Problem 22 - Intermediate Algebra

Question

Solve each proportion. $$ \frac{4}{y}=\frac{6}{27} $$

Step-by-Step Solution

Answer

The solution of the proportion is \( y = 18 \).

Step 1: Cross Multiply

To solve the proportion \( \frac{4}{y} = \frac{6}{27} \), we start by cross multiplying. This means we multiply the numerator of the first fraction by the denominator of the second fraction, and the numerator of the second fraction by the denominator of the first fraction. Set these two products equal to each other:\[ 4 \times 27 = 6 \times y \] which simplifies to \[ 108 = 6y \].

Step 2: Solve for y

Next, solve for \( y \) by isolating it on one side of the equation. Divide both sides of the equation by 6:\[ y = \frac{108}{6} \].

Step 3: Simplify the expression

Simplify the expression \( \frac{108}{6} \) to find the value of \( y \). Calculate the division:\[ y = 18 \].

Key Concepts

Cross Multiplication

Cross multiplication is a fantastic technique used when solving proportions. A proportion is an equation that states two ratios are equal. You can identify these ratios by their fraction form, like in the given exercise: \( \frac{4}{y} = \frac{6}{27} \). To apply cross multiplication:

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Then, multiply the numerator of the second fraction by the denominator of the first fraction.
• Set these two products equal to each other.

For instance, with our proportion, you would do:
\[4 \times 27 = 6 \times y\]This results in \( 108 = 6y \).Cross multiplication is especially handy because it simplifies the process of solving proportions to basic algebra. Using this method helps avoid errors compared to other techniques, and allows you to solve for the unknown easily.

Basic Algebra

Basic algebra is the foundation for solving equations like these. After we've applied cross multiplication and obtained an equation such as \( 108 = 6y \), our goal is to find the value of the unknown variable, which is \( y \) in this case.To do this:

• Isolate the variable on one side of the equation. You can achieve this by performing operations such as addition, subtraction, multiplication, or division to both sides of the equation.
• In this example, you need to get \( y \) alone, so you will divide both sides by 6:

\[y = \frac{108}{6}\]These steps boil down to a simple motto of solving for \( y \): perform the same operation to both sides of the equation. This keeps the equation balanced and is a key principle in algebraic manipulation.

Fraction Simplification

Fraction simplification is the final step in solving our proportion. After determining that:\[y = \frac{108}{6}\]We simplify this fraction to find the most straightforward answer.Here's how you can simplify a fraction:

• Check if you can divide both the numerator and the denominator by the same number, also called the greatest common divisor.
• In our case, 108 and 6 can both be divided evenly by 6.

Performing the division:
\[y = \frac{108 \div 6}{6 \div 6} = \frac{18}{1} = 18\]This step ensures you get the simplest form of your variable. Simplifying fractions helps in making answers more comprehensible and neatly presented.