Question

Write the rate 20 leaves to 8 twigs as a fraction in lowest terms. Write your answer in the box.

Short answer

The rate is \(\frac{5}{2}\) in lowest terms.

Step-by-step solution

Identify the Given Ratio

The problem gives the ratio of 20 leaves to 8 twigs. First, write this ratio as a fraction: \(\frac{20}{8}\).

Find the Greatest Common Divisor (GCD)

To reduce the fraction, determine the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 20 and 8 is 4.

Divide Both Numerator and Denominator by the GCD

To reduce the fraction to its lowest terms, divide both the numerator and the denominator by the GCD. \(\frac{20 \text{ (numerator)}}{8 \text{ (denominator)}} = \frac{20 \text{ / } 4}{8 \text{ / } 4} = \frac{5}{2}\)

Write the Simplified Fraction

Now that both the numerator and the denominator have been divided by their GCD, the fraction in its lowest terms is \(\frac{5}{2}\).

Key concepts

greatest common divisor

Understanding the greatest common divisor (GCD) is essential for reducing fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder.
For instance, in the exercise, we had to reduce the fraction \(\frac{20}{8}\). To find the GCD of 20 and 8, you can list out the factors of each number:

• Factors of 20: 1, 2, 4, 5, 10, 20
• Factors of 8: 1, 2, 4, 8

The greatest factor common to both lists is 4, which is the GCD. This understanding helps you to correctly simplify the fraction by dividing the numerator and the denominator by the GCD, making calculations simpler and more accurate in further mathematical operations.

ratios and proportions

Ratios and proportions are foundational concepts in math. A ratio is a way to compare two quantities. For example, in the problem, the ratio of leaves to twigs is given as 20 to 8. This can be written as a fraction \(\frac{20}{8}\).
When you simplify this fraction, you're finding the proportionate relationship between the two quantities in their simplest form. Proportions, on the other hand, state that two ratios are equal. For example, if another pair of leaves and twigs maintains the same ratio, say 40 leaves to 16 twigs, we would have \(\frac{40}{16}\) = \(\frac{5}{2}\). Understanding these concepts helps in solving real-world problems easily and accurately.
Ratios and proportions are used in various fields, from cooking recipes to scaling maps.

fraction simplification

Fraction simplification is the process of reducing a fraction to its simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD). For example, to simplify \(\frac{20}{8}\), you divide both 20 and 8 by their GCD, which is 4, giving you \(\frac{5}{2}\).
Why should we simplify fractions? Simplified fractions are easier to work with in calculations and provide a clearer understanding of the relationship between numbers. They are especially useful in measuring and comparing quantities.
Here's a quick step-by-step way to simplify any fraction:

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• Write down the result.

Practicing these steps will make fraction simplification become second nature.