Problem 85 - Beginning and Intermediate Algebra

Question

Yelena Isinbaeva currently holds the women's pole vault world record at \(5 \frac{1}{50}\) meters The men's pole vault world record is currently held by Sergei Bubka, at \(1 \frac{3}{25}\) meters higher than the women's record. What is the current men's pole vault record?

Step-by-Step Solution

Answer

The men's pole vault record is 6 \(\frac{7}{50}\) meters.

Step 1: Convert Yelena's Record

First, convert Yelena Isinbaeva's pole vault record from a mixed number to an improper fraction. Yelena's record is reported as \(5 \frac{1}{50}\) meters. To convert, take the whole number (5), multiply it by the denominator of the fraction (50), and then add the numerator (1). This gives:\[5 \times 50 + 1 = 250 + 1 = 251\]So, her record as an improper fraction is \(\frac{251}{50}\) meters.

Step 2: Convert Sergei's Height Difference

Next, convert the height difference of \(1 \frac{3}{25}\) meters from a mixed number to an improper fraction. Multiply the whole number (1) by the denominator of the fraction (25) and add the numerator (3):\[1 \times 25 + 3 = 25 + 3 = 28\]So, the height difference as an improper fraction is \(\frac{28}{25}\) meters.

Step 3: Add the Records

To find the men's record, add the improper fractions for Yelena's record and Sergei's height difference:\[\frac{251}{50} + \frac{28}{25}\]To add these fractions, they must have a common denominator. The least common denominator of 50 and 25 is 50. Convert \(\frac{28}{25}\) to an equivalent fraction with the denominator of 50:\[\frac{28}{25} \times \frac{2}{2} = \frac{56}{50}\]Now, add the fractions:\[\frac{251}{50} + \frac{56}{50} = \frac{307}{50}\]

Step 4: Convert Back to a Mixed Number

Convert \(\frac{307}{50}\) back to a mixed number. Divide 307 by 50:- 307 divided by 50 is 6 with a remainder of 7.Therefore, the mixed number is:\[6 \frac{7}{50}\] meters.

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This means the fraction represents a value equal to or greater than 1. Improper fractions are handy when dealing with complex mathematical operations because they are easier to calculate with than mixed numbers.
For example, if you have a mixed number like \(5 \frac{1}{50}\), converting it to an improper fraction involves:

• Multiplying the whole number by the denominator: \(5 \times 50 = 250\)
• Adding the numerator: \(250 + 1 = 251\)
• The result is the improper fraction: \(\frac{251}{50}\)

This conversion allows for straightforward addition or subtraction with other fractions.

Mixed Numbers

Mixed numbers consist of a whole number and a proper fraction, such as \(5 \frac{1}{50}\). They are often used because they are easier to interpret when dealing with real-world measurements but can be less practical for calculations.
To convert a mixed number to an improper fraction:

• Multiply the whole number by the fraction's denominator.
• Add the numerator of the fraction to this product.
• Place the result over the original denominator.

This method allows for seamless transitions between easily readable numbers and computationally friendly ones. For example, \(5 \frac{1}{50}\) becomes \(\frac{251}{50}\), which can be easily used in mathematical calculations.

Adding Fractions

Adding fractions requires the fractions to have the same denominator. If they don't, you'll need to convert one or both fractions.
For example, if you need to add \(\frac{251}{50}\) and \(\frac{28}{25}\), you must:

• Find a common denominator, in this case, 50.
• Convert \(\frac{28}{25}\) to \(\frac{56}{50}\), so both fractions share the same denominator.
• Once the fractions have the same denominator, add the numerators: \(251 + 56 = 307\).

The result is \(\frac{307}{50}\).
By using a common denominator, adding fractions becomes straightforward.

Least Common Denominator

The least common denominator (LCD) is the smallest number that can be used as a common denominator for two or more fractions. Finding the LCD is essential when you need to add, subtract, or compare fractions that have different denominators.
Here's a step-by-step way to find the LCD:

• List the multiples of each denominator.
• Identify the smallest multiple that is common to both lists.
• This number becomes the LCD.

For example, with the denominators 50 and 25, the multiples are:

• 50: 50, 100, 150, 200, etc.
• 25: 25, 50, 75, 100, etc.

The LCD here is 50, allowing you to convert and add fractions accurately.