Question

How many hours are there in \(31 / 2\) days? ______

Short answer

There are 372 hours in \(\frac{31}{2}\) days.

Step-by-step solution

Interpret the Fraction

The exercise asks for the number of hours in \(\frac{31}{2}\) days. This means we need to first understand what \(\frac{31}{2}\) represents in terms of whole numbers and fractions. \(\frac{31}{2}\) can be expressed as a mixed number: \(15\) full days and \(\frac{1}{2}\) of a day.

Calculate Hours in Full Days

We know that there are 24 hours in a full day. To find out the number of hours in 15 full days, we multiply:\[15 \text{ days} \times 24 \text{ hours/day} = 360 \text{ hours}\]

Calculate Hours in Half a Day

Now, calculate the hours in \(\frac{1}{2}\) of a day. Since a full day has 24 hours, half a day has:\[\frac{1}{2} \times 24 \text{ hours} = 12 \text{ hours}\]

Combine Total Hours

Add the hours from the full days and the half day together to find the total hours over \(\frac{31}{2}\) days:\[360 \text{ hours} + 12 \text{ hours} = 372 \text{ hours}\]

Key concepts

Fraction to Mixed Number

When you're given a fraction like \( \frac{31}{2} \) and need to convert it to a mixed number, you're essentially breaking the fraction into a whole number and a leftover fraction.
This is especially essential when dealing with time interpretations, such as in the exercise.

• Divide the numerator (31) by the denominator (2). The result here is 15 with a remainder of 1.
• The quotient (15) is the whole number in the mixed number.
• The remainder (1) becomes the new numerator, and the denominator remains the same (2).

Putting it all together, we convert \( \frac{31}{2} \) to the mixed number 15 and \( \frac{1}{2} \). Understanding this conversion helps you easily interpret fractions into more workable numbers, just like the days into hours.

Hours Calculation

Calculating hours involves basic multiplication, which stems from knowing that one full day is equal to 24 hours.
When converting days into hours, the key step is to multiply the number of days by 24.

• For 15 days, the calculation becomes: \( 15 \times 24 = 360 \) hours.
• It's a straightforward process where you multiply the number of full days by the hours in a full day.

Handling partial days, like in this exercise, involves multiplying the fraction part of the day by 24 as well. For half a day (\( \frac{1}{2} \)), you'd compute \( \frac{1}{2} \times 24 = 12 \) hours. Adding these two results together gives the total hours in the specified number of days.

Mixed Number Conversion

Converting a mixed number back to an improper fraction or understanding it in terms of decimals can provide additional insights.
Here's how you maneuver through these conversions:

• Start with a mixed number, like 15 \( \frac{1}{2} \).
• Convert the whole number (15) into a fraction by multiplying it by the denominator (2) to get 30.
• Add the numerator of the fractional part (1) to this result, yielding 31.
• The improper fraction is thus \( \frac{31}{2} \).
• Alternatively, you convert \( \frac{1}{2} \) into a decimal, which is 0.5, and add it to 15 for a total of 15.5 days.

Mastering these conversions ensures you can switch seamlessly between mixed numbers, improper fractions, and decimals.

Basic Arithmetic

Basic arithmetic forms the backbone of solving time-related problems like this one.
Understanding operations like addition, multiplication, and division provides the foundation.

• With addition, you're combining the hours from the full days and the fractional day. For example, 360 (full day hours) + 12 (half day hours) results in 372 hours.
• Multiplication helps calculate totals from repeated additions, like determining the hours in multiple days (e.g., \( 15 \times 24 \)).
• Division is useful in breaking down fractions to convert them to mixed numbers.

By mastering these operations, you can approach a wide variety of math problems with confidence and ease.