Simplify 10$\frac{2}{5} - 6 \frac{2}{3} + 3$

Analyze the question: We need to evaluate 10 2/5 − 6 2/3 + 3. The expression has mixed numbers, so we can work with the whole numbers and fractions separately.

Concept: Subtract and add the whole parts first, then handle the fractions using a common denominator (LCM of 5 and 3 = 15).

Solution:

Step 1: Convert the fractions to a common denominator.

• 2/5 = 6/15
• 2/3 = 10/15

Step 2: Rewrite the expression. 10 6/15 − 6 10/15 + 3

Step 3: Whole numbers: 10 − 6 + 3 = 7

Step 4: Fractions: 6/15 − 10/15 = −4/15

Step 5: Combine. 7 − 4/15 = 6 15/15 − 4/15 = 6 11/15

Check using improper fractions:

• 10 2/5 = 52/5 = 156/15
• 6 2/3 = 20/3 = 100/15
• 3 = 45/15

156 − 100 + 45 = 101, so 101/15 = 6 11/15 (since 6 × 15 = 90, and 101 − 90 = 11). ✓

Answer: 6 11/15, which is option B.

Tip: When the fraction part of a subtraction is smaller than the one being subtracted (6/15 < 10/15), you will end up borrowing 1 from the whole number. Converting everything to improper fractions is a safe way to double-check.