Problem 58 - Fundamentals of Mathematics

Question

For the following problems, determine the missing numerator or denominator. $$\frac{19}{20}=\frac{1045}{?}$$

Step-by-Step Solution

Answer

The missing denominator is 1100.

Step 1: Understand Equivalent Fractions

Equivalent fractions are fractions that express the same value even if they have different numerators and denominators. In the equation \( \frac{19}{20} = \frac{1045}{?} \), both fractions represent the same value.

Step 2: Set Up the Equation

The fractions \( \frac{19}{20} \) and \( \frac{1045}{?} \) are equivalent. To find the missing denominator, set up the equation \[ \frac{19}{20} = \frac{1045}{x} \] where \( x \) is the unknown denominator.

Step 3: Cross-Multiply to Solve for x

To solve for \( x \), cross-multiply the fractions. Multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the product of the denominator of the first fraction and the numerator of the second fraction: \[ 19 imes x = 20 imes 1045 \].

Step 4: Perform the Multiplication

Calculate the product of \( 20 \times 1045 \): \[ 20 \times 1045 = 20900 \]. Now the equation is \( 19x = 20900 \).

Step 5: Solve for x

To find \( x \), divide both sides of the equation by 19: \[ x = \frac{20900}{19} \].

Step 6: Calculate the Division

Perform the division: \[ \frac{20900}{19} = 1100 \]. Thus, \( x = 1100 \).

Key Concepts

Missing numerator or denominator

When dealing with fractions, you may often find yourself facing a problem where either the numerator or the denominator is missing. This challenge is especially interesting when you need to determine one of these missing components to equate two fractions. These types of problems revolve around understanding the concept of equivalent fractions. Equivalent fractions have different numerators and denominators, but they represent the same value. Thus, if you have a fraction equation like \( \frac{19}{20} = \frac{1045}{?} \), you're essentially trying to find the missing denominator that makes these two fractions equivalent.

To solve these, you need a strategy that allows for the determination of the unknown part. In this case, we can use the method of cross multiplication, a powerful tool that will be explained in detail below.

Cross multiplication

Cross multiplication is a technique often used to find a missing value in a fraction equation, particularly when dealing with proportionate or equivalent fractions. It involves a simple process that helps in quickly finding the unknown numerator or denominator. In our example with the equation \( \frac{19}{20} = \frac{1045}{x} \), cross multiplication lets us eliminate the fractions.

**How Cross Multiplication Works**
1. Multiply the numerator of the first fraction by the denominator of the second fraction.
2. Do the same for the denominator of the first fraction and the numerator of the second fraction.

This setup creates an equation from those cross products: \( 19 \times x = 20 \times 1045 \). Simplifying these products is then the next step, which will directly lead us to the solution for \( x \). Cross multiplication strategically equalizes the fractions, providing a clear path to determine the exact missing numerator or denominator.

Solving for variables in fractions

Once you have set up your equation via cross multiplication, the next task is to solve for the variable. This is essential in finding the value of the missing numerator or denominator. Sticking with our ongoing example, after applying cross multiplication, you arrive at the equation \( 19x = 20900 \).

**Steps to Solve for the Variable**
- Begin by isolating the variable. In the equation \( 19x = 20900 \), this means you need to get \( x \) alone on one side of the equation.
- You do this by dividing both sides of the equation by the coefficient of the variable, which in this case is 19.

After dividing, you find \( x = \frac{20900}{19} \). By calculating this division, we find \( x \) to be 1100. Thus, the missing denominator of the fraction is 1100. Solving for variables in such fraction equations is crucial as it allows you to determine exact values based on known equivalent fractions.