Problem 65 - Prealgebra

Question

Simplify the given expression. \(-\frac{7}{10}+1.2\)

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{19}{10}\).

Step 1: Convert Decimal to Fraction

First, we convert the decimal number 1.2 into a fraction. The number 1.2 is equal to \( \frac{12}{10} \) because 1.2 can be written as \( \frac{12}{10} \) when multiplied by \( \frac{10}{10} \) to remove the decimal.

Step 2: Find a Common Denominator

Next, we need to subtract the two fractions: \(-\frac{7}{10}\) and \(\frac{12}{10}\). Both fractions already have a common denominator of 10, so we can move on to step 3.

Step 3: Subtract the Numerators

Since the denominators are the same, subtract the numerators: \(-7 - 12 = -19\). Thus, the expression becomes \(-\frac{19}{10}\).

Step 4: Simplify the Fraction (if possible)

Check if the fraction \(-\frac{19}{10}\) can be simplified. Since 19 and 10 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Converting Decimals to Fractions

When dealing with math problems, you often come across decimal numbers that need to be converted into fractions. For example, when we have the decimal 1.2, we can convert it into a fraction. To do this, we look at the decimal's place value. Here, 1.2 has a tenths value, making it 12 tenths.
Converting it into a fraction results in \( \frac{12}{10} \). This is because moving the decimal one place to the right transforms it into a whole number, thus multiplying it by \( \frac{10}{10} \) as numerically equivalent.

• The numerator is 12.
• The denominator, which represents the place value, is 10.

This step ensures you have converted the decimal into a consistent format for performing further operations like addition or subtraction.

Common Denominators

Finding a common denominator is crucial in operations like adding or subtracting fractions. A denominator is the bottom number of a fraction and determines the 'part' size of the whole. For example, when you have fractions with different denominators, you can't directly add or subtract them. They need to be compatible or 'common.'
In our problem, both fractions \(-\frac{7}{10}\) and \(\frac{12}{10}\) already share the same denominator: 10. This makes our job much simpler.

• Check if both fractions have the same denominator.
• If they do, you can directly proceed with the arithmetic operation.
• If not, find a common multiple and adjust the fractions accordingly.

Remember, having common denominators is the foundation for smooth fraction operations.

Subtracting Fractions

Once you've confirmed common denominators, you can subtract the fractions with ease. The process is straightforward:

• Subtract the numerators while keeping the denominator the same.

In our example, the fractions \(-\frac{7}{10}\) and \(\frac{12}{10}\) share a denominator of 10, so we subtract the numerators:

• \(-7 - 12 = -19\)

Thus, resulting in the fraction \(-\frac{19}{10}\).
After subtracting, always check if the result can be simplified. Here, 19 is a prime number, and since 10 isn't divisible by 19, the fraction \(-\frac{19}{10}\) is already in its simplest form. Taking time to practice these steps will enable you to subtract fractions without a hitch.