Problem 46 - Fundamentals of Mathematics

Question

For the following problems, determine the missing numerator or denominator. $$\frac{1}{2}=\frac{4}{?}$$

Step-by-Step Solution

Answer

The missing denominator is 8.

Step 1: Identify the Problem Structure

We are given a fraction \(\frac{1}{2}\) and we need to determine the missing denominator in the fraction \(\frac{4}{?}\) such that the two fractions are equivalent.

Step 2: Cross-Multiply to Set Up Equation

Set the two fractions equal and cross-multiply to establish an equation: \(1 \times ? = 2 \times 4\).

Step 3: Solve the Equation

Simplify the equation from the previous step: \(? = 8\).

Step 4: Conclusion

The missing denominator that makes \(\frac{4}{8}\) equivalent to \(\frac{1}{2}\) is \(8\).

Key Concepts

Missing Numerator

When we talk about the missing numerator in fractions, we refer to the top number of a fraction that we need to find to ensure two fractions are equivalent. For example, if we have the fraction \(\frac{?}{6} = \frac{4}{12}\), to find the missing numerator (?), we must determine what number makes the two fractions equal.
To solve this, we can use the method of cross-multiplication. This involves multiplying the numerator of one fraction by the denominator of the other. Performing this operation will give us an equation that we can solve for the missing number. In this case, \(? \times 12 = 4 \times 6\).
Once you do the math, you solve \(? = \frac{24}{12}\), resulting in \( ? = 2 \). Thus, the missing numerator making the fractions equivalent is 2.

Missing Denominator

Understanding a missing denominator is crucial when working with fractions, as it refers to finding the bottom number that makes two fractions equivalent. Let’s say we have: \(\frac{5}{?} = \frac{10}{20}\). To find the missing denominator, we again employ cross-multiplication.
We set up the equation by multiplying across: \(5 \times 20 = 10 \times ?\). Solving this equation gives us \(100 = 10 \times ?\), which means \(? = \frac{100}{10}\), thus \(? = 10\).
In this example, the missing denominator that will create an equivalent fraction is 10. This approach helps ensure that the value of both fractions remains the same.

Cross-Multiplication

Cross-multiplication is a powerful technique used to find missing elements in equivalent fractions. It involves multiplying the numerator of one fraction by the denominator of the other, creating an equation that maintains the equality between fractions.
Follow these steps to cross-multiply:

• Set both fractions equal to each other.
• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Multiply the numerator of the second fraction by the denominator of the first fraction.
• Set these two products equal to each other, forming an equation.

Solving this equation will reveal the missing numerator or denominator, making the fractions equivalent. For example, in our exercise, when solving for the missing denominator with \(\frac{1}{2} = \frac{4}{?}\), we used \(1 \times ? = 2 \times 4\). This straightforward calculation can solve many fraction problems involving missing numbers.

Equivalent Fractions

Equivalent fractions are fractions that may look different but have the same value when simplified. They represent the same part of a whole. Understanding this is key to solving problems that involve missing numerators or denominators.
For example, \(\frac{1}{2}\) and \(\frac{4}{8}\) are equivalent because when you simplify \(\frac{4}{8}\) by dividing both the numerator and the denominator by 4, you end up with \(\frac{1}{2}\). This property of equivalency allows us to manipulate fractions without changing their value.
When you're trying to find a missing part of a fraction, you use the idea of equivalent fractions. You set the given fraction equal to the one with the missing number. Then, use cross-multiplication to create an equation and solve for the missing value. This technique ensures the fractions remain equal, maintaining their equivalent status.