Problem 71 - Fundamentals of Mathematics

Question

Determine the missing numerator or denominator. $$ \frac{15}{16}=\frac{180}{?} $$

Step-by-Step Solution

Answer

The missing denominator is 192.

Step 1: Identify Relationship Between Fractions

Observe the given fractions \( \frac{15}{16} \) and \( \frac{180}{?} \). The problem requires the two fractions to be equivalent.

Step 2: Find Multiplication Factor

To determine the relationship between the numerators, divide the second numerator by the first numerator: \( \frac{180}{15} = 12 \). This means the numerator of the second fraction is 12 times the numerator of the first fraction.

Step 3: Apply Multiplication Factor to Denominator

Since the fractions are equivalent, the denominator must also increase by the same factor. Multiply the first denominator by this factor: \( 16 \times 12 = 192 \).

Step 4: State the Missing Denominator

Thus, the missing denominator in \( \frac{180}{?} \) is 192.

Key Concepts

Numerators

In a fraction, the numerator is the number placed above the horizontal fraction line. It indicates how many parts of a whole are under consideration. In an equivalent fraction, despite having different numerators and denominators, the overall value or size of the fraction remains the same.
Understanding numerators is a foundational concept in math. It allows us to determine how the value of a fraction changes when the numerator is altered.

• For example, consider the fractions \( \frac{15}{16} \) and \( \frac{180}{192} \).
• In these fractions, while the numerators (15 and 180) differ, the fractions are still equivalent because of their proportional relationship through the multiplication factor.

If we multiply the numerator of a fraction by a certain number, the whole fraction becomes larger or smaller depending on whether that number is greater than 1 or less than 1. However, if we wish for two fractions to remain equivalent when altering the numerator, we must adjust the denominator by the same multiplication factor.

Denominators

Denominators play a crucial role in defining the fraction's actual size by showing into how many parts the whole is divided. The denominator is the number below the fraction line. When two fractions are equivalent, their denominators must hold a consistent relationship similar to their numerators.
To find a missing denominator in equivalent fractions, recognizing the relationship between original and new numerators can guide us.

• In the problem \( \frac{15}{16} = \frac{180}{?} \), we discovered that the denominator must be multiplied by the same factor as the numerators to maintain equivalence.
• Here, the multiplication factor was found to be 12, so the denominator was calculated as \( 16 \times 12 = 192 \).

By scaling the denominator by this factor, the balance of the fraction is preserved, ensuring that both fractions express the same value despite their differing numerical parts.

Multiplication Factor

The multiplication factor is vital for determining equivalence between fractions. It's the number by which we multiply both the numerator and denominator to transform a fraction. This ensures that while the numbers in the fraction change, its actual value does not.
To find the multiplication factor:

• Compare the numerators of the equivalent fractions. For \( \frac{15}{16} \) and \( \frac{180}{?} \), divide the second numerator by the first: \( \frac{180}{15} = 12 \).
• This factor of 12 signals that each part of the first fraction is multiplied to reach the second fraction's size.

Once you determine this factor, apply it equally to the denominator. Multiplying both parts by the same factor keeps the fractions equivalent. This balance ensures the fractions have different components yet represent an identical quantity or size in their simplest form.