Problem 72 - Fundamentals of Mathematics

Question

Determine the missing numerator or denominator. $$ \frac{21}{22}=\frac{336}{?} $$

Step-by-Step Solution

Answer

The missing denominator is 352.

Step 1: Understanding the Problem

We are given a proportion: \( \frac{21}{22} = \frac{336}{?} \). This means the two fractions are equivalent and we need to determine the missing denominator in the second fraction.

Step 2: Cross Multiplication

To solve for the missing denominator, use cross multiplication, which states that for two equal fractions \( \frac{a}{b} = \frac{c}{d} \), the equation \( a \times d = b \times c \) holds true. Apply this to find the unknown: \( 21 \times ? = 22 \times 336 \).

Step 3: Multiplying Known Values

Calculate the product on the right side of the equation: \( 22 \times 336 \). This equals \( 7392 \). Thus, the equation becomes \( 21 \times ? = 7392 \).

Step 4: Solving for the Unknown

To find the unknown number, divide both sides of the equation by 21: \( ? = \frac{7392}{21} \).

Step 5: Division Calculation

Perform the division \( 7392 \div 21 \) to find the missing number. This calculation results in \( 352 \).

Key Concepts

Equivalent Fractions

Equivalent fractions might sound like a complex mathematical term, but it simply means two fractions that represent the same part of a whole. In our problem, we are given two fractions: \( \frac{21}{22} \) and \( \frac{336}{?} \). These fractions are said to be equivalent because they represent the same proportion. This means that although the numbers are different, their values are the same.To identify equivalent fractions, you can multiply or divide the numerator and denominator of a fraction by the same non-zero number. For instance, if you take \( \frac{1}{2} \) and multiply both the numerator and denominator by 2, you get \( \frac{2}{4} \), which is equivalent to \( \frac{1}{2} \).Understanding equivalent fractions is important because it helps us solve problems like the one given, where elements of the fraction are missing.

Cross Multiplication

Cross multiplication is a key technique used to solve equations involving proportions and helps in finding missing values in equivalent fractions. It involves multiplying the numerator of one fraction by the denominator of the other to create an equation.In our example, we have \( \frac{21}{22} = \frac{336}{?} \). Following the cross multiplication rule, we multiply across the fractions in an 'X' pattern: the numerator of the first fraction (21) is multiplied by the denominator of the second fraction (which we need to find), and the denominator of the first fraction (22) is multiplied by the numerator of the second fraction (336).

• This creates an equation: \( 21 \times ? = 22 \times 336 \).
• By calculating \( 22 \times 336 \), we obtain 7392.
• The equation now reads \( 21 \times ? = 7392 \).

Cross multiplication allows us to work with straightforward multiplications to eventually isolate and solve for the unknown.

Solving for Unknowns

Finding the missing number in a mathematical equation is often referred to as solving for the unknown. In our problem, the missing number is part of a fraction's denominator, and we have set up the equation through cross multiplication: \( 21 \times ? = 7392 \).Solving for unknowns involves rearranging the equation to isolate the unknown on one side. To get the unknown by itself:

• Divide both sides of the equation by 21.
• This simplifies to \( ? = \frac{7392}{21} \).
• Carrying out the division, we find \( ? = 352 \).

By simplifying the equation, the missing denominator is 352. Solving for unknowns is a fundamental skill in algebra and helps in numerous real-world applications.

Numerator and Denominator

Every fraction has two key parts: the numerator and the denominator. The numerator is the top number and shows how many parts we have. The denominator is the bottom number and shows the total number of equal parts the whole is divided into.In our equation \( \frac{21}{22} = \frac{336}{?} \), the numerators are 21 and 336. The denominator for the first fraction is 22, while the second denominator is what we're trying to find. Understanding these components is crucial for working with fractions as they define the fraction's quantity and its relation to the whole.

• Numerator (top number): Indicates the count of parts.
• Denominator (bottom number): Shows the total, defining the size of each part.

Knowing how numerators and denominators work helps in setting up equations properly, allows for cross multiplication, and ultimately solving problems involving fractions.