Problem 6 - Prealgebra

Question

Estimate. Explain which method you used to estimate. $$34 \% \text { of } 24$$

Step-by-Step Solution

Answer

34% of 24 is approximately 8.

Step 1: Understand the Problem

We need to estimate the value of 34% of 24. Estimation requires us to find an approximate value rather than the exact one, which usually involves rounding number calculations to make the estimate easier.

Step 2: Round the Numbers

To simplify the calculation, we round both 34% and 24 to numbers that are easier to work with. We approximate 34% as 33%, which is closer to a whole number, 1/3, and 24 as 25.

Step 3: Convert Percentage to Fraction

Convert 33% to a fraction. 33% is equivalent to the fraction \( \frac{33}{100} \). For our estimate, we'll use \( \frac{1}{3} \), as it is a close representation.

Step 4: Calculate the Estimated Product

Now, multiply the fraction \( \frac{1}{3} \) by the approximated value of 25. This gives us \( \frac{1}{3} \times 25 \), which simplifies to approximately 8.33. For estimation purposes, we can round this to 8.

Step 5: Conclude the Estimation

Therefore, estimating 34% of 24 using rounding and simple fractions, we conclude that it is approximately 8.

Key Concepts

Rounding Numbers

Rounding is a valuable mathematical skill that simplifies numbers, making calculations easier and quicker. Essentially, rounding is about adjusting a number to a nearby and simpler value. In estimation, rounding helps us to work with numbers that are less cumbersome.

• **Why Round?** Rounding a number allows us to focus on the important parts and eliminate excessive insignificant digits. It makes mental calculations or rough estimates more manageable.
• **How to Round Numbers?** To round a number like 34% to a simpler form, consider the closest tens or logical approximation. For instance, here we choose 33%, which relates to \( \frac{1}{3} \), making the calculation straightforward.
• **Steps for Rounding** Rounding typically involves looking at the digit to the right of your desired place value. For numbers ending in 5 or higher, round up; otherwise, round down.

These steps ensure the numbers used in further calculations are more convenient.

Fraction to Percentage Conversion

Converting between percentages and fractions is a useful skill in estimation and various mathematical contexts. A percentage expresses a number as a fraction of 100. This makes the concept of percentages easier to visualize and compute with.

• **Understanding Percentages** Since a percentage is simply a part of 100, converting it means finding out how it would express as a fraction with 100 as the denominator.
• **Conversion Process** To convert 33% to a fraction, write this as \( \frac{33}{100} \). This is the exact equivalent of the percentage in fractional form.
• **Simplifying for Estimation** In estimation, finding a simpler fraction helps. For instance, 33% is approximated as \( \frac{1}{3} \), which is simpler to handle in calculation, especially when combined with rounded numbers like 25.

This conversion is key to bridging the gap between fractions and percentages in calculations.

Simple Fractions

A foundational concept in math is working with simple fractions. Simple fractions are ratios of two integers, typically one numerator and one denominator. These fractions offer a straightforward way of representing numbers, making complex calculations simpler.

• **Importance in Estimation** In estimation, simple fractions let us replace complicated percentages or larger numbers with easier components.
• **Example in Use** In our exercise, using \( \frac{1}{3} \) instead of 33% simplifies multiplying by the rounded number 25. This fraction is simpler to compute mentally or on paper.
• **Simplification** Simplifying fractions like \( \frac{33}{100} \) to a more manageable form ensures calculations stay manageable during estimations.

Opting for simple fractions wherever possible keeps estimations and calculations efficient and less error-prone.