Problem 37 - Elementary Algebra

Question

Set up a proportion and then solve. If 4 out of every 5 voters support the governor, then how many of the 1,200 people surveyed support the governor?

Step-by-Step Solution

Answer

960 people support the governor.

Step 1: Understand the Proportion

We know that 4 out of every 5 voters support the governor. We can express this as a proportion: \( \frac{4}{5} \).

Step 2: Set Up the Equation

We need to find how many out of 1,200 people support the governor. Let's call this number \( x \). The proportion is represented as: \( \frac{4}{5} = \frac{x}{1200} \).

Step 3: Cross-Multiply to Clear the Fraction

To solve the equation, cross-multiply to eliminate the fraction. This gives \( 5x = 4 \times 1200 \).

Step 4: Simplify the Equation

Calculate the right side of the equation: \( 4 \times 1200 = 4800 \). So, we have \( 5x = 4800 \).

Step 5: Solve for \( x \)

Divide both sides of the equation by 5 to solve for \( x \): \( x = \frac{4800}{5} = 960 \).

Step 6: Conclusion

Thus, 960 out of the 1,200 people surveyed support the governor.

Key Concepts

Cross Multiplication

Cross multiplication is a handy process used to solve equations involving proportions. It allows us to eliminate fractions by multiplying across the equals sign in a proportion, making calculations simpler. In a proportion, you have two fractions that are equal to each other, like \( \frac{a}{b} = \frac{c}{d} \).

To cross-multiply, you take the top number of one fraction and multiply it by the bottom number of the other fraction. Then, repeat for the other pair. The equation becomes \( a \times d = b \times c \).

• This method lets us work without fractions, simplifying our calculations.
• Cross multiplication is only applicable to equations set up as proportions.
• Remember to perform the operation carefully to maintain balance in the equation.

Cross multiplying is especially useful when dealing with word problems that translate into proportions, like determining support for a governor based on a sample survey.

Solving Equations

Solving equations is about finding the value of an unknown variable that makes an equation true. Once you have an equation set up, the goal is to find what the variable equals. You do this by isolating the variable on one side of the equation.

In our context, after setting up the equation from the proportion \( \frac{4}{5} = \frac{x}{1200} \), we used cross multiplication to remove fractions, resulting in \( 5x = 4800 \). Now we need to solve for \( x \).

Here are the steps:

• Simplify each side of the equation if necessary, as we did by calculating \( 4 \times 1200 = 4800 \).
• Divide each side by the number that multiplies the variable to get \( x \) alone. For our equation, divide both sides by 5 to find \( x = \frac{4800}{5} \).
• Simplify to reach the final answer, which is \( x = 960 \).

These steps are essential for solving equations efficiently. They form the backbone of numerous mathematical solutions.

Fractions

Fractions represent parts of a whole. They're crucial when working with proportions, as they help express relationships between numbers.

In our problem, \( \frac{4}{5} \) represents the fraction of voters who support the governor. This fraction can be seen as a ratio expressing how many parts, out of 5, are in favor.

• Numerator: The top number, here "4," indicates how many parts we are interested in.
• Denominator: The bottom number, "5," shows the total parts that make up one whole.
• Fractions are everywhere in math, from ratios and proportions to complex calculations.

Understanding fractions is essential because they allow us to express and work with ratios, crucial for solving proportion problems.

Ratio and Proportion

Ratios and proportions are mathematical tools that compare quantities. A ratio is a way to show the relative sizes of two or more values. A proportion is an equation that states that two ratios are equal.

In our survey problem, the initial ratio \( \frac{4}{5} \) shows that 4 out of every 5 voters support the governor. The proportion \( \frac{4}{5} = \frac{x}{1200} \) helps us find how many people out of 1,200 support him.

• Ratios are the stepping stones to creating proportions.
• Proportions let us find unknown values when we know part of the relationship.
• They're used extensively in real-world applications, from cooking recipes to statistical predictions.

Mastering ratios and proportions allows you to solve problems logically and methodically, giving you an edge in both academics and daily life.