Problem 117 - Elementary Algebra

Question

For exercises 115-118, evaluate. On a scientific calculator, use the fraction key. For graphing calculators, use the Frac command to rewrite the decimal answer as a fraction. $$ \left(\frac{5}{34}\right)\left(\frac{8}{21}\right) $$

Step-by-Step Solution

Answer

The product is \( \frac{20}{357} \).

Step 1 - Multiply the Numerators

First, multiply the numerators of the fractions. For \( \frac{5}{34} \) and \( \frac{8}{21} \), multiply 5 and 8.\[ 5 \times 8 = 40 \]

Step 2 - Multiply the Denominators

Next, multiply the denominators of the fractions. For \( \frac{5}{34} \) and \( \frac{8}{21} \), multiply 34 and 21.\[ 34 \times 21 = 714 \]

Step 3 - Write the Fraction

Now, write the product as a fraction using the results of the previous steps.\[ \frac{40}{714} \]

Step 4 - Simplify the Fraction

Simplify the fraction by finding the greatest common divisor (GCD) of 40 and 714 and dividing both the numerator and the denominator by this GCD. The GCD of 40 and 714 is 2.\[ \frac{40 \text{ ÷ 2}}{714 \text{ ÷ 2}} = \frac{20}{357} \]

Key Concepts

Numerator Multiplication

Numerator multiplication refers to the process of multiplying the top numbers of two fractions. When you multiply fractions, you deal with the numerators first.
For instance, in our exercise: \( \frac{5}{34} \times \frac{8}{21} \), we multiply the numerators 5 and 8.
This gives us:
\[ 5 \times 8 = 40 \]
This result, 40, will be the numerator of the product of these two fractions.

Denominator Multiplication

Denominator multiplication involves multiplying the bottom numbers of two fractions. After dealing with the numerators, you need to handle the denominators.
For the exercise: \( \frac{5}{34} \times \frac{8}{21} \), we multiply the denominators 34 and 21.
This results in:
\[ 34 \times 21 = 714 \]
This result, 714, will be the denominator of the product of these two fractions.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1.
In our example, after we multiply the numerators and denominators, we get:
\[ \frac{40}{714} \]
To simplify this fraction, we need to find a number that both 40 (numerator) and 714 (denominator) can be divided by without leaving a remainder.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder.
To simplify our fraction \[ \frac{40}{714} \], we must identify the GCD of 40 and 714.
Using the Euclidean algorithm or prime factorization, we determine that the GCD of 40 and 714 is 2.
We then divide both the numerator and the denominator by this GCD:

• \( 40 \div 2 = 20 \)
• \( 714 \div 2 = 357 \)

Thus, the simplified fraction becomes: \[ \frac{20}{357} \]