Problem 145 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{1}{2}+\frac{1}{5}=\frac{2}{7}$$

Step-by-Step Solution

Answer

The statement is false. The correct statement should be \( \frac{1}{2} + \frac{1}{5} = \frac{7}{10} \)

Step 1: Identify the Fractions

The fractions that need to be added together are \( \frac{1}{2} \) and \( \frac{1}{5} \)

Step 2: Find the Least Common Denominator (LCD)

The least common denominator of 2 and 5 is 10. It's the smallest number that both 2 and 5 can divide into evenly.

Step 3: Convert the Fractions

Convert the fractions to have the LCD as the denominator. So \( \frac{1}{2} \) becomes \( \frac{5}{10} \) and \( \frac{1}{5} \) becomes \( \frac{2}{10} \)

Step 4: Add the Fractions

Add the resulting fractions together: \( \frac{5}{10} + \frac{2}{10} = \frac{7}{10} \)

Step 5: Compare and Correct the Given Statement

The correct sum is \( \frac{7}{10} \) not \( \frac{2}{7} \), so the given statement is false. To make the statement true, it should be changed to: \( \frac{1}{2} + \frac{1}{5} = \frac{7}{10} \)

Key Concepts

Least Common Denominator

When adding fractions, one of the key steps is finding a common denominator. Without a common denominator, you can't directly add fractions because their denominators are different. The least common denominator (LCD) is the smallest number that each of the original denominators can divide into without leaving a remainder. In our example, we're working with the denominators 2 and 5. To find the LCD of these numbers, we look for the smallest multiple they both share. In this case, it is 10. This means that 10 is the smallest number that both 2 and 5 can divide into evenly.

Finding the LCD requires factoring each of the denominators into primes and using the highest power of each prime. Since 2 and 5 are prime numbers themselves, the LCD is simply their product, which is 10.

Once the LCD is found, it's used to convert each fraction to an equivalent fraction that has the common denominator. This makes the addition process straightforward.

Fractions

Fractions are a way of expressing numbers that are not whole. They consist of two parts: a numerator, which is the top part, and a denominator, which is the bottom part. The numerator represents how many parts we have, while the denominator represents how many total parts make up a whole. For example, in \( \frac{1}{2} \) we have 1 part out of a total of 2 parts.

When adding fractions, fractions must first have a common denominator so that the numerators can be directly added. Until both fractions share the same denominator, their sums cannot be combined straightforwardly. The actual arithmetic is performed only on the numerators, leaving the common denominator the same.

Understanding and manipulating fractions is an essential math skill, enabling you to perform operations like addition, subtraction, multiplication, and division with non-whole numbers.

Equivalent Fractions

Equivalent fractions are fractions that represent the same value or proportion of a whole, even though they may look different at first glance. For example, \( \frac{1}{2} \) and \( \frac{5}{10} \) are equivalent because they both represent half of a whole.

To find an equivalent fraction, you multiply or divide both the numerator and the denominator by the same number. This doesn't change the value of the fraction, just its appearance. When converting fractions to have a common denominator, you're using the concept of equivalent fractions. For instance, in our task, \( \frac{1}{2} \) was transformed into \( \frac{5}{10} \). This was done by multiplying both the numerator and the denominator by 5, the factor needed to reach the least common denominator of 10.

Working with equivalent fractions is crucial when adding, subtracting, or comparing fractions, as it allows us to work with like terms. This understanding is fundamental for accurately performing mathematical operations on fractions.