🧮 Part 1: Complex Operations

1. Calculate: 3¼ + 2⅝ - 1⅓

Solution: Convert to improper fractions: 3¼ = 13/4, 2⅝ = 21/8, 1⅓ = 4/3.

Find a common denominator (8): 13/4 = 26/8, 4/3 = 32/24. Now calculate: 26/8 + 21/8 - 32/24 = 47/8 - 4/3 = 47/8 - 32/24 = 47/8 - 32/24 = 47/8 - 32/24 = 23/24.

2. Find the product: 2⅗ × 1¾

Solution: Convert to improper fractions: 2⅗ = 13/5, 1¾ = 7/4.

Multiply: (13/5) × (7/4) = 91/20 = 4⅑.

3. Solve: ⅝ ÷ ¾ + ½

Solution: ⅝ ÷ ¾ = ⅝ × 4/3 = 4/6 = ⅔.

Now add: ⅔ + ½ = ⅔ + 3/6 = 4/6 = ⅔.

4. Which expression equals ⅚?

½ + ⅓

1⅙ - ⅓

¾ × 1⅑

1⅔ ÷ 2

Solution: Check each option: ½ + ⅓ = 5/6, 1⅙ - ⅓ = 5/6, ¾ × 1⅑ = 15/8, 1⅔ ÷ 2 = 5/6. The correct answers are ½ + ⅓ and 1⅙ - ⅓.

📊 Part 2: Multi-Step Word Problems

5. Sarah is making anzac biscuits. She uses 2¼ cups of oats, 1⅓ cups of flour, and ¾ cup of coconut. If her mixing bowl holds 5½ cups, how much extra space does she have after adding all ingredients?

Solution: Total ingredients = 2¼ + 1⅓ + ¾ = 2.25 + 1.33 + 0.75 = 4.33 cups.

Space left = 5½ - 4⅓ = 1⅙ cups.

6. A running track is 2⅝ kilometres long. Jamie runs 3¾ laps on Monday, 2½ laps on Wednesday, and 4⅙ laps on Friday. What total distance did Jamie run this week?

Solution: Total laps = 3¾ + 2½ + 4⅙ = 3.75 + 2.5 + 4.1667 = 10.4167 laps.

Total distance = 10.4167 × 2⅝ = 27.5 km.

7. At a school fair, ⅔ of the students bought food, ¼ bought games tickets, and ⅛ bought both food and games. What fraction of students bought either food OR games (but not both)?

Solution: Use the formula: P(A or B) = P(A) + P(B) - P(A and B).

So, (⅔ + ¼ - ⅛) = 5/8.

🎯 Part 3: Problem Solving Challenge

8. A recipe serves 6 people and calls for 1⅝ cups of milk. How much milk is needed to serve 15 people?

Solution: Milk needed = (1⅝ cups) × (15/6) = 4 cups.

9. Three friends share a pizza. Alex eats ⅜ of the pizza, Sam eats ¼, and Jordan eats the rest. If Jordan's portion weighs 180 grams, what was the weight of the whole pizza?

Solution: Total eaten = ⅜ + ¼ = 5/8. Jordan eats the rest, so 3/8 = 180 grams.

Whole pizza = 180 / (3/8) = 480 grams.

10. Challenge: Find three different mixed numbers that when added together equal exactly 7½. Show your working.

Solution: Example: 2⅓ + 2⅔ + 2 = 7½.