Problem 62 - Fundamentals of Mathematics

Question

(Section 5.3) Find the difference: \(\frac{7}{10}-\frac{5}{16}\).

Step-by-Step Solution

Answer

The difference is \(\frac{31}{80}\).

Step 1: Find the Least Common Denominator (LCD)

To subtract fractions, they need a common denominator. The denominators are 10 and 16. The least common multiple of these is 80. So, the LCD is 80.

Step 2: Convert Fractions to Have the LCD

Convert each fraction to an equivalent fraction with the denominator of 80. Multiply both the numerator and the denominator of \(\frac{7}{10}\) by 8 to get \(\frac{56}{80}\). For \(\frac{5}{16}\), multiply both the numerator and the denominator by 5 to get \(\frac{25}{80}\).

Step 3: Subtract the Fractions

Now that the fractions have the same denominator, subtract the numerators: \(56 - 25 = 31\). Keep the common denominator: \(\frac{56}{80} - \frac{25}{80} = \frac{31}{80}\).

Step 4: Simplify the Result (if necessary)

Check if the fraction \(\frac{31}{80}\) can be simplified. Since 31 is a prime number and does not divide evenly into 80, \(\frac{31}{80}\) is already in its simplest form.

Key Concepts

Least Common Denominator

When subtracting fractions, it's necessary for the fractions to have the same denominator. The least common denominator (LCD) is the smallest number that both denominators divide evenly into. In our example, we need to subtract \(\frac{7}{10}\) and \(\frac{5}{16}\).

To find the LCD for 10 and 16, you look for the least common multiple (LCM).

• List the multiples of each denominator.
• For 10: 10, 20, 30, 40, 50, 60, 70, 80...
• For 16: 16, 32, 48, 64, 80...

The smallest number that appears in both lists is 80. This means our LCD is 80, which serves as the common denominator we will use to make direct subtraction easier.

Finding the LCD ensures that the fractions represent similar parts of a whole, allowing for straightforward subtraction of the numerators.

Equivalent Fractions

To subtract fractions like \(\frac{7}{10}\) and \(\frac{5}{16}\), you first convert them to equivalent fractions with the least common denominator of 80. Equivalent fractions are different fractions that represent the same value or proportion of the whole.

For instance:

• For \(\frac{7}{10}\), we multiply both the numerator and the denominator by 8, which turns it into \(\frac{56}{80}\).
• For \(\frac{5}{16}\), we multiply both the numerator and denominator by 5, turning it into \(\frac{25}{80}\).

By doing this, you maintain the value of the fractions but change their denominators to match, allowing for straightforward subtraction. It’s essential because it gives both fractions a common base so that direct comparison and calculation of the difference is possible.

Simplifying Fractions

After subtracting the fractions with a common denominator, you are left with a new fraction: \(\frac{31}{80}\). Then, the next step would usually be to simplify this fraction. Simplifying a fraction means making it as simple as possible by ensuring the numerator and denominator have no common factors other than 1.

In our case:

• Look at 31. It's a prime number, which implies it only can be divided by 1 and itself.
• Since 31 does not divide 80 evenly, \(\frac{31}{80}\) is already in its simplest form.

Simplification is useful because it presents the fraction in an easier-to-understand form, ensuring clarity in communication of the answer. Even though no simplification was needed in this example, always check each time you subtract or add fractions.