Problem 36
Question
The problems below will allow you to review subtraction of fractions and mixed numbers. $$\frac{9}{10}-\frac{7}{10}$$
Step-by-Step Solution
Verified Answer
\(\frac{1}{5}\)
1Step 1: Identify the Common Denominator
Both fractions in the problem have the same denominator, which is 10. This makes it easy to proceed with simple subtraction.
2Step 2: Subtract the Numerators
Since the denominators are the same, subtract the numerator of the second fraction from the numerator of the first fraction: \(9 - 7 = 2\).
3Step 3: Write the Result as a Fraction
The solution to the subtraction is the result of the numerators over the common denominator: \(\frac{2}{10}\).
4Step 4: Simplify the Fraction
Simplify the fraction \(\frac{2}{10}\) by dividing both the numerator and denominator by their greatest common divisor, which is 2: \(\frac{2 \div 2}{10 \div 2} = \frac{1}{5}\).
Key Concepts
Common DenominatorNumeratorSimplifying Fractions
Common Denominator
When subtracting fractions, having a common denominator is essential. The denominator is the bottom number in a fraction and represents the total number of equal parts the whole is divided into.
In order to subtract fractions, both fractions need to share the same denominator, known as a common denominator.
This makes the subtraction process straightforward because the size of the parts we're subtracting remains consistent.
In order to subtract fractions, both fractions need to share the same denominator, known as a common denominator.
This makes the subtraction process straightforward because the size of the parts we're subtracting remains consistent.
- For example, in the fractions \( \frac{9}{10} \) and \( \frac{7}{10} \), the common denominator is already 10. This allows us to directly subtract the numerators.
- Having a common denominator means you don't need to adjust the fractions before subtracting which makes the calculation cleaner and simpler.
Numerator
The numerator is the top number in a fraction and indicates how many parts of the whole are being considered.
When subtracting fractions with a common denominator, the numerators are subtracted directly, while the denominator remains the same.
This step is crucial because changing only the numerators preserves the relation between the fractions and their parts.
When subtracting fractions with a common denominator, the numerators are subtracted directly, while the denominator remains the same.
This step is crucial because changing only the numerators preserves the relation between the fractions and their parts.
- Looking at our example, the numerators are 9 and 7.
- Subtract these to get \(9 - 7 = 2\).
Simplifying Fractions
Once the subtraction gives a new fraction, the next important step is simplifying it.
Simplifying a fraction means reducing it to its smallest possible denominator, keeping its value the same.
This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).
For instance, in the fraction \(\frac{2}{10}\), the GCD of 2 and 10 is 2. By dividing both by 2, we simplify \(\frac{2}{10}\) to \(\frac{1}{5}\).
Simplifying a fraction means reducing it to its smallest possible denominator, keeping its value the same.
This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).
For instance, in the fraction \(\frac{2}{10}\), the GCD of 2 and 10 is 2. By dividing both by 2, we simplify \(\frac{2}{10}\) to \(\frac{1}{5}\).
- Simplifying makes the fraction easier to understand and use in further calculations.
- Additionally, it often reveals the fraction in its most elegant form, showing the true proportion.