Problem 11 - Prealgebra

Question

Because there are 3 feet in every yard, the formula \(F=3 \cdot Y\) will convert \(Y\) yards into \(F\) feet. find \(F\). $$Y=2 \frac{2}{3} \text { yards }$$

Step-by-Step Solution

Answer

There are 8 feet in 2 2/3 yards.

Step 1: Convert Mixed Number to Improper Fraction

First, we need to change the mixed number \(2 \frac{2}{3}\) into an improper fraction. To do this, multiply the whole number part (2) by the denominator of the fraction (3) and add the numerator (2). This gives us \(2 \times 3 + 2 = 6 + 2 = 8\). So the improper fraction is \(\frac{8}{3}\).

Step 2: Use the Formula

Next, use the formula \(F = 3 \cdot Y\) to convert yards into feet. Substitute the improper fraction for \(Y\): \(F = 3 \cdot \frac{8}{3}\).

Step 3: Simplify the Expression

Simplify the expression \(F = 3 \cdot \frac{8}{3}\). By multiplying 3 by \(\frac{8}{3}\), we get \(F = \frac{24}{3}\).

Step 4: Divide to Find Feet

Now, divide 24 by 3 to simplify \(\frac{24}{3}\). This gives us \(F = 8\). Therefore, the number of feet is 8.

Key Concepts

Mixed Numbers

A mixed number combines both whole numbers and fractions. It is used particularly when the size exceeds a whole number but less than the next whole number. For instance, the mixed number \(2 \frac{2}{3}\) consists of the whole number 2 and the fraction \(\frac{2}{3}\). When working with mixed numbers, you'll often pretend the whole and part are multiplied together.

To convert a mixed number into an improper fraction, follow these steps:

• First, multiply the whole number by the denominator of the fraction.
• Add this result to the numerator of the fraction.
• The sum becomes the new numerator, while the denominator remains the same.

This process simplifies calculations for multiplying, adding, or subtracting with other fractions. For example, \(2 \frac{2}{3}\) converts to the improper fraction \(\frac{8}{3}\) by computing \(2 \times 3 + 2\). Such conversions are vital when performing operations in math exercises and applications.

Improper Fractions

An improper fraction is a type of fraction where the numerator is equal to or greater than the denominator. This results in a value equal to or greater than 1. Improper fractions are often more convenient in mathematical operations than mixed numbers.

For example, \(\frac{8}{3}\) is an improper fraction which represents the same value as the mixed number \(2 \frac{2}{3}\). Changing a mixed number into an improper fraction can make multiplication and division much easier.

• The numerator represents the total number of equal parts.
• The denominator still represents the number of parts in a whole.

When using improper fractions, it is crucial to understand that it provides a clearer representation suitable for algebraic manipulations. This way, they simplify arithmetic processes, especially when it comes to multiplying fractions.

Multiplication of Fractions

Multiplying fractions is a straightforward process, and it involves basic arithmetic. To multiply fractions, you simply multiply the numerators together and the denominators together.

Consider multiplying \(3\) (in its fraction form \(\frac{3}{1}\)) by the fraction \(\frac{8}{3}\):

• Multiply the numerators: \(3 \times 8 = 24\).
• Multiply the denominators: \(1 \times 3 = 3\).

This results in the fraction \(\frac{24}{3}\), which is the intermediate step before simplification.

Such operations are commonly used in various applications, including converting measurements like yards to feet. Multiplying fractions demands precision, yet with practice, students can find it quite performing.

Simplification of Fractions

Simplification of fractions aims to reduce a fraction to its smallest terms, making comprehension and further computation easier. Simplifying involves dividing both the numerator and denominator by their greatest common divisor (GCD).

In our example, the fraction \(\frac{24}{3}\) needs simplification. Since both 24 and 3 are divisible by 3, you divide the numerator by 3 to get 8, and the denominator by 3 to get 1. Hence, \(\frac{24}{3}\) simplifies to 8, which confirms the answer in simpler terms.

Key points to remember when simplifying fractions include:

• Identify the greatest common factor of the numerator and denominator.
• Perform division on both the numerator and the denominator by this factor.
• Express the fraction in its simplest form.

Simplifying fractions provides clarity in problem-solving and is crucial for maintaining accuracy across mathematical expressions and equations.