Problem 11 - Prealgebra

Question

Use the percent proportion to solve each problem. Round to the nearest tenth. $$ 17 \text { is what percent of } 85 ? $$

Step-by-Step Solution

Answer

17 is 20% of 85.

Step 1: Set up the percent proportion

The percent proportion can be expressed as \( \frac{a}{b} = \frac{x}{100} \), where \( a \) is the part (17), \( b \) is the whole (85), and \( x \) is the percent we need to find. Substituting the given values, we have \( \frac{17}{85} = \frac{x}{100} \).

Step 2: Cross multiply to solve for x

Cross multiply to eliminate the fractions: \( 17 \times 100 = 85 \times x \). This simplifies to \( 1700 = 85x \).

Step 3: Solve for x

To find \( x \), divide both sides by 85: \( x = \frac{1700}{85} \).

Step 4: Simplify the division

Perform the division: \( x = 20 \).

Step 5: Conclude and round the answer

Since \( x = 20 \), 17 is 20% of 85. No rounding is necessary as the result is already a whole number.

Key Concepts

Cross Multiplication

Cross multiplication is a crucial technique used to solve equations, especially when dealing with proportions. A proportion is an equation that shows two ratios are equal. In a problem like "17 is what percent of 85?", we set up the proportion as \( \frac{17}{85} = \frac{x}{100} \), where 17 is the part, 85 is the whole, and \( x \) is the unknown percent.To cross multiply, you multiply diagonally across the equal sign. This means multiplying 17 by 100, and 85 by \( x \). The equation formed from cross multiplication would be:

• 17 times 100 equals 1700
• 85 times \( x \) equals 85\( x \)

You then equate the two products to form the equation: 1700 = 85\( x \). Cross multiplication effectively helps in eliminating the fractions and makes it easier to solve for the unknown variable.

Solving for x

Once you have used cross multiplication, the next important step is solving for the unknown variable, \( x \). This means isolating \( x \) on one side of the equation, which in our case is 1700 = 85\( x \).To solve for \( x \), you need to perform division to undo the multiplication by 85. So you divide both sides of the equation by 85:

• \( x = \frac{1700}{85} \)

This step simplifies the equation, allowing you to find the value of \( x \). Solving for \( x \) helps in finding out what percent 17 is of 85. It's essential to perform these calculations carefully to ensure accuracy in getting the correct percentage.

Division in Fractions

Division in fractions is a fundamental arithmetic operation often used in solving equations like proportions. In the context of our exercise, after the cross multiplication step, we end up with the equation \( 1700 = 85x \).To isolate \( x \), you perform division. You divide 1700 by 85, which is the final step, as follows:

• Perform the division: \( x = \frac{1700}{85} \)
• This yields \( x = 20 \)

After performing the division, you conclude that 17 is 20% of 85. Understanding division in fractions is crucial not only in percentage problems but also in various mathematical operations, allowing you to simplify expressions and solve equations efficiently.