Problem 55 - Prealgebra

Question

A promotional poster is printed on 16 -inch by \(24 \frac{1}{2}\) -inch posterboard and the space between the three sections and both top and bottom of the poster are shown. If the total length of the three sections is \(18 \frac{1}{2}\) inches, how long are each of the three equal sections? A \(2 \frac{1}{3}\) in B \(3 \frac{4}{15}\) in. C \(4 \frac{5}{8}\) in. D \(6 \frac{1}{6}\) in.

Step-by-Step Solution

Answer

The length of each section is \(6 \frac{1}{6}\) inches (Option D).

Step 1: Understand the Problem

We need to find the length of each section. The poster is divided into three equal sections, and the total length of these sections is given as \(18 \frac{1}{2}\) inches.

Step 2: Convert Mixed Fractions to Improper Fractions

Convert the mixed fraction \(18 \frac{1}{2}\) to an improper fraction: \(18 \frac{1}{2} = \frac{37}{2}\). This conversion makes it easier to divide evenly among the three sections.

Step 3: Divide to Find Each Section's Length

Divide the total length by 3 to find the length of each section: \[\frac{37}{2} \div 3 = \frac{37}{2} \times \frac{1}{3} = \frac{37}{6}\]So, each section is \(\frac{37}{6}\) inches long.

Step 4: Convert the Improper Fraction to a Mixed Number

Convert \(\frac{37}{6}\) into a mixed number:\(\frac{37}{6}\) gives a quotient of 6 and a remainder of 1:\[\frac{37}{6} = 6 \frac{1}{6}\]Thus, each section is \(6 \frac{1}{6}\) inches long.

Step 5: Verify the Answer

Check that \(6 \frac{1}{6}\) multiplied by 3 gives the original total length:\[3 \times 6 \frac{1}{6} = 3 \times \frac{37}{6} = \frac{111}{6} = \frac{111}{6} = 18 \frac{1}{2}\]The calculation is correct, confirming that each section is exactly \(6 \frac{1}{6}\) inches long.

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a proper fraction. For example, in the exercise above, we have the mixed number \(18 \frac{1}{2}\). The whole number \(18\) represents the complete units we have, while the fraction \(\frac{1}{2}\) gives additional part of the next whole unit.
This type of number is very useful in everyday life when whole numbers alone are not enough to precisely describe a quantity. Mixed numbers provide a clear and simple way to understand just how much more than a whole something is.

• Example: If you have one full apple and half of another, you have \(1 \frac{1}{2}\) apples.
• Conversion: To work mathematically, especially in operations like division, we often convert mixed numbers into improper fractions.

Simply multiply the whole number by the fraction's denominator, add the numerator, and place that over the original denominator.

Improper Fractions

An improper fraction is where the numerator (the top number) is greater than or equal to the denominator (the bottom number). For instance, in our problem, \(\frac{37}{6}\) is an improper fraction.
Improper fractions make mathematical operations like division and multiplication straightforward because they can be easily manipulated without needing anything more complicated than arithmetic with whole numbers. Here’s a quick recap on how to convert and use them effectively:

• Conversion: For example, to convert a mixed number \(6 \frac{1}{6}\) to an improper fraction, multiply the whole number \(6\) by the denominator \(6\) to get \(36\), and add the numerator \(1\), making the fraction \(\frac{37}{6}\).
• Simplifying: When completing calculations, it is often easiest to work with improper fractions first and then convert to mixed numbers if needed in the final step.

This ease of conversion is why we prefer improper fractions for complex calculations.

Division of Fractions

The division of fractions might sound tricky, but it's just like multiplication with a simple twist. The key to dividing fractions is to multiply by the reciprocal.
In the context of our exercise, we needed to divide \(\frac{37}{2}\) by \(3\). This division is the same as multiplying by the reciprocal of \(3\), which is \(\frac{1}{3}\).

• Here's how it works: Take the fraction you’re dividing, \(\frac{37}{2}\), and multiply it by the reciprocal of the divisor, which is \(\frac{1}{3}\).
• Calculation: This gives us \(\frac{37}{2} \times \frac{1}{3} = \frac{37}{6}\).

By converting a division problem into multiplication with a reciprocal, this process simplifies even complex division scenarios, making it manageable and clear, giving a correct value like \(6 \frac{1}{6}\) in our mixed number format. This approach is clean and always leads to the correct solution when tackling division in fraction form.