Problem 98 - Intermediate Algebra

Question

Write a numerical statement to represent the problem. Then simplify the numerical expression to answer the question. After dieting for 30 days, Ignacio has lost 18 pounds. What number describes his average weight change per day?

Step-by-Step Solution

Answer

Ignacio's average weight change per day is 0.6 pounds.

Step 1: Understanding the Problem

Ignacio has lost a total of 18 pounds over the course of 30 days. To find his average weight change per day, we need to divide the total weight loss by the number of days.

Step 2: Write the Numerical Statement

To find the average weight change per day, write the numerical statement as: \[\frac{18}{30}\] This represents dividing the total pounds lost by the total number of days.

Step 3: Simplify the Fraction

To simplify the fraction \(\frac{18}{30}\), find the greatest common divisor of 18 and 30, which is 6. Divide both the numerator and the denominator by 6 to get the simplified expression:\[\frac{18 \div 6}{30 \div 6} = \frac{3}{5}\]

Step 4: Convert the Simplified Fraction to Decimal

Convert \(\frac{3}{5}\) to a decimal by dividing 3 by 5. This gives us:\[3 \div 5 = 0.6\] Thus, the average weight change per day is 0.6 pounds.

Key Concepts

Understanding Numerical Expressions

Numerical expressions are just like mathematical sentences that express calculations using numbers and operation symbols. In the exercise, to find Ignacio's average weight change per day, a numerical expression is written. The numerical expression here involves division, which is denoted by the fraction \( \frac{18}{30} \). This fraction indicates that 18 pounds are divided by 30 days.
Creating a numerical expression allows you to model real-world problems mathematically, enabling you to solve them systematically. Writing and interpreting these expressions can look different based on the context they represent. The key is understanding that in a numerical expression:

• Numbers signify the values you're working with.
• Operations (like addition, subtraction, multiplication, division) show how these values interact.

In our problem, the division reflects the 'average weight change per day', simplifying 18 pounds distributed evenly across 30 days.

Simplifying Fractions

Simplifying fractions is about reducing the fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. For the fraction \( \frac{18}{30} \), both numbers can be divided evenly by their greatest common divisor (GCD).
Here, the GCD of 18 and 30 is 6. Reducing the fraction means you divide both the numerator (18) and the denominator (30) by 6:

• \( 18 \div 6 = 3 \)
• \( 30 \div 6 = 5 \)

So, \( \frac{18}{30} \) simplifies to \( \frac{3}{5} \). Simplifying fractions helps in understanding and comparing fractions more easily. This also streamlines calculations and provides precise solutions while maintaining equivalency.

Decimal Conversion

Converting fractions to decimals offers a different perspective on numeric values, often making them easier to interpret and use in diverse mathematical contexts. In the case of the simplified fraction \( \frac{3}{5} \), conversion to decimal involves straightforward division.
You take the numerator (3) and divide it by the denominator (5):

• \( 3 \div 5 = 0.6 \)

The result, 0.6, is a decimal representation of the fraction, illustrating the same value in a different form. Decimal conversion is particularly helpful:

• In real-life calculations where precise decimal values are needed (like money, measurements, etc.).
• When comparing and performing operations with other decimal numbers.

Hence, decimals concurrently validate the accuracy of fractions and make math tasks more manageable.