Problem 28 - Elementary Algebra

Question

For exercises \(23-74\), evaluate. $$ \frac{4}{9} \cdot \frac{3}{16} $$

Step-by-Step Solution

Answer

The result is \( \frac{1}{12} \).

Step 1: Multiply the Numerators

First, multiply the numerators of both fractions. Here, the numerators are 4 and 3. Thus, we calculate: \[ 4 \times 3 = 12 \]

Step 2: Multiply the Denominators

Next, multiply the denominators of both fractions. Here, the denominators are 9 and 16. Thus, we calculate: \[ 9 \times 16 = 144 \]

Step 3: Form the New Fraction

Now place the product of the numerators over the product of the denominators to get the new fraction: \[ \frac{12}{144} \]

Step 4: Simplify the Fraction

Finally, simplify the fraction \( \frac{12}{144} \). Find the greatest common divisor of 12 and 144. The greatest common divisor is 12. Thus, simplify: \[ \frac{12 \div 12}{144 \div 12} = \frac{1}{12} \]

Key Concepts

multiply numerators

When multiplying fractions, the first step is to multiply the numerators. The numerator is the top number in a fraction. For example, in the exercise, the fractions are \(\frac{4}{9}\) and \(\frac{3}{16}\). The numerators here are 4 and 3.

To multiply these numerators, simply perform the multiplication as you would with any whole numbers:
\[ 4 \times 3 = 12 \]
The result, 12, will become the numerator of your new fraction.

If you have trouble multiplying, you can break it down further. For instance, 4 times 3 can be seen as adding 4 three times:
\[ 4 + 4 + 4 = 12 \]
Which gives you the same result!

multiply denominators

The second step in fraction multiplication is to multiply the denominators – the bottom numbers in the fraction. For the given exercise, the fractions \(\frac{4}{9}\) and \(\frac{3}{16}\) have denominators 9 and 16 respectively.

To multiply these denominators, follow the same steps as multiplying the numerators.
\[ 9 \times 16 = 144 \]
The result is 144, which will be the denominator of the new fraction.

If you find larger multiplications challenging, it helps to know some tricks, such as breaking down numbers. In this case, you could split 9 and 16 into smaller parts:
\[ 9 = 3 \times 3 \]
\[ 16 = 4 \times 4 \] Then you can recombine:
\[ (3 \times 3) \times (4 \times 4) = 3 \times 3 \times 4 \times 4 = 144 \]

simplify fractions

The final step is to simplify the fraction. After multiplying the numerators and denominators, we obtained \(\frac{12}{144}\). To simplify this, we need to find the greatest common divisor (GCD) of the numerator and the denominator.

For 12 and 144, the GCD is 12. So, we divide both the numerator and the denominator by 12:
\[ \frac{12 \, / \, 12}{144 \, / \, 12} = \frac{1}{12} \]
When simplifying, always look for common factors. You can list the factors like this:
\[ 12 = 1, 2, 3, 4, 6, 12 \]
\[ 144 = 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 \] The highest common factor is 12.

Remember, simplifying makes fractions easier to understand and work with in future calculations!