Problem 4
Question
For exercises \(3-6\), evaluate or simplify. $$ \frac{4}{15} \cdot \frac{5}{12} $$
Step-by-Step Solution
Verified Answer
\(\frac{1}{9}\)
1Step 1 - Multiply the Numerators
Multiply the numerators of the fractions: \(4 \times 5 = 20\)
2Step 2 - Multiply the Denominators
Multiply the denominators of the fractions: \(15 \times 12 = 180\)
3Step 3 - Simplify the Fraction
Simplify the fraction \(\frac{20}{180}\). The greatest common divisor (GCD) of 20 and 180 is 20. Divide both the numerator and the denominator by 20: \(\frac{20 \div 20}{180 \div 20} = \frac{1}{9}\).
Key Concepts
simplifying fractionsgreatest common divisornumerators and denominators
simplifying fractions
Simplifying fractions is a key concept when dealing with fraction multiplication. After multiplying the numerators and denominators, we often get a large fraction that needs to be simplified to make it more understandable.
To simplify a fraction, we divide both the numerator (top number) and the denominator (bottom number) by the same number. This special number is called the Greatest Common Divisor (GCD).
This process reduces the fraction to its smallest equivalent form.
For example, in the exercise given, after multiplying, we get \(\frac{20}{180}\). To simplify this fraction, we need to divide both 20 (numerator) and 180 (denominator) by their GCD. This results in a simpler, more manageable fraction.
To simplify a fraction, we divide both the numerator (top number) and the denominator (bottom number) by the same number. This special number is called the Greatest Common Divisor (GCD).
This process reduces the fraction to its smallest equivalent form.
For example, in the exercise given, after multiplying, we get \(\frac{20}{180}\). To simplify this fraction, we need to divide both 20 (numerator) and 180 (denominator) by their GCD. This results in a simpler, more manageable fraction.
greatest common divisor
The Greatest Common Divisor (GCD) is fundamental when simplifying fractions. The GCD of two numbers is the largest number that divides both numerators and denominators without a remainder.
Finding the GCD can involve listing the factors of each number and identifying the highest one they share.
For instance, the factors of 20 are 1, 2, 4, 5, 10, 20; the factors of 180 include 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.
The largest number common to both sets is 20.
Therefore, to simplify \(\frac{20}{180}\), we divide both by 20, resulting in \(\frac{1}{9}\). This produces a simplified, easy-to-work-with fraction.
Finding the GCD can involve listing the factors of each number and identifying the highest one they share.
For instance, the factors of 20 are 1, 2, 4, 5, 10, 20; the factors of 180 include 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.
The largest number common to both sets is 20.
Therefore, to simplify \(\frac{20}{180}\), we divide both by 20, resulting in \(\frac{1}{9}\). This produces a simplified, easy-to-work-with fraction.
numerators and denominators
Understanding numerators and denominators is crucial when working with fractions.
The numerator is the top number of a fraction and represents how many parts we have. The denominator is the bottom number and indicates the total number of equal parts the whole is divided into.
For example, in \(\frac{4}{15}\), 4 is the numerator, and 15 is the denominator.
When multiplying fractions, we multiply the numerators together and the denominators together separately.
In the given exercise, we multiply the numerators 4 and 5 to get 20.
Then, we multiply the denominators 15 and 12 to get 180.
This produces the fraction \(\frac{20}{180}\), which we then simplify by using the GCD.
This step-by-step understanding makes solving such problems more straightforward and understandable.
The numerator is the top number of a fraction and represents how many parts we have. The denominator is the bottom number and indicates the total number of equal parts the whole is divided into.
For example, in \(\frac{4}{15}\), 4 is the numerator, and 15 is the denominator.
When multiplying fractions, we multiply the numerators together and the denominators together separately.
In the given exercise, we multiply the numerators 4 and 5 to get 20.
Then, we multiply the denominators 15 and 12 to get 180.
This produces the fraction \(\frac{20}{180}\), which we then simplify by using the GCD.
This step-by-step understanding makes solving such problems more straightforward and understandable.
Other exercises in this chapter
Problem 4
For exercises 1-12, use prime factorization to find the least common denominator. $$ \frac{7}{15 y} ; \frac{1}{24 y^{2}} $$
View solution Problem 4
For exercises \(1-4\), evaluate. $$ \frac{16}{21}-\frac{4}{21} $$
View solution Problem 4
For exercises 1-66, simplify. $$ \frac{54 c^{2} d^{5}}{72 c d} $$
View solution Problem 5
The relationship of \(x\) and \(y\) is a direct variation. When \(x=2, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represen
View solution