Problem 10 - Prealgebra

Question

se the percent proportion to solve each problem. Round to the nearest tenth.10. 72 is what percent of \(160 ?\) 17 is what percent of \(85 ?\)

Step-by-Step Solution

Answer

72 is 45% of 160; 17 is 20% of 85.

Step 1: Understand the Percent Proportion Formula

The percent proportion formula is \( \frac{part}{whole} = \frac{percent}{100} \). We will use this formula to solve each problem by substituting the given values.

Step 2: Set up the first proportion for 72 out of 160

Substitute 72 for the part and 160 for the whole in the proportion formula: \( \frac{72}{160} = \frac{x}{100} \), where \( x \) is the percent we need to find.

Step 3: Solve for the Percent in the first problem

Cross-multiply to solve for \( x \). We have \( 72 \times 100 = 160 \times x \), which simplifies to \( 7200 = 160x \). Divide both sides by 160 to isolate \( x \): \( x = \frac{7200}{160} = 45 \). Hence, 72 is 45% of 160.

Step 4: Set up the second proportion for 17 out of 85

Substitute 17 for the part and 85 for the whole in the proportion formula: \( \frac{17}{85} = \frac{y}{100} \), where \( y \) is the percent we need to find.

Step 5: Solve for the Percent in the second problem

Cross-multiply to solve for \( y \). We have \( 17 \times 100 = 85 \times y \), which simplifies to \( 1700 = 85y \). Divide both sides by 85 to isolate \( y \): \( y = \frac{1700}{85} = 20 \). Hence, 17 is 20% of 85.

Key Concepts

Prealgebra Problems

Prealgebra is a foundational stage in mathematics that prepares students for algebra.
It focuses on arithmetic and introduces basic algebraic concepts. Understanding prealgebra is crucial since it lays down the groundwork for more advanced mathematical concepts, including dealing with proportions and percentages.

These "prealgebra problems" revolve around simple mathematical manipulations:

• Addition, subtraction, multiplication, and division of whole numbers, fractions, and decimals.
• Understanding of ratios and proportions to compare different quantities.
• Basic geometry, including properties of shapes and measurement.

Applying these skills can help students effectively tackle problems like finding what percent one number is of another, as shown in the provided exercise.
The goal is to help students gain familiarity and confidence in handling mathematical problems, making the transition to algebra smoother.

Cross-Multiplication

Cross-multiplication is a technique used to solve equations involving two fractions.
In a proportion problem, where two ratios are set equal, cross-multiplication helps find an unknown value, by coordinating the numerator of one fraction with the denominator of the other fraction.

Here is how it works:

• Consider two fractions set as a proportion, such as \( \frac{a}{b} = \frac{c}{d} \).
• Cross-multiplication involves multiplying "a" by "d" and "b" by "c".
• This yields an equation: \( a \times d = b \times c \), which can be solved for the unknown.

In our exercise, cross-multiplication was used to find the percentage by setting up equations like \( 72 \times 100 = 160 \times x \).
This method is efficient and reliable for solving proportions because it simplifies the equation to a straightforward multiplication and division problem.

Percent Calculations

Percent calculations are a common and practical application of mathematics in everyday life.
Percentages represent a part per hundred and are often used in discounts, interest rates, and statistical data.

To tackle percent calculations effectively, it's crucial to understand:

• A percentage formula is derived from a ratio, as shown: \( \frac{part}{whole} = \frac{percent}{100} \).
• In problems where you need to find what percent a number is of another, identifying the "part" and the "whole" correctly is key.
• Substitute these values into the percentage formula and solve for the unknown percent value, often using cross-multiplication.

In the exercise, for example, determining what percent 72 is of 160 involves setting up a proportion, \( \frac{72}{160} = \frac{x}{100} \), and solving for \( x \).
Understanding these steps can significantly help in making accurate and quick percent calculations regularly.