Problem 83 - Prealgebra

Question

Exercising According to national studies, childhood obesity is on the rise. Doctors recommend a minimum of 30 minutes of exercise three times a week to help keep us fit. Suppose during a given week you walk for \(\frac{1}{4}\) hour one day, \(\frac{2}{3}\) of an hour a second day and \(\frac{3}{4}\) of an hour on a third day. Find the total number of hours walked as a fraction.

Step-by-Step Solution

Answer

The total number of hours walked is \(\frac{5}{3}\) hours.

Step 1: Understand the Problem

We need to find the total amount of time spent walking over three days, where each day is represented as a fraction of an hour. The fractions are \(\frac{1}{4}\), \(\frac{2}{3}\), and \(\frac{3}{4}\).

Step 2: Find a Common Denominator

To add the fractions, we first need a common denominator. The denominators here are 4, 3, and 4. The least common multiple of these denominators is 12.

Step 3: Convert Fractions to Common Denominator

Convert each fraction to have a denominator of 12.\- For \(\frac{1}{4}\): Multiply both the numerator and denominator by 3 to get \(\frac{3}{12}\).\- For \(\frac{2}{3}\): Multiply both the numerator and denominator by 4 to get \(\frac{8}{12}\).\- For \(\frac{3}{4}\): Multiply both the numerator and denominator by 3 to get \(\frac{9}{12}\).

Step 4: Add the Fractions

Add the converted fractions: \(\frac{3}{12} + \frac{8}{12} + \frac{9}{12}\).\First, add \(\frac{3}{12}\) and \(\frac{8}{12}\) to get \(\frac{11}{12}\).\Then add \(\frac{11}{12} + \frac{9}{12}\) to get \(\frac{20}{12}\).

Step 5: Simplify the Result

Simplify \(\frac{20}{12}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 4, to get \(\frac{5}{3}\).

Key Concepts

Common Denominator

When working with fractions, especially for addition or subtraction, it’s crucial to find a common denominator. A common denominator is a shared multiple of the denominators of two or more fractions. This shared multiple allows you to effectively combine the fractions into one cohesive expression.

• The common denominator is often the least common multiple (LCM) of the denominators.
• In our problem, we had denominators 4, 3, and 4. The LCM of these numbers is 12.
• Finding this common ground among fractions is like speaking the same language, making addition or subtraction seamless.

By rewriting the fractions with the common denominator, we're able to align the parts of the fractions to make operations straightforward. This keeps the fractions consistent, enabling us to add or subtract without errors. Remember, the key is to find a common denominator that all terms can share and work equally well with.

Simplifying Fractions

Simplifying fractions is akin to reducing a recipe to only its essential ingredients. It's about finding the most straightforward representation of a fraction by dividing both the numerator and denominator by their greatest common divisor (GCD).

• In the exercise, we simplified \( \frac{20}{12} \) to \( \frac{5}{3} \).
• We achieved this by recognizing that the greatest common divisor of 20 and 12 is 4.

To simplify, divide both the numerator and the denominator of the fraction by this GCD. Simplification makes fractions easier to understand and use, much like reducing complex expressions to their simplest form in algebra. Once simplified, fractions are more manageable and reveal their core value at a glance. It's a handy tool in your math toolkit for keeping calculations clean and clear.

Childhood Obesity

Childhood obesity is a growing concern worldwide. It can affect both the physical and emotional well-being of children, leading to health issues like diabetes, high blood pressure, and increased risk of heart disease. Several factors can contribute to rising obesity rates among children.

• Unhealthy eating patterns, with high-calorie and low-nutrient foods.
• Lack of physical activity, often due to sedentary lifestyles or reduced playtime.
• Genetic factors or hormonal imbalances.

Tackling childhood obesity requires a collaborative effort involving schools, parents, and community programs. By promoting healthier lifestyle choices and encouraging active living, we can support children in maintaining a healthy weight and improving their overall well-being.

Exercise Recommendations

Doctors and health experts universally agree on the significant benefits of regular exercise. For children, it is essential not only for maintaining a healthy weight but also for fostering physical development, emotional well-being, and social skills. The general recommendation is at least 30 minutes of moderate to vigorous exercise most days of the week.

• This can include walking, running, playing sports, or even dancing.
• Incorporating a variety of activities can keep exercise fun and engaging.
• Physical activity helps kids build strong bones, maintain a healthy weight, and reduce the risk of many diseases.

Making exercise a family affair can increase adherence and make it a lifestyle choice rather than a chore. The goal is to create lifetime habits that promote health and happiness.