Question 1: Write 3 equivalent fractions and simplify
Part A: Write 3 equivalent fractions
What to do: Find fractions that are equal to the given fraction by multiplying or dividing both numerator and denominator by the same number.
a) 1/3
- Answer: 2/6, 3/9, 4/12
- Steps: Multiply both top and bottom by 2, 3, and 4
b) 1/2
- Answer: 2/4, 3/6, 4/8
- Steps: Multiply both top and bottom by 2, 3, and 4
c) 1/6
- Answer: 2/12, 3/18, 4/24
- Steps: Multiply both top and bottom by 2, 3, and 4
d) 1/10
- Answer: 2/20, 3/30, 4/40
- Steps: Multiply both top and bottom by 2, 3, and 4
Part B: Simplify fractions
What to do: Find the largest number that divides both numerator and denominator, then divide both by that number.
e) 6/8
- Answer: 3/4
- Steps: Both 6 and 8 can be divided by 2 → 6÷2 = 3, 8÷2 = 4
f) 10/15
- Answer: 2/3
- Steps: Both 10 and 15 can be divided by 5 → 10÷5 = 2, 15÷5 = 3
g) 9/12
- Answer: 3/4
- Steps: Both 9 and 12 can be divided by 3 → 9÷3 = 3, 12÷3 = 4
h) 16/24
- Answer: 2/3
- Steps: Both 16 and 24 can be divided by 8 → 16÷8 = 2, 24÷8 = 3
Question 2: Convert between fractions, decimals, and percentages
Part A: Write equivalent fractions and decimals
What to do: Convert percentages to fractions (over 100) and decimals (divide by 100).
a) 10%
- Fraction: 10/100 = 1/10
- Decimal: 0.1
b) 50%
- Fraction: 50/100 = 1/2
- Decimal: 0.5
c) 75%
- Fraction: 75/100 = 3/4
- Decimal: 0.75
d) 20%
- Fraction: 20/100 = 1/5
- Decimal: 0.2
Part B: Write equivalent fractions and percentages
What to do: Convert decimals to fractions and percentages.
e) 0.64
- Fraction: 64/100 = 16/25
- Percentage: 64%
f) 0.32
- Fraction: 32/100 = 8/25
- Percentage: 32%
g) 0.09
- Fraction: 9/100
- Percentage: 9%
h) 0.8
- Fraction: 8/10 = 4/5
- Percentage: 80%
Question 3: Measure percentages and convert to different units
What to do: Calculate percentages of 1.2 litres and convert to millilitres. Note: 1 litre = 1000 millilitres
a) 50% of 1.2L
- Calculation: 1.2 × 0.5 = 0.6L
- In millilitres: 0.6 × 1000 = 600ml
b) 25% of 1.2L
- Calculation: 1.2 × 0.25 = 0.3L
- In millilitres: 0.3 × 1000 = 300ml
c) 10% of 1.2L
- Calculation: 1.2 × 0.1 = 0.12L
- In millilitres: 0.12 × 1000 = 120ml
d) 20% of 1.2L
- Calculation: 1.2 × 0.2 = 0.24L
- In millilitres: 0.24 × 1000 = 240ml
e) 5% of 1.2L
- Calculation: 1.2 × 0.05 = 0.06L
- In millilitres: 0.06 × 1000 = 60ml
f) 1% of 1.2L
- Calculation: 1.2 × 0.01 = 0.012L
- In millilitres: 0.012 × 1000 = 12ml
Question 4: Egyptian Fractions Investigation
What to do: Prove that 1/3 + 1/6 = 1/2 and find patterns in unit fractions.
Prove that 1/3 + 1/6 = 1/2
Steps:
- Find common denominator: 6
- Convert 1/3 to sixths: 1/3 = 2/6
- Add: 2/6 + 1/6 = 3/6
- Simplify: 3/6 = 1/2 ✓
Find 2 unit fractions that add up to make these:
For any unit fraction 1/n, we want to find unit fractions 1/a and 1/b such that: 1/a + 1/b = 1/n
1/n = 1/(n+1) + 1/[n(n+1)]
For 1/4:
- Answer: 1/5 + 1/20 = 1/4
- Check: 4/20 + 1/20 = 5/20 = 1/4 ✓
For 1/5:
- Answer: 1/6 + 1/30 = 1/5
- Check: 5/30 + 1/30 = 6/30 = 1/5 ✓
For 1/6:
- Answer: 1/7 + 1/42 = 1/6
- Check: 6/42 + 1/42 = 7/42 = 1/6 ✓