Problem 36 - Prealgebra

Question

Solve each proportion. $$\frac{a}{15}=\frac{12}{60}$$

Step-by-Step Solution

Answer

The solution is \(a = 3\).

Step 1: Understand the Proportion

The given proportion is \( \frac{a}{15} = \frac{12}{60} \). This means that the ratio of \(a\) to 15 is the same as the ratio of 12 to 60.

Step 2: Simplify the Known Ratio

Simplify the ratio on the right side of the equation. The fraction \( \frac{12}{60} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 12. This gives \( \frac{12 \div 12}{60 \div 12} = \frac{1}{5} \).

Step 3: Set the Proportions Equal

After simplifying, the proportion becomes \( \frac{a}{15} = \frac{1}{5} \). Now, these two fractions are set equal to each other again.

Step 4: Solve for the Variable

To find \(a\), cross-multiply the fractions. This gives: \(a \times 5 = 1 \times 15\). Carry out the multiplication to get \(5a = 15\).

Step 5: Isolate the Variable

Solve for \(a\) by dividing both sides of the equation \(5a = 15\) by 5, which gives: \(a = \frac{15}{5} = 3\).

Key Concepts

Simplifying Ratios

Simplifying ratios is a fundamental skill in solving proportions, and it can make complex problems easier to handle. Ratios are often presented as fractions, and simplifying them involves reducing the fraction to its lowest terms. This process requires identifying the greatest common divisor (GCD) of the numerator and the denominator.

• For example, consider the ratio \( \frac{12}{60} \). The GCD of 12 and 60 is 12.
• We simplify the fraction by dividing both numerator and denominator by 12: \( \frac{12 \div 12}{60 \div 12} = \frac{1}{5} \).

This simplification shows that the original ratio is equivalent to a more straightforward \( \frac{1}{5} \). By simplifying ratios, you not only make calculations more manageable but also clarify the relationships between the quantities involved.

Cross-Multiplication

Cross-multiplication is a powerful tool for solving proportions. It allows you to find an unknown number in a proportion by converting the proportion into a simple equation. In cross-multiplication, you multiply diagonally across the equal sign.

• Consider the proportion \( \frac{a}{15} = \frac{1}{5} \).
• Cross-multiply by multiplying each pair of opposite numerators and denominators: \( a \times 5 = 1 \times 15 \).
• This creates the equation \( 5a = 15 \).

This step translates the proportion into a solvable equation, making it straightforward to find the value of the unknown variable. Cross-multiplication is especially useful in eliminating fractions and bringing clarity to the problem.

Isolate Variable

To solve for the unknown variable after using cross-multiplication, you need to isolate the variable. Isolating the variable means getting the variable alone on one side of the equation. In our equation \( 5a = 15 \), here's how you can isolate \( a \):

• Divide both sides of the equation by 5, the coefficient of \( a \).
• Thus, \( a = \frac{15}{5} \).
• Calculate the division to find \( a = 3 \).

This process ensures that you have successfully determined the value of \( a \). Isolating the variable is crucial in solving equations and involves reversing the operations applied to the variable. This step confirms your solution and transforms an abstract equation into a clear answer.