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Chapter 6: Problem 16

Write the ratio in lowest terms. 10 lb to \(14 \mathrm{lb}\)

Short Answer

Expert verified
5:7

Step by step solution

01

- Identify the given quantities

The first quantity given is 10 pounds and the second quantity is 14 pounds.
02

- Write the ratio

The ratio of 10 pounds to 14 pounds can be written as a fraction: \ \ \( \frac{10}{14} \).
03

- Find the Greatest Common Divisor (GCD)

Determine the GCD of 10 and 14. The GCD of 10 and 14 is 2.
04

- Simplify the fraction

Divide both the numerator and the denominator by the GCD. \ \ \( \frac{10 \div 2}{14 \div 2} = \frac{5}{7} \).
05

- Write the simplified ratio

The simplified ratio of 10 pounds to 14 pounds is 5:7.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Divisor
The Greatest Common Divisor (GCD) is the largest number that can divide two given numbers without leaving a remainder. Understanding the GCD is crucial for simplifying ratios and fractions.
To find the GCD of 10 and 14, list the factors of each number:
  • Factors of 10: 1, 2, 5, 10
  • Factors of 14: 1, 2, 7, 14
The common factors are 1 and 2, so the GCD is 2.
Once the GCD is found, it becomes easier to simplify fractions or ratios, as you divide both the numerator and the denominator by this number.
Simplifying Fractions
Simplifying fractions involves reducing them to their lowest terms. In essence, this means dividing both the numerator and the denominator by their Greatest Common Divisor (GCD).
For our exercise, the fraction we started with was \( \frac{10}{14} \). Using the GCD of 2:
  • 10 divided by 2 is 5
  • 14 divided by 2 is 7
This results in the simplified fraction \( \frac{5}{7} \).
Simplifying fractions makes them easier to understand and compare. It's a fundamental skill in math that helps keep calculations and results neat and manageable.
Mathematical Ratios
Ratios are a way to compare two quantities. For example, the ratio of 10 pounds to 14 pounds can be written as 10:14 or as a fraction \( \frac{10}{14} \).
Simplifying ratios, like simplifying fractions, involves reducing to the smallest possible whole numbers by dividing both parts of the ratio by their GCD. In our exercise, this leads from 10:14 to 5:7.
Simplified ratios are better for easier comparison and understanding. They can be used in various real-world applications like recipes, maps, and more.
  • A ratio of 5:7 means for every 5 units of one quantity, there are 7 of the other.
  • Simplifying ratios ensures clarity and precision.

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