Problem 72 - Elementary Algebra

Question

\(\frac{11}{12} h-\frac{5}{8} h+\frac{4}{5} k-\frac{1}{9} k\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{7}{24} h + \frac{31}{45} k \).

Step 1: Combine Like Terms for 'h'

Identify and combine the like terms involving 'h'. The terms are \(\frac{11}{12} h \) and \(\frac{5}{8} h \). To combine them, find a common denominator for the fractions. The common denominator for 12 and 8 is 24.

Step 2: Convert 'h' Terms to Common Denominator

Convert each fraction to have the common denominator of 24. \(\frac{11}{12} = \frac{11 \times 2}{12 \times 2} = \frac{22}{24}\) and \(\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}\). So, the expression becomes \(\frac{22}{24} h - \frac{15}{24} h \).

Step 3: Simplify 'h' Terms

Subtract the fractions: \(\frac{22}{24} h - \frac{15}{24} h = \frac{22 - 15}{24} h = \frac{7}{24} h \).

Step 4: Combine Like Terms for 'k'

Identify and combine the like terms involving 'k'. The terms are \(\frac{4}{5} k \) and \(\frac{1}{9} k \). To combine these, find a common denominator for the fractions. The common denominator for 5 and 9 is 45.

Step 5: Convert 'k' Terms to Common Denominator

Convert each fraction to have the common denominator of 45. \(\frac{4}{5} = \frac{4 \times 9}{5 \times 9} = \frac{36}{45}\) and \(\frac{1}{9} = \frac{1 \times 5}{9 \times 5} = \frac{5}{45}\). So, the expression becomes \(\frac{36}{45} k - \frac{5}{45} k \).

Step 6: Simplify 'k' Terms

Subtract the fractions: \(\frac{36}{45} k - \frac{5}{45} k = \frac{36 - 5}{45} k = \frac{31}{45} k \).

Step 7: Combine Simplified Terms

Combine the simplified terms from Steps 3 and 6: \(\frac{7}{24} h + \frac{31}{45} k \).

Key Concepts

common denominators

When working with fractions, it’s important to have a common denominator if you want to add or subtract them. The common denominator is a shared multiple of the denominators of the fractions you’re working with. For instance, in our exercise, to combine \( \frac{11}{12} h \) and \( \frac{5}{8} h \), we needed to find a common denominator for 12 and 8. We found it to be 24, so we converted both fractions to have this denominator:

• \( \frac{11}{12} = \frac{22}{24} \)
• \( \frac{5}{8} = \frac{15}{24} \)

Converting fractions to have a common denominator makes the addition or subtraction process simpler and more straightforward. Knowing how to quickly find common denominators is a powerful tool in algebra.

fraction subtraction

When you need to subtract fractions, the first step is often to convert your fractions so they all have a common denominator. This makes the subtraction process much simpler.
Step-by-step process for fraction subtraction:

• Find a common denominator: Identify a number that both denominators divide into evenly.
• Convert the fractions: Modify each fraction to have the common denominator.
• Subtract the numerators: Perform the subtraction operation on the numerators while keeping the common denominator the same.

In our exercise, look at the subtraction involving the 'h' terms:

• Convert: \( \frac{11}{12}h - \frac{5}{8}h = \frac{22}{24}h - \frac{15}{24}h \)
• Subtract: \( \frac{22}{24} - \frac{15}{24} = \frac{7}{24}h \)

The process is the same for 'k' terms, where we got

• Convert: \( \frac{4}{5}k - \frac{1}{9}k = \frac{36}{45}k - \frac{5}{45}k \)
• Subtract: \( \frac{36}{45} - \frac{5}{45} = \frac{31}{45}k \)

These steps make combining fractions involving variables easier to follow.

simplifying expressions

Simplifying algebraic expressions means making them as straightforward as possible. It often involves combining like terms and reducing fractions to their simplest forms. For our example, we took the terms involving 'h' and 'k' separately to simplify them.
Steps involved in simplifying expressions:

• Combine like terms: Group terms with the same variable.
• Use common denominators: Make sure the fractions have common denominators if you need to add or subtract them.
• Simplify the fractions: Perform the arithmetic operations and simplify the fraction if possible.

For the 'h' terms:

• Combine like terms: \( \frac{22}{24}h - \frac{15}{24}h = \frac{7}{24} h \)

For the 'k' terms:

• Combine like terms: \( \frac{36}{45}k - \frac{5}{45}k = \frac{31}{45}k \)

Finally, we combine the results to get the simplified expression:
\( \frac{7}{24} h + \frac{31}{45} k \) This process shows how to take a complex expression and make it more manageable by breaking it down and simplifying step by step.