Problem 97 - Prealgebra

Question

A column on horse racing in the Daily News in Los Angeles reported that the horse Action This Day ran 3 furlongs in 35 \(\frac{1}{5}\) seconds and another horse, Halfbridled, went two-fifths of a second faster. How many seconds did it take Halfbridled to run 3 furlongs?

Step-by-Step Solution

Answer

It took Halfbridled 34.8 seconds to run 3 furlongs.

Step 1: Convert mixed number to improper fraction

First, convert 35 \(\frac{1}{5}\) seconds into an improper fraction. The whole number 35 can be represented as \(\frac{35 \times 5}{5} = \frac{175}{5}\). Adding the fraction part \(\frac{1}{5}\), we get the improper fraction \(\frac{176}{5}\) seconds for the time Action This Day took.

Step 2: Calculate the time for Halfbridled

Halfbridled ran two-fifths of a second faster than Action This Day. Therefore, we need to subtract \(\frac{2}{5}\) seconds from \(\frac{176}{5}\) seconds. This calculation is performed as follows: \(\frac{176}{5} - \frac{2}{5} = \frac{176 - 2}{5} = \frac{174}{5}\) seconds.

Step 3: Convert the improper fraction to a decimal

Now, convert \(\frac{174}{5}\) into a decimal to find the time in seconds. Perform the division \(174 \div 5 = 34.8\). Thus, the time it took Halfbridled is 34.8 seconds.

Key Concepts

Improper Fractions

Improper fractions are actually quite simple. They occur when the numerator, that's the top number in a fraction, is larger than the denominator, which is the bottom number. Let's look at our exercise to understand this better. In the case of the time it took for the horse Action This Day, we started with a mixed number: 35 \( \frac{1}{5} \) seconds. We needed to make this into an improper fraction for easier calculation. Here’s how it goes:

• First, multiply the whole number by the denominator of the fraction: 35 times 5 equals 175.
• Then, add the numerator: 175 plus 1 equals 176.
• Finally, put it all over the original denominator: \( \frac{176}{5} \). Now, that's our improper fraction!

Improper fractions are super handy in subtraction or addition because they make the math straightforward, without mixing whole numbers with fractions. Once you understand this conversion, solving fraction problems becomes much easier!

Subtraction of Fractions

Subtracting fractions can be a breeze if you break down the process. The key is to have a common denominator, which simplifies subtraction. In our exercise, Halfbridled needed to be faster by two-fifths of a second. That's where subtraction came in:

• Both fractions, \( \frac{176}{5} \) and \( \frac{2}{5} \), already share the same denominator.
• This simplicity means we can directly subtract the numerators: 176 minus 2 gives us 174.
• The result is an improper fraction: \( \frac{174}{5} \).

The subtraction of the numerators is straightforward because the denominators are identical. Always remember: if the denominators were different, you'd need to find a common denominator before subtracting. Fortunately, skipping this step makes the process faster in our current example!

Decimal Conversion

Converting fractions into decimals may sound tricky, but it’s mostly about division. After we subtracted the fractions for each horse, we had an improper fraction: \( \frac{174}{5} \). Here’s the conversion to a decimal:

• Divide the numerator by the denominator: 174 divided by 5.
• Doing the math, you get 34.8.

So, 34.8 seconds is how long Halfbridled took to complete the distance. The beauty of decimal conversion is that it turns fractions into more familiar numbers, which can be easier to grasp and compare. Remember, whenever you see a fraction, it can always be converted to a decimal by dividing the top number by the bottom number!