EXPLAIN two different ways to calculate 2 1/2 x 1 2/5.
Sure, there are several ways to perform this multiplication involving mixed numbers, which are numbers consisting of an integer and a proper fraction. Here are two different methods:
**Method 1: Convert to Improper Fractions and Multiply**
Step 1: Convert each mixed number to an improper fraction.
- For 2 1/2, multiply the whole number 2 by the denominator 2 and add the numerator 1 to get 5/2.
- For 1 2/5, multiply the whole number 1 by the denominator 5 and add the numerator 2 to get 7/5.
Now we have 5/2 and 7/5 as improper fractions.
Step 2: Multiply the improper fractions.
- Multiply the numerators: 5 * 7 = 35.
- Multiply the denominators: 2 * 5 = 10.
So, the product of the numerators over the product of the denominators gives us 35/10.
Step 3: Simplify the resulting fraction.
- Divide both numerator and denominator by their greatest common divisor, which is 5 in this case: 35/5 = 7 and 10/5 = 2.
So, the simplified answer is 7/2, which is an improper fraction.
Step 4: If required, convert back to a mixed number.
- Divide the numerator by the denominator: 7 ÷ 2 = 3 with a remainder of 1.
So, the answer in mixed number form is 3 1/2.
**Method 2: Use Whole Numbers and Fractions Separately**
Step 1: Divide both mixed numbers into their whole and fractional parts.
- For 2 1/2, we have the whole part "2" and the fraction "1/2".
- For 1 2/5, we have the whole part "1" and the fraction "2/5".
Step 2: Multiply the mixed numbers by distributing the multiplication across the whole and fractional parts.
- First, multiply the whole numbers together: 2 * 1 = 2.
- Then, multiply the whole number of the first by the fraction of the second: 2 * 2/5 = 4/5.
- Multiply the fraction of the first by the whole number of the second: 1/2 * 1 = 1/2.
- Finally, multiply the fractions: 1/2 * 2/5 = 2/10, which simplifies to 1/5.
Step 3: Combine the results.
- Start with the result of the whole number multiplication: 2.
- Add the results of the mixed multiplications: 4/5 + 1/2.
To add 4/5 and 1/2, find a common denominator, which is 10 in this case:
- Convert 1/2 to 5/10
- Now add 4/5 (which is 8/10) and 5/10 to get 13/10.
Step 4: Combine all parts.
- Add the whole number 2 to the fractional sum 13/10: 2 + 13/10 = 2 + 1 3/10.
- Simplify this to a single mixed number by adding the whole numbers together: 2 + 1 = 3, and keep the fraction part as 3/10.
So, the answer using this method is also 3 1/2 (since 3 3/10 can be further simplified to 3 1/2).
Both methods yield the same result: 3 1/2.