Problem 33 - Prealgebra

Question

Write each of the following fractions as an equivalent fraction with denominator 6. $$\frac{55}{66}$$

Step-by-Step Solution

Answer

\(\frac{55}{66}\) simplifies to \(\frac{5}{6}\), which already has the denominator 6.

Step 1: Simplify the Initial Fraction

Start by simplifying the given fraction \( \frac{55}{66} \). Find the greatest common divisor (GCD) of 55 and 66. The GCD is 11 because 11 divides both 55 and 66 evenly. Divide both the numerator and the denominator by 11: \( \frac{55 \div 11}{66 \div 11} = \frac{5}{6} \).

Step 2: Confirm the Denominator

Ensure that the simplified fraction \( \frac{5}{6} \) already has the desired denominator of 6. In this case, it does match, so we don't need to make further adjustments.

Key Concepts

Simplifying Fractions

Simplifying fractions is like cleaning up a messy room. You want to make the fraction look as neat and simple as possible. This means taking a fraction and reducing it to its simplest form. To do this, we need to find a number that divides both the numerator and the denominator exactly, without leaving any remainder.
Let's break it down:

• Look at fraction \( \frac{55}{66} \).
• Check if there is a number that divides 55 and 66 evenly. That number is their greatest common divisor (GCD).
• For \(\frac{55}{66}\), 11 is the GCD because both 55 and 66 can be divided by 11.
• Divide both the top (55) and bottom (66) of the fraction by 11.
• You get \( \frac{5}{6} \) after dividing both by 11.

After simplification, \( \frac{55}{66} \) becomes \( \frac{5}{6} \), which is much simpler, just like a tidy room.

Greatest Common Divisor

The greatest common divisor (GCD) is a key player in simplifying fractions. It's the largest number that can divide both the numerator and the denominator with no leftovers. Finding the GCD is a crucial first step when breaking down a fraction to its simplest form.
To find the GCD:

• List out the factors of each number. For 55, these are 1, 5, 11, and 55. For 66, they are 1, 2, 3, 6, 11, 22, 33, and 66.
• Spot the largest number that appears in both lists. Here, it is 11.

With \( \frac{55}{66} \), using 11 as the GCD lets us tidy up the fraction to \( \frac{5}{6} \). Knowing about the GCD is like having a special tool to make tidying tasks easier.

Denominator Adjustment

Denominator adjustment means changing the bottom number of a fraction to a specified number, usually to find equivalent fractions. When you simplify a fraction, you might end up with the exact denominator you are looking for.
Here’s how it works:

• When we simplified \( \frac{55}{66} \) to \( \frac{5}{6} \), the new denominator was already 6.
• In problems where you need to change the denominator, you generally multiply or divide the existing denominator to reach the desired number.
• Yet, since \( \frac{5}{6} \) already has 6 as its denominator, we didn't need to make any changes here.

Denominator adjustment ensures the fraction fits perfectly into a set of fractions with the same bottom number, making them easier to compare or add together. Since our denominator was already 6, our job was done with no further adjustments needed.