Let’s go step by step through each section. We’ll solve every problem carefully and check our work.
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Section A: Reciprocals
Reciprocal means “flip the fraction” — so for a fraction like 3/4, its reciprocal is 4/3. When you multiply a number by its reciprocal, you always get 1.
Problem 1:
Prove that 3/4 × 4/3 = 1
→ Multiply numerators: 3 × 4 = 12
→ Multiply denominators: 4 × 3 = 12
→ So, 12/12 = 1
✔
Problem 2: Fill in the blanks
a) 2/3 × ___ = 1 → reciprocal of 2/3 is 3/2 → answer:
3/2
b) ___ × 5/7 = 1 → reciprocal of 5/7 is 7/5 → answer:
7/5
c) 1 = 1/2 × ___ → reciprocal of 1/2 is 2 → answer:
2
d) ___ × 8 = 1 → reciprocal of 8 (which is 8/1) is 1/8 → answer:
1/8
Also: Any number multiplied by its
reciprocal is equal to 1.
Problem 3: Find the reciprocal
a) 6/11 → flip it →
11/6
b) -2/3 → flip it →
-3/2 (keep the negative sign!)
c) 5 → write as 5/1 → flip →
1/5
d) 1/2 → flip →
2/1 or just 2
e) 8/19 → flip →
19/8
f) 4 2/3 → first convert to improper fraction:
4 2/3 = (4×3 + 2)/3 = 14/3 → flip →
3/14
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Section B: Dividing integers by fractions
Dividing by a fraction means multiplying by its reciprocal.
Problem 1: Explain diagram
The big box is labeled “1”. It’s split into 3 equal parts, each labeled 1/3.
So, how many 1/3s are in 1? → 3 of them!
That’s why 1 ÷ 1/3 = 3.
Problem 2: Calculate
Remember: dividing by a fraction = multiply by its reciprocal.
a) 2 ÷ 1/3 = 2 × 3/1 =
6
b) 2 ÷ 2/3 = 2 × 3/2 = 6/2 =
3
c) 10 ÷ 2/3 = 10 × 3/2 = 30/2 =
15
d) 10 ÷ 2/5 = 10 × 5/2 = 50/2 =
25
e) 10 ÷ 3/5 = 10 × 5/3 = 50/3 =
16 2/3 (or leave as 50/3 if mixed numbers not required)
f) 21 ÷ 2 1/3 → first convert 2 1/3 to improper: (2×3+1)/3 = 7/3
Then: 21 ÷ 7/3 = 21 × 3/7 = 63/7 =
9
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Section C: Dividing any pair of fractions
Same rule: divide = multiply by reciprocal.
a) 1/3 ÷ 1/3 = 1/3 × 3/1 = 3/3 =
1
b) 2/3 ÷ 1/2 = 2/3 × 2/1 = 4/3 =
1 1/3
c) 4 2/3 ÷ 1/2 → convert 4 2/3 to 14/3 → 14/3 × 2/1 = 28/3 =
9 1/3
d) 5/7 ÷ 5/12 = 5/7 × 12/5 = (5×12)/(7×5) = 60/35 → simplify: divide numerator and denominator by 5 →
12/7 or 1 5/7
e) -5/12 ÷ 4/9 = -5/12 × 9/4 = (-5×9)/(12×4) = -45/48 → simplify: divide by 3 →
-15/16
f) 2 1/8 ÷ 9/10 → convert 2 1/8 to 17/8 → 17/8 × 10/9 = 170/72 → simplify: divide by 2 → 85/36 =
2 13/36
g) 9/11 ÷ 9/11 = 9/11 × 11/9 = 99/99 =
1
h) 7/12 ÷ 3/4 ÷ 1/2 → do left to right:
First: 7/12 ÷ 3/4 = 7/12 × 4/3 = 28/36 = 7/9
Then: 7/9 ÷ 1/2 = 7/9 × 2/1 = 14/9 =
1 5/9
i) 3 1/7 ÷ 5 1/2 → convert both:
3 1/7 = 22/7, 5 1/2 = 11/2
So: 22/7 ÷ 11/2 = 22/7 × 2/11 = 44/77 → simplify: divide by 11 →
4/7
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Final Answer:
Section A
1) Proven: 3/4 × 4/3 = 12/12 = 1
2) a) 3/2, b) 7/5, c) 2, d) 1/8; blank: reciprocal
3) a) 11/6, b) -3/2, c) 1/5, d) 2, e) 19/8, f) 3/14
Section B
1) The diagram shows 1 whole divided into 3 equal parts of 1/3 each → so there are 3 thirds in 1 → 1 ÷ 1/3 = 3
2) a) 6, b) 3, c) 15, d) 25, e) 50/3 or 16⅔, f) 9
Section C
1) a) 1, b) 4/3 or 1⅓, c) 28/3 or 9⅓, d) 12/7 or 1⁵⁄₇, e) -15/16, f) 85/36 or 2¹³⁄₆, g) 1, h) 14/9 or 1⁵₉, i) 4/7
Parent Tip: Review the logic above to help your child master the concept of 6th grade math fractions worksheet.
