Problem 61 - Fundamentals of Mathematics

Question

For problems 61-72, determine the missing numerator or denominator. $$\frac{3}{7}=\frac{?}{35}$$

Step-by-Step Solution

Answer

The missing numerator is 15.

Step 1: Identify the Relationship

In the given problem, we have \( \frac{3}{7} = \frac{?}{35} \). Our task is to identify the missing numerator. To do this, first observe how the denominator has changed from 7 to 35.

Step 2: Finding the Multiplier

Determine the factor by which the original denominator has been multiplied: \( 7 \times x = 35 \). Divide 35 by 7 to find \( x \): \( x = \frac{35}{7} = 5 \). This tells us that the denominator has been multiplied by 5.

Step 3: Apply the Same Multiplier to the Numerator

To preserve the equality of the two fractions, apply the same multiplier to the numerator: \( 3 \times 5 = 15 \). Thus, the missing numerator is 15.

Step 4: Verify the Proportions

To ensure the fractions are equal, compare them: \( \frac{3}{7} = \frac{15}{35} \). Simplify \( \frac{15}{35} \) by dividing both the numerator and the denominator by 5, resulting in \( \frac{3}{7} \). The fractions are indeed equal.

Key Concepts

Numerator

In fractions, the numerator is the top number that indicates how many parts out of a whole are being considered. In the fraction \( \frac{3}{7} \), 3 is the numerator. This shows that we are talking about 3 parts of a whole that is divided into 7 equal parts.
The numerator plays a crucial role as it shows the actual quantity or count of components out of the total number provided by the denominator. The numerator can be any integer, representing a count for that particular situation. Anytime you see a fraction and wonder 'how much' or 'how many', look to the numerator.
When finding a missing numerator, as in our example, you must ensure that whatever adjustments are made (multiplication, addition, etc.) reflect a similar change to maintain the equivalent fraction when compared with another fraction.

Denominator

The denominator is the bottom number in a fraction and it shows how many equal parts the whole is divided into. For example, in \( \frac{3}{7} \), the denominator is 7. This indicates that the entire object or group is split into 7 equal portions.
The denominator determines the size of each part, providing context to the numerator. It is a critical component because it ensures that we understand the scale or magnitude of what each fraction part represents.
When the denominator changes, as it did from 7 to 35 in our example, you must analyze how that conversion affects the fraction. This is often done through multiplication or division, which is vital for finding an equivalent fraction, as you see in problems requiring you to find a missing numerator or denominator to keep the fractions equal.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same quantity or value. They may look different, but after simplifying or through multiplication or division, they reflect the same part of a whole.
For example, \( \frac{3}{7} \) and \( \frac{15}{35} \) are equivalent fractions. Even though their numerators and denominators have changed, these fractions signify the same quantity. You can check this by simplifying \( \frac{15}{35} \) to \( \frac{3}{7} \), achieved by dividing both the numerator and the denominator by 5.
Understanding how to find equivalent fractions is essential when comparing them or when finding missing numerators or denominators, as in our exercise. Both the numerator and the denominator must change in a consistent manner to maintain equality between the fractions.

Proportions

Proportions are equations that signify two ratios or fractions are equal. In mathematical terms, two fractions are proportional if their cross-products are equal.
In our problem, we worked with the proportion \( \frac{3}{7} = \frac{15}{35} \). To check if these two fractions are proportions of one another, you multiply the numerator of one fraction by the denominator of the other and compare the results. They should be the same if the fractions are equivalent.
Understanding proportions helps in solving problems where you're required to find missing elements in fractions, ensuring that the overall equality is maintained. Proportional reasoning is a powerful tool in mathematics that applies to various real-world situations, from scaling recipes to understanding similar triangles in geometry.