Problem 25 - Prealgebra

Question

Find the missing term in each of the following proportions. Set up each problem like the examples in this section. Write your answers as fractions in lowest terms. $$\frac{10}{20}=\frac{20}{n}$$

Step-by-Step Solution

Answer

The missing term \( n \) is 40.

Step 1: Cross-Multiply the Proportions

To solve the proportion \( \frac{10}{20} = \frac{20}{n} \), start by cross-multiplying the terms. Multiply 10 by \( n \) and 20 by 20, setting the equations as equal: \[10 \times n = 20 \times 20\] This results in \[10n = 400\]

Step 2: Solve for n

Now, isolate \( n \). Divide both sides of the equation \( 10n = 400 \) by 10 to solve for \( n \): \[n = \frac{400}{10}\]Simplify the division on the right side: \[n = 40\]

Step 3: Verify Solution

Let's verify our solution by checking if \( \frac{10}{20} \) is indeed equal to \( \frac{20}{40} \). Simplify \( \frac{20}{40} \) by dividing both the numerator and the denominator by 20: \[\frac{20}{40} = \frac{1}{2} \] \And \( \frac{10}{20} = \frac{1}{2} \) \Therefore, both fractions are equal, confirming that the solution is correct.

Key Concepts

Cross-Multiplication in Proportions

Cross-multiplication is a handy technique used to solve proportions effortlessly. We often come across equations that look like two fractions set to be equal, such as \( \frac{10}{20} = \frac{20}{n} \). To find the unknown variable here, we apply cross-multiplication. This means we multiply diagonally: one fraction's numerator with the other fraction's denominator, and vice versa. For this exercise, we multiply 10 by \( n \), and 20 by 20. Simplifying, this gives us \( 10n = 400 \). This step lays the groundwork for isolating the unknown variable in the next steps.

• 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 the two results equal to each other to form an equation.

Cross-multiplication transforms a proportion into a simple equation that is easier to solve, often the go-to method for equations of this form. It's an essential tool for quickly solving problems involving proportions.

Simplifying Fractions

Once we have isolated the variable, like in the equation \( 10n = 400 \), the next step is simplification. Solving for \( n \) involves dividing both sides by the coefficient of \( n \), in this case, 10. This gives us \( n = \frac{400}{10} \).

Simplifying fractions means to change them into their simplest form, where the numerator and denominator have no common factors other than 1. For \( n = \frac{400}{10} \), dividing both the numerator and the denominator by their greatest common divisor simplifies the fraction: \( n = 40 \).

• Simplification involves reducing the numerator and denominator by their greatest common factor.
• This ensures the fraction is in the lowest possible terms.

Reducing fractions to their simplest form is crucial as it provides clarity and ensures the accuracy of the final answer.

Verifying Solutions in Proportions

After arriving at a solution, it’s crucial to verify it by ensuring that both sides of the original proportion are equal when substituted back in. In this exercise, once we determine that \( n = 40 \), we substitute back into the proportion to check: \( \frac{10}{20} = \frac{20}{40} \).
Simplifying the right-hand side, \( \frac{20}{40} \), into \( \frac{1}{2} \), confirms it is indeed equal to the left-hand side, \( \frac{1}{2} \). This verification confirms our solution is correct and reliable.

• Substitute the found value back into the original proportion.
• Simplify both sides to ensure they are equal.
• If they match, the solution is verified as correct.

Verifying solutions gives confidence that the answer is correct, and helps in catching any calculation errors along the way.