Problem 62 - Elementary Algebra

Question

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{c}{3}+\frac{d}{2}}{\frac{c}{2}+\frac{d}{7}} $$

Step-by-Step Solution

Answer

\( \frac{7(2c + 3d)}{3(7c + 2d)} \)

Step 1 - Identify the Numerator

The numerator of the given expression is \ \( \frac{c}{3} + \frac{d}{2} \).

Step 2 - Identify the Denominator

The denominator of the given expression is \ \( \frac{c}{2} + \frac{d}{7} \).

Step 3 - Find a Common Denominator for the Numerator

For the numerator \( \frac{c}{3} + \frac{d}{2} \), the least common multiple of 3 and 2 is 6. Rewrite the fractions with this common denominator: \ \( \frac{c}{3} = \frac{2c}{6} \) and \( \frac{d}{2} = \frac{3d}{6} \). Thus, the numerator becomes \ \( \frac{2c}{6} + \frac{3d}{6} = \frac{2c + 3d}{6} \).

Step 4 - Find a Common Denominator for the Denominator

For the denominator \( \frac{c}{2} + \frac{d}{7} \), the least common multiple of 2 and 7 is 14. Rewrite the fractions with this common denominator: \ \( \frac{c}{2} = \frac{7c}{14} \) and \( \frac{d}{7} = \frac{2d}{14} \). Thus, the denominator becomes \ \( \frac{7c}{14} + \frac{2d}{14} = \frac{7c + 2d}{14} \).

Step 5 - Divide the Numerator by the Denominator

Write the expression as: \ \( \frac{\frac{2c + 3d}{6}}{\frac{7c + 2d}{14}} \). To divide by a fraction, multiply by its reciprocal: \ \( \frac{2c + 3d}{6} \times \frac{14}{7c + 2d} \).

Step 6 - Simplify the Expression

Multiply the numerators and the denominators: \ \( \frac{(2c + 3d) \times 14}{6 \times (7c + 2d)} = \frac{14(2c + 3d)}{6(7c + 2d)} \). Simplify the fractions if possible: \ \( \frac{14(2c + 3d)}{6(7c + 2d)} = \frac{7(2c + 3d)}{3(7c + 2d)} \).

Key Concepts

common denominators

In algebra, when you add or subtract fractions, they need to have the same denominator, called a common denominator. For instance, in the exercise, the fractions \( \frac{c}{3} + \frac{d}{2} \) must be combined under a common denominator. The least common multiple (LCM) of 3 and 2 is 6.
To rewrite the fractions with this common denominator, we convert them as follows:

• \( \frac{c}{3} = \frac{2c}{6} \)
• \( \frac{d}{2} = \frac{3d}{6} \)

So now, the numerator becomes \( \frac{2c + 3d}{6} \).
Similarly, for the denominator \( \frac{c}{2} + \frac{d}{7} \), the LCM of 2 and 7 is 14. So, we rewrite the fractions as follows:

• \( \frac{c}{2} = \frac{7c}{14} \)
• \( \frac{d}{7} = \frac{2d}{14} \)

So now, the denominator becomes \( \frac{7c + 2d}{14} \).

fraction division

Dividing fractions involves multiplying by the reciprocal. When we have \( \frac{\frac{2c + 3d}{6}}{\frac{7c + 2d}{14}} \), we switch from division to multiplication by flipping the denominator:
Firstly, rewrite the expression:

• \( \frac{\frac{2c + 3d}{6}}{\frac{7c + 2d}{14}} \)
• Reciprocal of \( \frac{7c + 2d}{14} \) is \( \frac{14}{7c + 2d} \)
• Thus, it becomes \( \frac{2c + 3d}{6} \times \frac{14}{7c + 2d} \)

After rewriting, it's much easier to see how we can continue simplifying the fraction.

simplifying fractions

Simplifying fractions means reducing them to their simplest form. For example, after we've written \( \frac{2c + 3d}{6} \times \frac{14}{7c + 2d} \), we need to simplify:

• Multiply the numerators: \( (2c + 3d) \times 14 \)
• Multiply the denominators: \( 6 \times (7c + 2d) \)
• So, we get \( \frac{14(2c + 3d)}{6(7c + 2d)} \)

Next, we can see if there is any common factor that can make this expression simpler. Here, we can divide top and bottom by 2:

• Divide the numerator: \( 14(2c + 3d) / 2 = 7(2c + 3d) \)
• Divide the denominator: \( 6(7c + 2d) / 2 = 3(7c + 2d) \)

Thus, the simplified form is \( \frac{7(2c + 3d)}{3(7c + 2d)} \).

least common multiple

The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of them.
In this exercise, we need the LCM to find common denominators:

• For 3 and 2, the multiples of 3 are 3, 6, 9, etc., and the multiples of 2 are 2, 4, 6, etc. The smallest common multiple is 6.
• For 2 and 7, the multiples of 2 are 2, 4, 6, 8, 10, 12, 14, etc., and the multiples of 7 are 7, 14, 21, etc. The smallest common multiple is 14.

This helps in combining fractions like \( \frac{c}{3} + \frac{d}{2} \) efficiently under a shared denominator.