Problem 33 - 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. $$\frac{0.3}{0.18}=\frac{n}{0.6}$$

Step-by-Step Solution

Answer

The missing term, \( n \), is 1 (or \( \frac{1}{1} \) as a fraction).

Step 1: Set Up the Equation

The given proportion is \( \frac{0.3}{0.18} = \frac{n}{0.6} \). This tells us that the two fractions are equivalent, so we can set them equal to each other to solve for \( n \).

Step 2: Cross-Multiply

To solve for \( n \), cross-multiply the fractions. This means multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa. This gives:\[ 0.3 \times 0.6 = 0.18 \times n \]

Step 3: Simplify the Equation

Calculate the left side of the equation:\[ 0.3 \times 0.6 = 0.18 \]Thus, the equation becomes:\[ 0.18 = 0.18 \times n \]

Step 4: Solve for n

To isolate \( n \), divide both sides of the equation by 0.18:\[ n = \frac{0.18}{0.18} = 1 \]

Step 5: Write the Answer as a Fraction

The value of \( n \) as a fraction is simply \( 1 \), which is already in its lowest term as \( \frac{1}{1} \).

Key Concepts

Cross-Multiplication

Cross-multiplication is a powerful tool used in solving proportions. A proportion is simply an equation that states two ratios or fractions are equivalent. For example, if you have the proportion \( \frac{a}{b} = \frac{c}{d} \), you can find an unknown value by cross-multiplying.

This means you multiply the numerator of one fraction by the denominator of the other fraction. So, in this example, you would multiply \( a \times d \) and \( b \times c \). This results in an equation: \( a \times d = b \times c \). You can use this step to easily solve for any unknown term within the proportion.

Using cross-multiplication simplifies the comparison between the ratios and helps establish equality, making it an efficient method to find an unknown variable in proportions.

Fractions in Lowest Terms

When dealing with proportions, it is essential to write your answer as fractions in their lowest terms. Simplifying a fraction means reducing it to a form where the numerator and the denominator have no common factors other than 1. This makes the fraction easier to understand and compare.

To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator. Then, you divide both by this GCD.

For example, the fraction \( \frac{8}{12} \) can be simplified. The GCD of 8 and 12 is 4, so divide both numbers by 4, resulting in \( \frac{2}{3} \). This is the fraction in its lowest terms.

In the original solution, the answer was \( n = 1 \), which is already simplified since the only factors are 1 and itself.

Solving Equations

Solving equations involves finding the value of the unknown variable, in this case, \( n \). Once you've set up the problem using cross-multiplication, the next step is to simplify the equation to isolate the variable.

After cross-multiplying, you might end up with an equation like \( 0.18 = 0.18 \times n \). To solve for \( n \), divide both sides by the coefficient of \( n \) (which is the number multiplied by \( n \)). This simplifies the equation to \( n = 1 \).

Always remember to check your calculations to ensure accuracy, and if needed, simplify your final answer to a fraction in the lowest terms. This provides clarity and precision in your solutions.