Problem 3 - Prealgebra

Question

Express each ratio as a fraction in simplest form. 10 inches to 3 feet

Step-by-Step Solution

Answer

The ratio 10 inches to 3 feet is \(\frac{5}{18}\) in simplest form.

Step 1: Convert feet to inches

First, we need to ensure both quantities are in the same units. Since 1 foot equals 12 inches, convert 3 feet to inches by multiplying: \(3 \text{ feet} \times 12 \text{ inches/foot} = 36 \text{ inches}\). Now we have 10 inches to 36 inches as our ratio.

Step 2: Write as a fraction

Express the ratio of 10 inches to 36 inches as a fraction: \(\frac{10}{36}\).

Step 3: Simplify the fraction

Simplify \(\frac{10}{36}\) by finding the greatest common divisor (GCD) of 10 and 36, which is 2. Divide both the numerator and the denominator by 2: \(\frac{10 \div 2}{36 \div 2} = \frac{5}{18}\).

Key Concepts

Simplifying Fractions

Fractions often appear in many math problems, and simplifying fractions means reducing them to their smallest form. This makes them easier to work with and understand.
When we simplify a fraction, we're looking to make both the numerator (top number) and the denominator (bottom number) as small as possible while still maintaining the same value. This is done by finding the greatest common divisor (GCD), which is the largest number that evenly divides both the numerator and the denominator.
Here's how it works step by step:

• Take the fraction you want to simplify, like \(\frac{10}{36}\).
• Find the GCD of the numerator and denominator. For example, with 10 and 36, the GCD is 2.
• Divide both the numerator and the denominator by the GCD: \(10 \div 2 = 5\) and \(36 \div 2 = 18\).
• Write the simplified fraction, \(\frac{5}{18}\).

By reducing fractions in this way, calculations become more straightforward and comparisons between numbers become clearer.

Converting Units

Unit conversion is an important skill in mathematics, especially when dealing with measurements. It allows us to accurately compare and compute quantities by ensuring they're in the same unit.
Converting units can seem daunting at first, but with a basic understanding, it’s quite straightforward. You just need to know the conversion factor, which is a number used to change one set of units to another. For example, when converting feet to inches, the conversion factor is 12 (since 1 foot = 12 inches).
To convert units:

• Identify the conversion factor (e.g., 3 feet × 12 inches/foot = 36 inches).
• Multiply the number of units you have by this conversion factor to get the equivalent quantity in the new unit.
• Adjust your calculations accordingly, like ensuring a ratio such as 10 inches to 36 inches is now correctly set for further steps.

By mastering unit conversions, you ensure accurate computations in both real-world and mathematical problems. It’s all about making sure everything is "speaking the same language."

Mathematics Problem Solving

When tackling any math problem, having a step-by-step approach is essential. This structured way of solving problems ensures that we cover all the bases and reach the correct solution.
The process of solving a problem can be broken down into these steps:

• Understand the problem: What is being asked? In our example, it was to express a ratio as a fraction in simplest form.
• Plan the solution: Decide on a strategy. Here, we decided to first convert units before proceeding to other steps.
• Carry out the plan: Follow the steps methodically. Start by achieving unit consistency, then express and simplify the fraction.
• Review the solution: Check that the answer makes sense and re-evaluate if necessary. This involves verifying if the final fraction (\(\frac{5}{18}\)) is in its simplest form.

By employing this structured method, you develop a clear pathway for solving even complex math problems. This reduces errors and builds confidence in your mathematical abilities.