Operations with Fractions: A Step-by-Step Guide
In this article, we will explore how to evaluate the expression 2/3 + 1/4 × 2/5 - 3/4 ÷ 5/2 by following the order of operations (PEMDAS).
Step 1: Multiply 1/4 and 2/5
To evaluate the expression, we need to follow the order of operations. The first operation we encounter is the multiplication of 1/4 and 2/5. To multiply fractions, we multiply the numerators (numbers on top) and multiply the denominators (numbers on the bottom), like this:
1/4 × 2/5 = (1 × 2) / (4 × 5) = 2/20
Step 2: Simplify the Fraction
We can simplify the fraction 2/20 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us:
2/20 = 1/10
Step 3: Add 2/3 and 1/10
Next, we add 2/3 and 1/10. To add fractions with different denominators, we need to find the least common multiple (LCM) of the denominators, which is 30. We can convert both fractions to have a denominator of 30, like this:
2/3 = (2 × 10) / (3 × 10) = 20/30
1/10 = (1 × 3) / (10 × 3) = 3/30
Now we can add the fractions:
20/30 + 3/30 = 23/30
Step 4: Divide 3/4 by 5/2
The next operation is the division of 3/4 by 5/2. To divide fractions, we invert the second fraction (i.e., flip the numerator and denominator) and then multiply:
3/4 ÷ 5/2 = 3/4 × 2/5
Now we can multiply the numerators and multiply the denominators:
(3 × 2) / (4 × 5) = 6/20
Step 5: Simplify the Fraction
We can simplify the fraction 6/20 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us:
6/20 = 3/10
Step 6: Subtract 3/10 from 23/30
Finally, we subtract 3/10 from 23/30. To subtract fractions with different denominators, we need to find the least common multiple (LCM) of the denominators, which is 30. We can convert both fractions to have a denominator of 30, like this:
3/10 = (3 × 3) / (10 × 3) = 9/30
23/30 remains the same
Now we can subtract the fractions:
23/30 - 9/30 = 14/30
Step 7: Simplify the Final Answer
We can simplify the fraction 14/30 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us:
14/30 = 7/15
Therefore, the final answer is:
2/3 + 1/4 × 2/5 - 3/4 ÷ 5/2 = 7/15
