Problem 21 - Fundamentals of Mathematics

Question

Determine the missing numerator or denominator. $$ \frac{3}{10}=\frac{45}{?} $$

Step-by-Step Solution

Answer

The missing denominator is 150.

Step 1: Understand Equivalent Fractions

Equivalent fractions represent the same value or portion of a whole. To determine the missing numerator or denominator in equivalent fractions, we use the property that both fractions must represent the same proportional relationship.

Step 2: Identify the Relationship

Look at the fractions given: \( \frac{3}{10} = \frac{45}{?} \). We need to find the number that, when combined with 45, makes the fraction equivalent to \( \frac{3}{10} \).

Step 3: Find the Scaling Factor for the Numerator

Identify how the numerator of the first fraction scales to the numerator of the second fraction. From 3 to 45, the scaling factor is \( \frac{45}{3} = 15 \). This means each part of the fraction was multiplied by 15.

Step 4: Apply the Scaling Factor to the Denominator

To maintain equivalence, multiply the denominator of \( \frac{3}{10} \) by the same scaling factor. Calculate \( 10 \times 15 = 150 \).

Step 5: Verify the Solution

By placing the calculated denominator, 150, into the fraction, we check: Does \( \frac{45}{150} = \frac{3}{10} \)? Simplify \( \frac{45}{150} \) to \( \frac{3}{10} \) by dividing both the numerator and denominator by 15.

Key Concepts

Understanding Numerator and Denominator in Fractions

In fractions, the numerator and the denominator are critical components. The numerator is the top number and it indicates how many parts of the whole are being considered. The denominator is the bottom number and shows the total number of equal parts in the whole.
For instance, in the fraction \( \frac{3}{10} \), 3 is the numerator, and 10 is the denominator. This fraction denotes 3 parts out of a total of 10 equal parts. If we understand that, comparing fractions becomes an exciting way to see how different-sized pieces relate to each other.

• The numerator tells us 'how many.' For example, in recipes, it might tell us how many cups of sugar are needed.
• The denominator answers 'of what.' It can help us understand proportions, like the parts that make up an entire pie.

Understanding numerators and denominators helps build a strong foundation for working with equivalent fractions.

Scaling Factor in Equivalent Fractions

When we talk about equivalent fractions, one important idea is the scaling factor. A scaling factor tells us how much we multiply or divide both the numerator and the denominator of a fraction.
In our example, \( \frac{3}{10} = \frac{45}{?} \), the numerator 3 is scaled up to 45. To figure out the scaling factor, divide 45 by 3:
\[\text{Scaling Factor} = \frac{45}{3} = 15\]This indicates that both the numerator and denominator are multiplied by 15 to maintain the balance.

• If you increase the numerator 3 times, you must also increase the denominator 3 times to keep both fractions equal.
• Notice how this does not change the value of the fraction, just how it's expressed.

Scaling factors are like zooming in or out in a picture. They help us understand how much bigger or smaller the pieces become while the whole stays constant.

Grasping Proportional Relationships in Fractions

The notion of proportional relationships in fractions means these fractions give the same result when simplified. They display the same ratio or division.
Proportional relationships are powerful. They help in predicting quantities or comparing several fractions, even when their numerators and denominators vary widely.
In our solved problem, \( \frac{3}{10} = \frac{45}{150} \), both share the same proportional relationship of 3 to 10, meaning they represent the same slice of a pie or segment of a bar in different-sized frames.

• Recognizing proportional relationships simplifies tasks like cooking in smaller or larger batches.
• In budgeting and time management, they allow easier adaptations when conditions change, like adjusting budgets or schedules based on circumstances.

Understanding proportionality helps in making reasoning in real-life scenarios more intuitive and straightforward.