Problem 352 Simplify. $$ \frac{182}{294}... [FREE SOLUTION]

Chapter 1: Problem 352

Simplify. $$ \frac{182}{294} $$

Short Answer

Expert verified

The simplified form is $\frac{13}{21}$

Step by step solution

- Find the Greatest Common Divisor (GCD)

First, determine the greatest common divisor (GCD) of the numerator (182) and the denominator (294). The GCD can be found by using the Euclidean algorithm.

- Apply the Euclidean Algorithm

To find the GCD of 182 and 294, perform the following steps:1. Divide 294 by 182 and find the remainder: \[ 294 \text{ mod } 182 = 112 \]2. Divide 182 by 112 and find the remainder: \[ 182 \text{ mod } 112 = 70 \]3. Continue dividing by the last remainder until the remainder is 0: \[ 112 \text{ mod } 70 = 42 \] \[ 70 \text{ mod } 42 = 28 \] \[ 42 \text{ mod } 28 = 14 \] \[ 28 \text{ mod } 14 = 0 \]Hence, the GCD is 14.

- Simplify the Fraction

Divide both the numerator and the denominator by their GCD (14): \[ \frac{182 \,/\, 14}{294 \,/\, 14} = \frac{13}{21} \]

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Key Concepts

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

Greatest Common Divisor

When simplifying fractions, the first step is to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of two numbers is the largest number that evenly divides both of them. Knowing the GCD helps in reducing the fraction to its simplest form. For instance, in the problem $\frac{182}{294}$, finding the GCD allows us to break it down easily. The GCD of 182 and 294 is 14.

Euclidean Algorithm

To find the greatest common divisor efficiently, we use the Euclidean algorithm. It鈥檚 a method that breaks down the problem into smaller steps:
1. Divide the larger number by the smaller number and get the remainder.
2. Replace the larger number with the smaller number and the smaller number with the remainder.
3. Repeat the process until the remainder is zero.
This helps in quickly finding the GCD. For example, to find the GCD of 182 and 294:
1. \[ 294 \mod 182 = 112 \]2. \[ 182 \mod 112 = 70 \]3. \[ 112 \mod 70 = 42 \]4. \[ 70 \mod 42 = 28 \]5. \[ 42 \mod 28 = 14 \]6. \[ 28 \mod 14 = 0 \]
Thus, the GCD is 14.

Simplifying Fractions

Once we have the GCD, simplifying a fraction becomes straightforward. We divide both the numerator and the denominator by the GCD to get the simplest form. Let鈥檚 apply this to our fraction $\frac{182}{294}$:
1. GCD of 182 and 294 is 14.
2. Now divide both numbers by 14: \[ \frac{182 \, / \, 14}{294 \, / \, 14} = \frac{13}{21} \]
This means $\frac{182}{294}$ simplifies to $\frac{13}{21}$. This fraction is now in its simplest form.

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91影视

Simplify. $$ \frac{182}{294} $$

Short Answer

Step by step solution

- Find the Greatest Common Divisor (GCD)

- Apply the Euclidean Algorithm

- Simplify the Fraction

Key Concepts

Greatest Common Divisor

Euclidean Algorithm

Simplifying Fractions

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