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Chapter 2: Problem 34

Simplify. $$ \frac{1000}{1080} $$

Short Answer

Expert verified
The simplified fraction is \(\frac{25}{27}\).

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

To simplify the fraction, first find the greatest common divisor (GCD) of the numerator, 1000, and the denominator, 1080. The GCD is the largest number that divides both numbers without leaving a remainder.
02

Calculate the GCD

The prime factorization of 1000 is: 1000 = 2^3 * 5^3The prime factorization of 1080 is:1080 = 2^3 * 3^3 * 5The common factors are: 2^3 and 5Thus, the GCD is 2^3 * 5 = 40.
03

Divide the Numerator and Denominator by the GCD

Divide both the numerator (1000) and the denominator (1080) by the GCD (40):\[\frac{1000}{40} = 25\frac{1080}{40} = 27\]So, the simplified fraction is \(\frac{25}{27}\).

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Key Concepts

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

Greatest Common Divisor (GCD)
To simplify fractions effectively, it's crucial to understand the concept of the Greatest Common Divisor (GCD). The GCD is the largest number that divides two numbers without leaving a remainder. For example, to simplify the fraction \( \frac{1000}{1080} \), first find the GCD of 1000 and 1080.

Identifying the GCD helps you find the largest number by which both the numerator and the denominator can be divided.

In other words, the GCD acts like a critical key to unlock the 'simplest' version of a fraction. This process makes your calculations and further mathematical tasks much easier.
Prime Factorization
Prime factorization is a method of expressing a number as a product of its prime numbers. For example:
  • The prime factorization of 1000 is \( 1000 = 2^3 \cdot 5^3 \)
  • The prime factorization of 1080 is \( 1080 = 2^3 \cdot 3^3 \cdot 5 \)
Identifying the prime factors helps pinpoint common factors between the numerator and the denominator.

Once you've broken down both numbers into primes, look for the common prime factors. These shared primes are then used to calculate the GCD. In our example, the common factors are \(2^3\) and \(5\). Thus, the GCD is \(2^3 \cdot 5 = 40 \). This knowledge allows for smooth simplification.
Fraction Simplification
Simplifying a fraction means reducing it to its lowest terms. To do this, divide both the numerator and the denominator by their GCD.

Steps:
  • Identify the GCD of the numerator and the denominator
  • Divide both the numerator and the denominator by the GCD
For instance, with \( \frac{1000}{1080} \), we already determined the GCD is 40.

Following through:
  • \( \frac{1000}{40} = 25 \)
  • \( \frac{1080}{40} = 27 \)
The simplified fraction is \( \frac{25}{27} \). This final fraction is now in its simplest form, making it easier to work with in further mathematical problems.

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Most popular questions from this chapter

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