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Chapter 5: Problem 5

Simplify to lowest terms. $$\frac{26}{52}$$

Short Answer

Expert verified
Answer: The simplified fraction of \(\frac{26}{52}\) in the lowest terms is \(\frac{1}{2}\).

Step by step solution

01

Identify the GCD of 26 and 52

To do this, we will list the divisors of both the numerator (26) and the denominator (52) and determine the greatest common divisor among them. Divisors of 26: 1, 2, 13, 26 Divisors of 52: 1, 2, 4, 13, 26, 52 The greatest common divisor is 26.
02

Divide both the numerator and the denominator by the GCD

Divide both the numerator (26) and the denominator (52) by the GCD (26). Numerator: 26 梅 26 = 1 Denominator: 52 梅 26 = 2
03

Write the simplified fraction

The simplified fraction is: $$\frac{1}{2}$$ The simplified fraction is \(\frac{1}{2}\) in the lowest terms.

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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 an essential concept in simplifying fractions. It helps us find the largest number that can evenly divide both the numerator and denominator of a fraction. Here's how it works:

  • First, list all the divisors of the numerator. For example, with a numerator of 26, the divisors are 1, 2, 13, and 26.
  • Next, list the divisors of the denominator. For a denominator of 52, the divisors are 1, 2, 4, 13, 26, and 52.
  • Now, find the largest number common to both lists. This is the GCD. In our example, the GCD of 26 and 52 is 26.
This step is crucial because dividing both parts of the fraction by the GCD ensures we simplify the fraction to its lowest terms without changing its value. Getting a good grasp of the GCD is the first step in making simplification straightforward.
Numerator and Denominator
Understanding the roles of the numerator and denominator is key to simplifying fractions. The numerator is the top part of a fraction, indicating how many parts we have.

The denominator, on the other hand, is the bottom part, showing the total number of equal parts the whole is divided into.

When simplifying a fraction, here's how the numerator and denominator play their roles:

  • You divide both the numerator and the denominator by their GCD, which reduces them proportionally.
  • This action maintains the same value for the fraction, even though it appears smaller.
  • In the example with \(\frac{26}{52}\), the GCD is 26, so we divide both 26 (numerator) and 52 (denominator) by 26.
After doing this division, the roles of numerator and denominator are maintained while reflecting the smallest form of the fraction, \(\frac{1}{2}\). This process makes fractions easier to understand and work with.
Simplified Fraction
The goal of fraction simplification is to express the fraction in its simplest form. Simplification involves reducing the fraction so that the numerator and denominator are as small as possible while still retaining the same value.

Here鈥檚 why simplifying fractions is useful:

  • It makes the fraction easier to understand and use in arithmetic operations.
  • Simplified fractions are more readable and can help in comparing the sizes of different fractions.
  • They are essential in algebra where keeping fractions simple can simplify further calculations.
In our exercise, simplifying \(\frac{26}{52}\) to \(\frac{1}{2}\) means we used the GCD to reduce it to the simplest terms. This ensures the fraction is presented in the clearest possible way. It is like writing in a shorthand that saves space and makes problems more approachable.

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Most popular questions from this chapter

Solve and check. $$\frac{1}{4}+y=\frac{5}{8}$$

Use \(d=r t\). If a family drove \(4 \frac{3}{4}\) hours at an average rate of 65 miles per hour, how far did they travel?

Translate to an equation and solve. \(3 \frac{1}{4}\) more than twice \(n\) is \(-\frac{1}{6}\)

Solve. Write all answers in lowest terms. A recipe calls for \(1 \frac{1}{3}\) cups of sugar. How much sugar should be used in making half of the recipe?

solve and check. $$x+\frac{3}{5}=\frac{7}{10}$$
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