]

### Step 3: Write the ratio as a fraction
Now we can write the ratio of `6 (2)/(3)` hours to 1 day as a fraction:
\[ \frac{20/3}{24} \]

### Step 4: Simplify the fraction
To simplify the fraction, we can multiply by the reciprocal of the denominator:
\[ \frac{20}{3} \times \frac{1}{24} = \frac{20}{3 \times 24} = \frac{20}{72} \]

### Step 5: Reduce the fraction to simplest form
Now we need to reduce \( \frac{20}{72} \) to its simplest form. We can find the greatest common divisor (GCD) of 20 and 72, which is 4.

Now divide both the numerator and the denominator by 4:
\[ \frac{20 \div 4}{72 \div 4} = \frac{5}{18} \]

### Step 6: Express the ratio
Finally, we can express this as a ratio:
\[ 5 : 18 \]

Thus, the simplest form of the ratio `6 (2)/(3)` hours to 1 day is:
\[ 5 : 18 \]

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