Problem 188 - Intermediate Algebra

Question

In the following exercises, add or subtract. (a) \(\frac{3 a}{8} \div \frac{7}{12}\) (b) \(\frac{3 a}{8}-\frac{7}{12}\)

Step-by-Step Solution

Answer

(a) \(\frac{9a}{14}\), (b) \(\frac{9a - 14}{24}\)

Step 1: Divide the fractions for part (a)

To divide the fractions \(\frac{3a}{8}\) and \(\frac{7}{12}\), multiply \(\frac{3a}{8}\) by the reciprocal of \(\frac{7}{12}\). The reciprocal of \(\frac{7}{12}\) is \(\frac{12}{7}\). Therefore, \(\frac{3a}{8} \div \frac{7}{12} = \frac{3a}{8} \times \frac{12}{7}\).

Step 2: Multiply the fractions for part (a)

Now, multiply the numerators and the denominators: \[ \frac{3a \cdot 12}{8 \cdot 7} = \frac{36a}{56} \].

Step 3: Simplify the fraction for part (a)

Divide the numerator and denominator by their greatest common divisor (GCD), which is 4: \[ \frac{36a}{56} \div \frac{4}{4} = \frac{9a}{14} \]. So, \(\frac{3a}{8} \div \frac{7}{12} = \frac{9a}{14}\).

Step 4: Find a common denominator for part (b)

To subtract the fractions \(\frac{3a}{8} - \frac{7}{12}\), find a common denominator. The least common multiple (LCM) of 8 and 12 is 24. Convert the fractions to have a common denominator: \[ \frac{3a}{8} = \frac{3a \cdot 3}{8 \cdot 3} = \frac{9a}{24} \] and \[ \frac{7}{12} = \frac{7 \cdot 2}{12 \cdot 2} = \frac{14}{24} \].

Step 5: Subtract the fractions for part (b)

Subtract the fractions now that they have a common denominator: \[ \frac{9a}{24} - \frac{14}{24} = \frac{9a - 14}{24} \]. So, \(\frac{3a}{8} - \frac{7}{12} = \frac{9a - 14}{24}\).

Key Concepts

fraction division

When dividing fractions, always remember to multiply by the reciprocal of the divisor. The reciprocal is simply the flipped version of the fraction. For instance, the reciprocal of \( \frac{7}{12} \) is \( \frac{12}{7} \).
To divide \( \frac{3a}{8} \) by \( \frac{7}{12} \), first convert the division into multiplication by the reciprocal:

• \( \frac{3a}{8} \times \frac{12}{7} \)

Once you have this setup, you multiply the numerators together and the denominators together:
Numerator: \( 3a \times 12 = 36a \)
Denominator: \( 8 \times 7 = 56 \)
So, the fraction becomes \( \frac{36a}{56} \).
Finally, simplify the fraction. Simplification involves dividing both the numerator and the denominator by their greatest common divisor (GCD). For this example, the GCD of 36 and 56 is 4. Divide both by 4 to get:
\( \frac{36a}{56} \rightarrow \frac{9a}{14} \).
So, \( \frac{3a}{8} \times \frac{12}{7} = \frac{9a}{14} \).

fraction subtraction

Subtracting fractions involves ensuring both fractions have a common denominator. Once that’s established, subtract the numerators directly. For example:

• Given \( \frac{3a}{8} - \frac{7}{12} \), we start by finding a common denominator (covered in the next section).

Suppose the common denominator is 24. Convert both fractions:
\( \frac{3a}{8} = \frac{3a \times 3}{8 \times 3} = \frac{9a}{24} \)
and
\( \frac{7}{12} = \frac{7 \times 2}{12 \times 2} = \frac{14}{24} \).
Now that both fractions have a common denominator, subtract the numerators:
\( \frac{9a}{24} - \frac{14}{24} = \frac{9a - 14}{24} \)
That’s your result: \( \frac{3a}{8} - \frac{7}{12} = \frac{9a - 14}{24} \).

finding common denominators

Finding a common denominator is crucial for both addition and subtraction of fractions. The common denominator is a multiple of the original denominators. The least common multiple (LCM) is often used.
For the fractions \( \frac{3a}{8} \) and \( \frac{7}{12} \), identify the LCM of 8 and 12. Factors of 8: \( 1, 2, 4, 8 \). Factors of 12: \( 1, 2, 3, 4, 6, 12 \). The smallest common factor both share is 24.
Now convert:

• \( \frac{3a}{8} = \frac{3a \times 3}{8 \times 3} = \frac{9a}{24} \)
• \( \frac{7}{12} = \frac{7 \times 2}{12 \times 2} = \frac{14}{24} \)

Both have a denominator of 24, making subtraction straightforward.
So, to subtract:
\( \frac{9a}{24} - \frac{14}{24} = \frac{9a - 14}{24} \)
Thus, finding common denominators simplifies your operations.