Problem 10 - Intermediate Algebra

Question

Use a graphing calculator to check each exercise. $$ \frac{7}{10}-\frac{4}{5} $$

Step-by-Step Solution

Answer

The result of \( \frac{7}{10} - \frac{4}{5} \) is \( \frac{-1}{10} \).

Step 1 - Finding a common denominator

To subtract fractions, they must have the same denominator. The fractions \( \frac{7}{10} \) and \( \frac{4}{5} \) have denominators 10 and 5 respectively. Find the least common multiple of 10 and 5, which is 10. Thus, we only need to convert \( \frac{4}{5} \) to have a denominator of 10.

Step 2 - Convert fractions to common denominators

Multiply the numerator and the denominator of \( \frac{4}{5} \) by 2 to convert it into a fraction with a denominator of 10. This gives the fraction \( \frac{8}{10} \).

Step 3 - Subtract the fractions

Now that both fractions have the same denominator, subtract the numerators: \( 7 - 8 = -1 \). Therefore, \( \frac{7}{10} - \frac{4}{5} = \frac{-1}{10} \).

Step 4 - Verify with a graphing calculator

Enter the expression \( \frac{7}{10} - \frac{4}{5} \) into a graphing calculator. The calculator should show the result as \( \frac{-1}{10} \), confirming our manual calculation.

Key Concepts

Common Denominator

To subtract fractions accurately, they must share a common denominator. Imagine trying to compare apples to oranges without a common unit. The common denominator acts like that unit, allowing fair comparison and calculation. In our example, the fractions \( \frac{7}{10} \) and \( \frac{4}{5} \) have different denominators: 10 and 5 respectively. We need to convert them to the same denominator to perform subtraction. By finding a common denominator, arithmetic operations on fractions become straightforward, ensuring accuracy.
To get these fractions to the same footing, we find the least common denominator, which leads to our next concept.

Least Common Multiple

Finding the least common multiple (LCM) is crucial for aligning denominators. The LCM is the smallest number that two or more numbers can divide without a remainder. For denominators 10 and 5, the LCM is 10. Why? Because 10 is the smallest number that both 10 and 5 can divide into evenly. This step is key in maintaining the integrity of our original values while allowing seamless arithmetic. Once determined, we elevate all fractions to this common platform. In this case, \( \frac{4}{5} \) is adjusted by multiplying its numerator and denominator by 2, transforming it into \( \frac{8}{10} \). Now, both fractions are poised for subtraction.

Graphing Calculator

A graphing calculator offers a visual and computational tool to verify our manual calculations. These calculators are equipped with an ability to handle complex arithmetic, including fraction subtraction. By entering \( \frac{7}{10} - \frac{4}{5} \), the calculator processes and displays \( \frac{-1}{10} \). This dual verification process, both manually and electronically, ensures the accuracy of our solution. Using this tool can solidify understanding and highlight any errors in manual calculations, bolstering confidence in the arithmetic process.

Fraction Subtraction Verification

Verification ensures the correctness of fraction subtraction. After computing \( \frac{7}{10} - \frac{4}{5} \) manually to get \( \frac{-1}{10} \), confirmation by a graphing calculator bolsters our results. It's essential as incorrect calculations can lead to misunderstandings in larger math problems. Using additional methods like calculators acts like a second opinion, catching any potential oversight. For students, it builds a habit of cross-checking work, a valuable practice for accuracy and precision in mathematical problem-solving.