Problem 4 - Elementary Algebra Within Reach

Question

Write the ratio as a fraction in simplest form. 27 to 63

Step-by-Step Solution

Answer

The simplified form of the ratio 27 to 63 as a fraction is \( \frac{3}{7} \).

Step 1: Write the Ratio as a Fraction

The given ratio is 27 to 63. Writing this ratio as a fraction, we get \( \frac{27}{63} \).

Step 2: Identify the Greatest Common Divisor (GCD)

We need to determine the greatest common divisor of 27 and 63 to simplify the fraction. Both 27 and 63 are divisible by 9, so the GCD is 9.

Step 3: Simplify the fraction

Divide both the numerator and the denominator by the GCD (9). This gives us \( \frac{27 \div 9}{63 \div 9} \), which simplifies to \( \frac{3}{7} \)

Key Concepts

Understanding Ratios

The concept of a ratio is foundational in mathematics. Simply put, a ratio is a way to compare two quantities by expressing how much of one there is compared to another. For instance, if you have 27 apples and 63 oranges, the ratio of apples to oranges is 27 to 63. To use this ratio meaningfully, it's often helpful to express it as a fraction. This is done by writing the ratio as \( \frac{27}{63} \).
Ratios are not just limited to numbers. They can compare virtually anything, as long as it's possible to relate their quantities:

• Objects (e.g., 4 cats to 7 dogs)
• Distances (e.g., 2 km to 5 km)
• Liquids (e.g., 5 liters of water to 10 liters of juice)

The flexibility of ratios makes them a powerful tool in various fields, such as finance, cooking, and science.

Calculating the Greatest Common Divisor (GCD)

To simplify a fraction, you need to find the greatest common divisor (GCD) of its numerator and denominator. The GCD is the largest number that divides both numbers without leaving a remainder. For example, with 27 and 63, you begin by identifying the factors of each number. 27 has factors of 1, 3, 9, and 27, while 63 has factors of 1, 3, 7, 9, 21, and 63. By comparing these lists, the greatest factor they share is 9, making it the GCD.
Understanding how to calculate the GCD is essential for simplifying fractions. Without it, you might simplify incorrectly. Here are simple steps to find the GCD of two numbers:

• List out the factors of each number.
• Find the largest number that appears in both lists.

If you're working with larger numbers, you might use the Euclidean algorithm to save time and effort. The GCD is a key part of simplifying ratios in fraction form.

Simplifying Fractions

Simplifying fractions is all about making them easier to understand without changing their value. When you simplify a fraction, you reduce it to its smallest possible numerator and denominator while retaining the original ratio's meaning. For the fraction \( \frac{27}{63} \), you simplify it by dividing both the numerator and the denominator by their GCD, which we've determined to be 9. Doing this math gives us \( \frac{27 \div 9}{63 \div 9} = \frac{3}{7} \).
Simplifying fractions involves a couple of key steps:

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

Once simplified, the fraction \( \frac{3}{7} \) conveys the same ratio as \( \frac{27}{63} \), but it's clearer and easier to use. Simplifying fractions is crucial in mathematics because it makes calculations and interpretations much more manageable.