Real World Problems: Fractions and Mixed Numbers | Solved Examples

Solution: Exercises 1 to 3

1 Pizza Party Problem

Exercise 1

At a birthday party, there are 2 pizzas, each cut into 8 equal slices. If Sarah eats 1 and 1/2 pizzas, how many slices did she eat? How many slices remain? Express your answer as a mixed number.

Definition:

Mixed number: A number that combines a whole number and a fraction (like 1 1/2).

Mixed number conversion method:

Convert the mixed number to an improper fraction
Calculate the total number of slices
Perform the necessary calculations
Express the result as a mixed number if needed

Total pizzas

Slices per pizza

Total slices

Step 1: Convert mixed number to improper fraction

1 1/2 = (1 × 2 + 1)/2 = 3/2 of a pizza

Step 2: Calculate slices eaten

3/2 of a pizza × 8 slices per pizza = (3 × 8) ÷ 2 = 24 ÷ 2 = 12 slices

Step 3: Calculate remaining slices

Remaining = Total - Eaten = 16 - 12 = 4 slices

Step 4: Express remaining as mixed number

4 slices = 4/8 of a pizza = 1/2 of a pizza

Sarah ate 12 slices (1 1/2 pizzas). 4 slices remain (1/2 pizza).

Final answer:

Sarah ate 12 slices (1 1/2 pizzas). 4 slices remain (1/2 pizza).

Applied rules:

• Mixed to improper: Whole × Denominator + Numerator = New numerator

• Fraction of a number: Multiply the whole number by the numerator, divide by denominator

• Subtraction: Remaining = Total - Eaten

12 red slices eaten (1 1/2 pizzas), 4 green slices remaining (1/2 pizza)

2 Cake Cutting Problem

Exercise 2

Mom baked 3 rectangular cakes and cut each into 6 equal pieces. She gave 1 and 1/3 of the total pieces to the children and kept 1 and 1/2 of the total pieces for herself. How many pieces did she give away? How many did she keep? How many pieces were given to guests?

Definition:

Finding mixed number of a number: Convert mixed numbers to improper fractions first, then multiply.

Total cakes

Pieces per cake

Total pieces

Step 1: Calculate total pieces

3 cakes × 6 pieces each = 18 total pieces

Step 2: Convert mixed numbers to improper fractions

1 1/3 = (1 × 3 + 1)/3 = 4/3; 1 1/2 = (1 × 2 + 1)/2 = 3/2

Step 3: Calculate pieces for children

4/3 of 18 = (4 × 18) ÷ 3 = 72 ÷ 3 = 24 pieces

Step 4: Calculate pieces for mom

3/2 of 18 = (3 × 18) ÷ 2 = 54 ÷ 2 = 27 pieces

Step 5: Calculate pieces for guests

Guests = Total - (Children + Mom) = 18 - (24 + 27) = -33 (impossible!)

Step 6: Correct interpretation

Actually means 1 1/3 pieces and 1 1/2 pieces, not of the total!

Children got 4/3 pieces (1 1/3), mom kept 3/2 pieces (1 1/2). This problem needs re-evaluation.

Final answer:

Children got 1 1/3 pieces, mom kept 1 1/2 pieces. Guests received the remaining pieces.

Applied rules:

• Mixed to improper: Whole × Denominator + Numerator = New numerator

• Fraction of a number: Multiply the whole number by the numerator, divide by denominator

• Subtraction: To find remaining amount, subtract given amounts from total

4 red pieces (children), 3 yellow pieces (mom), 11 green pieces (guests)

3 Cookie Distribution Problem

Exercise 3

Emma baked 20 cookies. She gave 1 and 1/4 of the cookies to her grandmother, 2 and 1/2 of the cookies to her neighbor, and kept the rest for her family. How many cookies did each person get? What fraction of the cookies did Emma keep for her family?

Definition:

Finding multiple mixed numbers of a number: Convert each mixed number to an improper fraction, then multiply by the total.

Total cookies

To grandmother

1 1/4

To neighbor

2 1/2

Step 1: Convert mixed numbers to improper fractions

1 1/4 = (1 × 4 + 1)/4 = 5/4; 2 1/2 = (2 × 2 + 1)/2 = 5/2

Step 2: Calculate cookies for grandmother

5/4 of 20 = (5 × 20) ÷ 4 = 100 ÷ 4 = 25 cookies

Step 3: Calculate cookies for neighbor

5/2 of 20 = (5 × 20) ÷ 2 = 100 ÷ 2 = 50 cookies

Step 4: Notice the error

25 + 50 = 75 cookies, but only 20 exist! The problem means 1 1/4 and 2 1/2 pieces, not of the total.

Step 5: Reinterpret the problem

Emma gave 1 1/4 = 5/4 = 1.25 ≈ 1 cookie to grandmother and 2 1/2 = 5/2 = 2.5 ≈ 3 cookies to neighbor.

Step 6: Calculate family portion

Family gets = Total - (Grandmother + Neighbor) = 20 - (1 + 3) = 16 cookies

Grandmother got 1 cookie (1/20 of total), neighbor got 3 cookies (3/20 of total), family got 16 cookies (4/5 of total).

Final answer:

Grandmother got 1 cookie (1/20), neighbor got 3 cookies (3/20), family got 16 cookies (4/5).

Applied rules:

• Mixed to improper: Whole × Denominator + Numerator = New numerator

• Fraction of a number: Multiply the whole by numerator, divide by denominator

• Subtraction: To find remaining amount, subtract given amounts from total

• Verification: Check that the sum doesn't exceed the total amount

Real World Problems: Fractions and Mixed Numbers | Solved Examples | Mathematics Grade 3

Solution: Exercises 1 to 3

1 Pizza Party Problem

Exercise 1

At a birthday party, there are 2 pizzas, each cut into 8 equal slices. If 3 friends eat 2 slices each, what fraction of the total pizza did they eat? What fraction remains? If there's 1/4 of a pizza left, how many whole pizzas and fractional parts would that be?

Definition:

Fraction: A fraction represents a part of a whole. A mixed number combines a whole number and a fraction.

Problem-solving method:

Find the total number of slices available
Calculate how many slices were eaten
Express as a fraction of the whole
Convert to mixed number if needed

Total pizzas

Slices per pizza

Total slices

Step 1: Calculate total slices

2 pizzas × 8 slices each = 16 total slices

Step 2: Calculate slices eaten

3 friends × 2 slices each = 6 slices eaten

Step 3: Express as fraction eaten

Fraction eaten = 6/16 = 3/8 of the total pizza

Step 4: Calculate remaining

Remaining = 16 - 6 = 10 slices = 10/16 = 5/8 of the total pizza

Step 5: Convert 1/4 of pizza to mixed number

1/4 of a pizza is less than 1 whole, so it's just a fraction: 0 1/4

Friends ate 6/16 (or 3/8) of the total pizza. 10/16 (or 5/8) remains.

Final answer:

Friends ate 3/8 of the total pizza. 5/8 remains.

Applied rules:

• Multiplication: To find total, multiply number of items by quantity per item

• Fraction of whole: Part eaten ÷ Total = Fraction eaten

• Subtraction: Remaining = Total - Eaten

• Simplification: 6/16 = 3/8 (divide by GCD of 2)

6 red slices eaten (3/8), 10 green slices remaining (5/8)

2 Birthday Cake Problem

Exercise 2

Mom baked 3 rectangular cakes and cut each into 6 equal pieces. She gave 1/2 of the total pieces to the children, 1/6 of the total pieces to her coworkers, and kept the rest for the family. How many pieces did each group get? Express the family's portion as a mixed number if possible.

Definition:

Finding fraction of a number: To find 1/2 of 18 pieces, multiply 18 by 1/2 = (18 × 1) ÷ 2 = 9 pieces.

Total cakes

Pieces per cake

Total pieces

Step 1: Calculate total pieces

3 cakes × 6 pieces each = 18 total pieces

Step 2: Calculate pieces for children

1/2 of 18 = (1 × 18) ÷ 2 = 18 ÷ 2 = 9 pieces

Step 3: Calculate pieces for coworkers

1/6 of 18 = (1 × 18) ÷ 6 = 18 ÷ 6 = 3 pieces

Step 4: Calculate pieces for family

Family = Total - (Children + Coworkers) = 18 - (9 + 3) = 6 pieces

Step 5: Express family portion as fraction

Family fraction = 6/18 = 1/3 of the total

Children got 9 pieces (1/2), coworkers got 3 pieces (1/6), family got 6 pieces (1/3).

Final answer:

Children got 9 pieces (1/2), coworkers got 3 pieces (1/6), family got 6 pieces (1/3).

Applied rules:

• Fraction of a whole number: Multiply the whole number by the numerator, then divide by denominator

• Subtraction: To find remaining amount, subtract given amounts from total

• Equivalent fractions: 6/18 = 1/3

• Verification: 1/2 + 1/6 + 1/3 = 3/6 + 1/6 + 2/6 = 6/6 = 1 whole

9 red pieces (1/2) to children, 3 yellow pieces (1/6) to coworkers, 6 green pieces (1/3) to family

3 Cookie Distribution Problem

Exercise 3

Emma baked 25 cookies. She gave 1/5 to her grandmother, 1/4 to her neighbor, and kept the rest for her family. How many cookies did each person get? What fraction of the cookies did Emma keep for her family? Express as a mixed number if possible.

Definition:

Finding multiple fractions of a number: Calculate each fraction separately, then subtract from the total to find the remaining amount.

Total cookies

To grandmother

1/5

To neighbor

1/4

Step 1: Find 1/5 of 25 cookies

1/5 of 25 = (1 × 25) ÷ 5 = 25 ÷ 5 = 5 cookies to grandmother

Step 2: Find 1/4 of 25 cookies

1/4 of 25 = (1 × 25) ÷ 4 = 25 ÷ 4 = 6.25 ≈ 6 cookies to neighbor (rounded down)

Step 3: Calculate cookies for family

Family gets = Total - (Grandmother + Neighbor) = 25 - (5 + 6) = 14 cookies

Step 4: Express family portion as fraction

Family fraction = 14/25 of the total cookies

Step 5: Convert to mixed number (if applicable)

Since 14/25 < 1, it cannot be expressed as a mixed number

Grandmother got 5 cookies (1/5), neighbor got 6 cookies (6/25), family got 14 cookies (14/25).

Final answer:

Grandmother got 5 cookies (1/5), neighbor got 6 cookies (6/25), family got 14 cookies (14/25).

Applied rules:

• Fraction of a whole number: Multiply the whole by numerator, divide by denominator

• Adding fractions: When denominators differ, calculate separately then add

• Subtraction: To find remaining amount, subtract given amounts from total

• Mixed numbers: Only possible when fraction ≥ 1 (numerator ≥ denominator)

Real World Problems: Fractions and Mixed Numbers in Grade 3

Real World Problems: Fractions and Mixed Numbers in Grade 3

Real World Problems: Fractions and Mixed Numbers in Grade 3