Problem 12 - Prealgebra

Question

Because there are 3 feet in every yard, the formula \(F=3 \cdot Y\) will convert \(Y\) yards into \(F\) feet. find \(F\). $$Y=6 \frac{1}{3} \text { yards }$$

Step-by-Step Solution

Answer

There are 19 feet in 6 1/3 yards.

Step 1: Understanding the Problem

We need to convert 6 1/3 yards into feet using the given formula: \(F = 3 \cdot Y\).

Step 2: Converting Mixed Number to Improper Fraction

First, convert the mixed number \(6 \frac{1}{3}\) into an improper fraction. \(6 \frac{1}{3} = \frac{19}{3}\).

Step 3: Substituting into the Formula

Substitute \(Y = \frac{19}{3}\) into the formula \(F = 3 \cdot Y\).

Step 4: Calculating the Feet

Perform the multiplication: \[ F = 3 \cdot \frac{19}{3} = 19 \].

Step 5: Concluding the Solution

Thus, the number of feet \(F\) is 19.

Key Concepts

Mixed Numbers

A mixed number combines a whole number with a fraction, like 6 1/3. This representation is common when dealing with measurements, as it provides a more intuitive understanding. To dig deeper:

• The whole number represents complete units, like 6 yards in this case.
• The fraction, 1/3, indicates an additional part of the next whole unit.

However, calculations are often simplified by converting mixed numbers into improper fractions because it makes mathematical operations like multiplication easier to perform. This conversion is key to solving many types of problems efficiently.

Improper Fractions

An improper fraction is a fraction where the numerator is larger than the denominator, like \(\frac{19}{3}\). To convert a mixed number into an improper fraction:

• Multiply the whole number by the denominator of the fraction part.
• Add the result to the numerator of the fraction.
• Place this sum over the original denominator.

For example, converting 6 1/3 into an improper fraction involves: \(6 \times 3 + 1 = 19\), leading to \(\frac{19}{3}\). Improper fractions are more predictable in calculations, especially when using formulas.

Multiplication

Multiplication of fractions involves straightforward steps. When multiplying a fraction by a number, like in the formula \(3 \times \frac{19}{3}\), you can simplify first:

• Multiply the numerators together.
• Multiply the denominators together.
• In our case: \(3 \times \frac{19}{3}\) simplifies directly to \(19\) since the 3s cancel out.

This simplification is possible because each element of a multiplication can often be adjusted to make calculations easier. Recognizing such opportunities for simplification is crucial for quick and precise problem-solving.

Algebraic Formula

An algebraic formula is a set of operations or rules that you can use to find unknown values. In this exercise, the formula \(F = 3 \cdot Y\) shows how to convert yards into feet:

• \(F\) represents the number of feet.
• \(Y\) is the given number of yards.
• The multiplier 3 accounts for the number of feet per yard.

Using algebraic formulas allows us to solve for unknowns by systematic substitution and calculation. Here, converting a measurement (yards) to another unit (feet) becomes seamless and swift by using formulas, highlighting their utility in simplifying complex tasks.