Problem 36
Question
Reduce each fraction to lowest terms. $$\frac{150 a b^{2}}{210 a b}$$
Step-by-Step Solution
Verified Answer
The fraction reduces to \(\frac{5b}{7}\).
1Step 1: Identifying Common Factors
To reduce a fraction to its lowest terms, first identify any common factors present in the numerator and the denominator. For \(150ab^2\) and \(210ab\), both parts have the factors \(a\) and \(b\) in common.
2Step 2: Cancel Out Common Variables
Cancel the common factors \(a\) and \(b\) from both the numerator \(150ab^2\) and the denominator \(210ab\), resulting in \(\frac{150b}{210}\).
3Step 3: Finding GCD of Coefficients
Next, focus on the coefficients 150 and 210. Determine the greatest common divisor (GCD) of these numbers. Use the prime factorizations: \(150 = 2 \times 3 \times 5^2\) and \(210 = 2 \times 3 \times 5 \times 7\). The GCD is \(2 \times 3 \times 5 = 30\).
4Step 4: Dividing by the GCD
Divide both 150 and 210 by their GCD (30) to simplify the fraction further: \(\frac{150b}{210} = \frac{\frac{150}{30}b}{\frac{210}{30}} = \frac{5b}{7}\).
5Step 5: Confirmation of Simplified Fraction
Check that the fraction \(\frac{5b}{7}\) is completely reduced with no remaining common factors between the numerator and the denominator. The fraction \(\frac{5b}{7}\) is now in its simplest form.
Key Concepts
Understanding Common FactorsExploring the Greatest Common Divisor (GCD)Prime Factorization DemystifiedCreating Reduced Fractions
Understanding Common Factors
The process of reducing fractions begins by identifying common factors in the numerator and the denominator. A common factor is any integer that divides both numbers without leaving a remainder. In mathematical terms, if you have two numbers, say 8 and 12, then 4 is a common factor because it divides both 8 and 12 equally.
In more complex expressions, like the fraction \(\frac{150ab^2}{210ab}\), common factors can also include variables such as \(a\) and \(b\). By identifying these common factors, you can start to simplify the fraction by canceling out these shared elements between the numerator and the denominator. This step is crucial for making the fraction easier to work with.
In more complex expressions, like the fraction \(\frac{150ab^2}{210ab}\), common factors can also include variables such as \(a\) and \(b\). By identifying these common factors, you can start to simplify the fraction by canceling out these shared elements between the numerator and the denominator. This step is crucial for making the fraction easier to work with.
Exploring the Greatest Common Divisor (GCD)
The greatest common divisor, or GCD, is the largest number that can be evenly divided into both the numerator and the denominator of a fraction. To find the GCD, you often use prime factorization, which involves breaking down a number into its prime components.
For example, let's look at the numbers 150 and 210. The prime factorization of 150 is \(2 \times 3 \times 5^2\) and for 210, it's \(2 \times 3 \times 5 \times 7\). The GCD is determined by multiplying the smallest power of all common prime numbers. In this case, it's \(2 \times 3 \times 5 = 30\). This number is pivotal in fraction simplification, enabling the cancelation of shared factors to achieve the most reduced form.
For example, let's look at the numbers 150 and 210. The prime factorization of 150 is \(2 \times 3 \times 5^2\) and for 210, it's \(2 \times 3 \times 5 \times 7\). The GCD is determined by multiplying the smallest power of all common prime numbers. In this case, it's \(2 \times 3 \times 5 = 30\). This number is pivotal in fraction simplification, enabling the cancelation of shared factors to achieve the most reduced form.
Prime Factorization Demystified
Prime factorization involves expressing a number as a multiplication of prime numbers. Prime numbers are numbers greater than 1 that can only be divided by 1 and themselves without leaving a remainder.
To prime factorize, you start with the smallest prime number, which is 2, and check if it divides the number evenly. If it does, you continue dividing by 2 until it no longer works. Then you proceed to the next smallest prime number like 3, 5, 7, and so on.
Take the number 150 as an instance. It breaks down to \(2 \times 3 \times 5^2\). By using prime factorization, you can determine the greatest common divisor when simplifying fractions, ensuring a streamlined reduction process.
To prime factorize, you start with the smallest prime number, which is 2, and check if it divides the number evenly. If it does, you continue dividing by 2 until it no longer works. Then you proceed to the next smallest prime number like 3, 5, 7, and so on.
Take the number 150 as an instance. It breaks down to \(2 \times 3 \times 5^2\). By using prime factorization, you can determine the greatest common divisor when simplifying fractions, ensuring a streamlined reduction process.
Creating Reduced Fractions
A reduced fraction is one in which the numerator and the denominator have no common factors other than 1. Simplifying a fraction to this elementary form makes it clearer and more concise, and often much simpler to work with in further calculations.
Using our example \(\frac{150b}{210}\), once we've determined the GCD to be 30, we divide both the numerator and the denominator by 30. This means \(\frac{150}{30}b\) and \(\frac{210}{30}\), leaving us with \(\frac{5b}{7}\). Now \(5b\) and \(7\) share no common factors apart from 1, making \(\frac{5b}{7}\) a reduced fraction.
Recognizing and creating reduced fractions can simplify mathematical operations, making them less error-prone and more straightforward to solve.
Using our example \(\frac{150b}{210}\), once we've determined the GCD to be 30, we divide both the numerator and the denominator by 30. This means \(\frac{150}{30}b\) and \(\frac{210}{30}\), leaving us with \(\frac{5b}{7}\). Now \(5b\) and \(7\) share no common factors apart from 1, making \(\frac{5b}{7}\) a reduced fraction.
Recognizing and creating reduced fractions can simplify mathematical operations, making them less error-prone and more straightforward to solve.
Other exercises in this chapter
Problem 36
Find the LCD for each of the following; then use the methods developed in this section to add or subtract as indicated. $$-\frac{1}{30}+\frac{9}{40}$$
View solution Problem 36
Cooking A recipe calls for \(3 \frac{1}{4}\) cups of flour. If Diane is using only half the recipe, how much flour should she use?
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Simplify each expression as much as possible. $$12 \div\left(\frac{1}{4}\right)^{2}$$
View solution Problem 36
Write each of the following fractions as an equivalent fraction with denominator 12. $$\frac{5}{6}$$
View solution