RS Aggarwal Solutions for Class 6 Chapter 6 Simplification

 

Exercise 6.1

 

Question 1. Write the fraction representing the shaded portion:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
Answer:
(i) \( \frac{2}{3} \)
(ii) \( \frac{11}{15} \)
(iii) \( \frac{8}{9} \)
(iv) \( \frac{3}{7} \)
(v) \( \frac{4}{9} \)
(vi) \( \frac{1}{2} \)
(vii) \( \frac{1}{2} \)
(viii) \( \frac{1}{5} \)
(ix) \( \frac{1}{4} \)

Exam Tip: To find the fraction of a shaded region, count the total number of equal parts and how many are shaded. The shaded parts form the numerator and the total parts form the denominator.

 

Question 2. Write the fraction representing the shaded parts:
(i)
(ii)
(iii)
(iv)
Answer:
(i) \( \frac{3}{9} = \frac{1}{3} \)
(ii) \( \frac{4}{8} = \frac{1}{2} \)
(iii) \( \frac{3}{12} = \frac{1}{4} \)
(iv) \( \frac{5}{10} = \frac{1}{2} \)

Exam Tip: Always reduce your fraction to its simplest form by dividing both numerator and denominator by their greatest common divisor.

 

Question 3. Write the fraction representing the shaded portion:
(i)
(ii)
Answer:
(i) \( \frac{1}{2} \)
(ii) \( \frac{4}{8} = \frac{1}{2} \)

Exam Tip: Visual shapes may look different, but fractions with the same value are equivalent - always simplify to verify.

 

Question 4. Colour the part according to the fraction given:
(i) \( \frac{1}{6} \)
(ii) \( \frac{2}{4} \)
(iii) \( \frac{1}{3} \)
(iv) \( \frac{3}{4} \)
(v) \( \frac{4}{9} \)
(vi) \( \frac{1}{4} \)
Answer: Colour the shaded regions as shown in the solution diagram.

Exam Tip: When colouring fractions, divide the shape into equal parts based on the denominator, then shade the number of parts equal to the numerator.

 

Question 5. What fraction of an hour is 20 minutes?
Answer: There are 60 minutes in an hour. So 20 minutes is \( \frac{20}{60} = \frac{1}{3} \) of an hour.

Exam Tip: Always identify the total quantity (here 60 minutes) and express the given quantity as a fraction with that total as the denominator.

 

Question 6. Write the natural numbers from 2 to 12. What fraction of them are prime numbers?
Answer: The natural numbers from 2 to 12 are: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. The prime numbers among them are: 2, 3, 5, 7, and 11. Out of 11 numbers in total, 5 are prime. Therefore, the fraction of prime numbers is \( \frac{5}{11} \).

Exam Tip: A prime number has exactly two factors - 1 and itself. Remember that 1 is not considered prime, and 2 is the only even prime number.

 

Question 7. Write the natural numbers from 102 to 113. What fraction of them are prime numbers.
Answer: The natural numbers from 102 to 113 are: 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, and 113. Among these, the prime numbers are: 103, 107, 109, and 113. Out of 12 numbers total, 4 are prime. Therefore, the fraction of prime numbers is \( \frac{4}{12} = \frac{1}{3} \).

Exam Tip: Check divisibility carefully for larger numbers - test small prime divisors first before concluding a number is prime.

 

Question 8. Mukesh has a box of 24 pencils. He gives half of them to sunita. How many does sunita get? How many does mukesh still have?
Answer: Mukesh has 24 pencils in total. He gives half to Sunita, which means Sunita gets \( \frac{24}{2} = 12 \) pencils. The number of pencils Mukesh still keeps is \( 24 - 12 = 12 \) pencils.

Exam Tip: "Half" always means dividing by 2. Always verify your answer by checking that the two parts add back to the original total.

 

Question 9. Kavita has 44 cassettes. She gives 3/4 of them to Sonia. How many does Sonia get? How many does kavita keep?
Answer: Kavita has 44 cassettes. She gives \( \frac{3}{4} \) of them to Sonia. To find \( \frac{3}{4} \) of 44, divide 44 by 4 to get 11, then multiply by 3 to get 33. So Sonia gets 33 cassettes. The number of cassettes Kavita keeps is \( 44 - 33 = 11 \) cassettes.

Exam Tip: To find a fraction of a whole number, first divide by the denominator, then multiply by the numerator. Always show both parts of the calculation.

 

Question 10. Shikas has three frocks that she wears when playing. The material is good, but the colours are faded. Her mother buys some blue dye and uses it on two of the frocks. What fraction of all of the shikas play frocks did her mother dye?
Answer: Shikha has 3 frocks in total. Her mother dyed 2 of them with blue dye. The fraction of frocks that were dyed is \( \frac{2}{3} \). Therefore, Shikha's mother dyed \( \frac{2}{3} \) of Shikha's frocks.

Exam Tip: When finding what fraction of something was affected, use the part affected as the numerator and the total quantity as the denominator.

 

Exercise 6.2

 

Question 1. Represent 2/5 on a number line.
Answer: On a number line, mark points from 0 to 2. Divide the space between 0 and 1 into 5 equal parts. The point \( \frac{2}{5} \) is located 2 divisions away from 0.

Exam Tip: The denominator tells you how many equal divisions to make between 0 and 1; the numerator shows which division to mark.

 

Question 2. Represent 0/10, 1/10, 5/10 and 10/10 on a number line.
Answer: On a number line from 0 to 1 (or beyond), divide the space into 10 equal parts. Mark \( \frac{0}{10} \) at 0, \( \frac{1}{10} \) at the first mark, \( \frac{5}{10} \) at the middle (which equals \( \frac{1}{2} \)), and \( \frac{10}{10} \) at 1.

Exam Tip: Notice that \( \frac{10}{10} = 1 \) and \( \frac{5}{10} = \frac{1}{2} \) - fractions can be reduced to simpler forms.

 

Question 3. Represent 2/7, 5/7 and 6/7 on a number line.
Answer: On a number line from 0 to 1, divide the space into 7 equal parts. Mark \( \frac{2}{7} \) at the second division, \( \frac{5}{7} \) at the fifth division, and \( \frac{6}{7} \) at the sixth division from 0.

Exam Tip: All three fractions lie between 0 and 1. The larger the numerator (with the same denominator), the farther right the fraction appears on the number line.

 

Question 4. How many fractions lie between 0 and 1?
Answer: There are infinitely many fractions between 0 and 1. This can be verified by observing that for any fraction, you can always find another fraction between it and 0 or 1 by taking numerators and denominators smaller or larger in appropriate ways.

Exam Tip: A fraction lies between 0 and 1 when the numerator is smaller than the denominator. No matter how close two fractions are, there are always more fractions between them.

 

Question 5. Represent 0/8 and 8/8 on a number line.
Answer: On a number line, mark \( \frac{0}{8} \) at point 0 and \( \frac{8}{8} \) at point 1, since \( \frac{8}{8} \) equals 1.

Exam Tip: When the numerator and denominator are equal, the fraction always equals 1. When the numerator is 0, the fraction always equals 0.

 

Exercise 6.3

 

Question 1. Write each of the following divisions as fraction:
1. 6 ÷ 3
2. 25 ÷ 5
3. 125 ÷ 50
4. 55 ÷ 11
Answer:
1. \( \frac{6}{3} \)
2. \( \frac{25}{5} \)
3. \( \frac{125}{50} \)
4. \( \frac{55}{11} \)

Exam Tip: Division can be written as a fraction by placing the dividend (first number) as the numerator and the divisor (second number) as the denominator.

 

Question 2. Write each of the following fractions as divisions:
1. 9/7
2. 3/11
3. 90/63
4. 1/5
Answer:
1. 9 ÷ 7
2. 3 ÷ 11
3. 90 ÷ 63
4. 1 ÷ 5

Exam Tip: Any fraction can be expressed as a division problem - the numerator is divided by the denominator. This shows that fractions and division are closely related concepts.

 

Question 1. Convert each of the following into a mixed fraction:
(i) 28/9
(ii) 226/15
(iii) 145/9
(iv) 128/5
Answer: To convert an improper fraction to a mixed fraction, divide the numerator by the denominator. The quotient becomes the whole number part, the remainder becomes the new numerator, and the denominator stays the same.
(i) 3 1/9
(ii) 15 1/15
(iii) 16 1/9
(iv) 25 3/5

Exam Tip: Always verify your mixed fraction by converting it back to an improper fraction - multiply the whole number by the denominator and add the numerator.

 

Question 2. Convert each of the following into an improper fraction:
(i) 7 1/4
(ii) 8 5/7
(iii) 5 3/10
(iv) 12 7/15
Answer: To change a mixed fraction to an improper fraction, multiply the whole number by the denominator, then add the numerator. Write this total as the new numerator with the same denominator.
(i) 29/9
(ii) 61/7
(iii) 53/10
(iv) 187/15

Exam Tip: Check your work by dividing the numerator by the denominator - you should get back the original mixed fraction.

 

Question 1. (i) Looking at the circles and rectangles shown, write the fraction that represents the shaded portion in each case. Are they equivalent fractions?
Answer: When we look at each diagram, we can determine the fraction by counting the shaded parts and total parts. The circle shows 1/2 shaded. The second circle shows 2/4 shaded, which equals 1/2. The third circle shows 3/6 shaded, which also equals 1/2. The final circle shows 4/8 shaded, which simplifies to 1/2. Since all four fractions reduce to 1/2, they are indeed equivalent to one another.
In simple words: All four diagrams show the same amount shaded - exactly half. Even though the numbers look different, they all mean the same thing.

Exam Tip: Equivalent fractions represent the same value. You can check by reducing each fraction to its simplest form - if they're the same, they're equivalent.

 

Question 1. (ii) Looking at the rectangular diagrams shown, write the fraction that represents the shaded portion in each case. Are they equivalent fractions?
Answer: The first rectangle shows 5/15 shaded. When we reduce this by dividing both parts by 5, we get 1/3. The second rectangle shows 3/9 shaded, which also reduces to 1/3. The third rectangle shows 2/6 shaded, which simplifies to 1/3. All three diagrams display the same simplified value, so they are equivalent to each other.
In simple words: All three rectangles show exactly one-third of the space filled. The numbers differ, but the amount is the same.

Exam Tip: To verify equivalence, divide both the numerator and denominator by their HCF - if you get the same result, the fractions are equivalent.

 

Question 2. Write the fractions and match fractions in column I with the equivalent fractions in column II.
Answer: By examining the shaded sections in each diagram, we can find what fraction is represented and then match it with its equivalent form. Looking at the visual representations and reducing them to their simplest forms allows us to identify which diagrams show equal quantities.
(i) (b)
(ii) (c)
(iii) (a)
(iv) (d)

Exam Tip: Draw grid lines or use cross-multiplication to match equivalent fractions quickly - if a × d = b × c, then a/b = c/d.

 

Question 3. Replace * in each of the following by the correct number:
(i) 2/7 = 6/*
(ii) 5/8 = 10/*
(iii) 4/5 = */20
(iv) 45/60 = 15/*
(v) 18/24 = */4
Answer:
(i) 2/7 = 6/21
(ii) 5/8 = 10/16
(iii) 4/5 = 16/20
(iv) 45/60 = 15/20
(v) 18/24 = 3/4

Exam Tip: Determine the scale factor by dividing the known parts of the original fraction into the known parts of the equivalent fraction, then apply this factor to find the missing number.

 

Question 4. Find the equivalent fraction of 3/5, having:
(i) Numerator 9
(ii) Denominator 30
(iii) Denominator 21
(iv) Numerator 40
Answer:
(i) To get numerator 9, we need to find what number, when multiplied by 3, gives 9. Since 3 × 3 = 9, we multiply both the top and bottom by 3: 3/5 × 3/3 = 9/15
(ii) To get denominator 30, we identify that 5 × 6 = 30, so we multiply both parts by 6: 3/5 × 6/6 = 18/30
(iii) To get denominator 21, we see that 5 × 7 = 21, so we multiply both numerator and denominator by 7: 3/5 × 7/7 = 21/35
(iv) To get numerator 40, we note that 3 × 8 = 24, but we need to reconsider. Actually, if the numerator should be 40 and we start with 3/5, we multiply by 8/8 to get a different result. Let me recalculate: Since 5 × 8 = 40, we multiply both parts by 8: 3/5 × 8/8 = 24/40

Exam Tip: Always identify the scale factor first by dividing the new value by the old value for either the numerator or denominator, then apply it to both parts consistently.

 

Question 5. Find the fraction equivalent to 45/60, having:
(i) Numerator 15
(ii) Denominator 4
(iii) Denominator 240
(iv) Numerator 135
Answer:
(i) Since 45 ÷ 3 = 15, we divide both the numerator and denominator by 3: 45/60 ÷ 3/3 = 15/20
(ii) Since 60 ÷ 15 = 4, we divide both parts by 15: 45/60 ÷ 15/15 = 3/4
(iii) Since 60 × 4 = 240, we multiply both numerator and denominator by 4: 45/60 × 4/4 = 180/240
(iv) Since 45 × 3 = 135, we multiply both the top and bottom by 3: 45/60 × 3/3 = 135/180

Exam Tip: When finding a larger denominator, multiply; when finding a smaller denominator, divide - the key is finding the correct scale factor.

 

Question 6. Find the fraction equivalent to 35/42, having:
(i) Numerator 15
(ii) Denominator 18
(iii) Denominator 30
(iv) Numerator 30
Answer: First, we reduce 35/42 to its simplest form. Since the HCF of 35 and 42 is 7, we divide: 35/42 ÷ 7/7 = 5/6
(i) To get numerator 15, since 5 × 3 = 15, we multiply both parts by 3: 5/6 × 3/3 = 15/18
(ii) To get denominator 18, since 6 × 3 = 18, we multiply numerator and denominator by 3: 5/6 × 3/3 = 15/18
(iii) To get denominator 30, since 6 × 5 = 30, we multiply both top and bottom by 5: 5/6 × 5/5 = 25/30
(iv) To get numerator 30, since 5 × 6 = 30, we multiply both parts by 6: 5/6 × 6/6 = 30/36

Exam Tip: Always reduce the fraction to its lowest terms first to make calculations simpler, then build up the equivalent fraction with the required numerator or denominator.

 

Question 7. Check whether the given fractions are equivalent:
(i) 5/9, 30/54
(ii) 2/7, 16/42
(iii) 7/13, 5/11
(iv) 4/11, 32/88
(v) 3/10, 12/50
(vi) 9/27, 25/75
Answer:
(i) We multiply 5 × 54 and 9 × 30. This gives 5 × 54 = 270 and 9 × 30 = 270. Since both products are equal, the fractions are equivalent.
(ii) We multiply 2 × 42 and 7 × 16. This gives 2 × 42 = 84 and 7 × 16 = 112. Since 84 ≠ 112, these fractions are not equivalent.
(iii) We multiply 7 × 11 and 13 × 5. This gives 7 × 11 = 77 and 13 × 5 = 65. Since 77 ≠ 65, these fractions are not equivalent.
(iv) We multiply 4 × 88 and 11 × 32. This gives 4 × 88 = 352 and 11 × 32 = 352. Since both are equal, these fractions are equivalent.
(v) We multiply 3 × 50 and 10 × 12. This gives 3 × 50 = 150 and 10 × 12 = 120. Since 150 ≠ 120, these fractions are not equivalent.
(vi) We multiply 9 × 75 and 27 × 25. This gives 9 × 75 = 675 and 27 × 25 = 675. These products are equal, so the fractions are equivalent.

Exam Tip: Use cross-multiplication to test equivalence quickly - if the products are equal, the fractions are equivalent; if not, they are different.

 

Question 8. Match the equivalent fractions and write another 2 for each:

Answer: By reducing each fraction on the left to its simplest form, we can match it with its equivalent on the right and then generate two additional equivalent fractions.
(i) Matches with (d) 5/8. Two more equivalents: 250/400 = 5/8 = 10/16 = 15/24
(ii) Matches with (e) 9/10. Two more equivalents: 180/200 = 9/10 = 18/20 = 27/30
(iii) Matches with (a) 2/3. Two more equivalents: 660/990 = 2/3 = 4/6 = 6/9
(iv) Matches with (c) 1/2. Two more equivalents: 180/360 = 1/2 = 2/4 = 3/6
(v) Matches with (b) 2/5. Two more equivalents: 220/550 = 2/5 = 4/10 = 6/15

Exam Tip: Simplify each fraction first by dividing by the HCF, then generate more equivalents by multiplying both numerator and denominator by the same number.

 

Question 9. Write some equivalent fractions which contain all digits from 1 to 9 once only.
Answer: These special fractions use each digit from 1 through 9 exactly once across the numerator and denominator combined. Examples include: 2/6 = 3/9 = 58/174 and 2/4 = 3/6 = 79/158. These satisfy the pandigital requirement where no digit repeats.

Exam Tip: For pandigital equivalent fractions, verify that each digit 1-9 appears exactly once across all three fractions, and confirm equivalence using cross-multiplication.

 

Question 10. Ravish had 20 pencils, Sikha had 50 pencils and Priya had 80 pencils. After 4 months, Ravish used up 10 pencils, Sikha used up 25 pencils and Priya used 40 pencils. What fraction did each use up? Check if each has used up an equal fraction of their pencils.
Answer: We need to find what fraction of their starting pencils each person used.

For Ravish: He had 20 pencils and used 10. The fraction used = 10/20. When we reduce this by dividing both parts by their HCF of 10, we get 1/2.

For Sikha: She had 50 pencils and used 25. The fraction used = 25/50. When we divide both the numerator and denominator by their HCF of 25, we get 1/2.

For Priya: She had 80 pencils and used 40. The fraction used = 40/80. Dividing both parts by their HCF of 40 gives us 1/2.

All three individuals reduced their fractions to 1/2, which means each person used exactly the same fraction of their pencils - half of what they started with.

Exam Tip: Always reduce fractions to simplest form before comparing to see if different fractions are actually equivalent.

 

Question 1. Reduce each of the following fractions to its lowest term (simplest form):
(i) 40/75
(ii) 42/28
(iii) 12/52
(iv) 40/72
(v) 80/24
(vi) 84/56
Answer:
(i) For 40/75: The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. The factors of 75 are 1, 3, 5, 15, and 75. The common factors are 1 and 5. The HCF is 5. When we divide both numerator and denominator by 5, we get 8/15.

(ii) For 42/28: The factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42. The factors of 28 are 1, 2, 4, 7, 14, 28. The common factors are 1, 2, 7, and 14. The HCF is 4. When we divide both parts by 4, we get 3/7. (Note: The correct HCF is actually 14, giving 3/2, but checking the working: 42 ÷ 14 = 3 and 28 ÷ 14 = 2, so the simplest form is 3/2.)

(iii) For 12/52: Factors of 12 are 1, 2, 3, 4, 6, 12. Factors of 52 are 1, 2, 4, 13, 26, 52. Common factors are 1, 2, and 4. The HCF is 4. Dividing both by 4 gives 3/13.

(iv) For 40/72: Factors of 40 are 1, 2, 4, 5, 8, 10, 20, 40. Factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. Common factors include 1, 2, 4, and 8. The HCF is 8. Dividing both by 8 gives 5/9.

(v) For 80/24: Factors of 80 are 1, 2, 4, 5, 8, 10, 16, 20, 40, 80. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Common factors are 1, 2, 4, and 8. The HCF is 8. Dividing both by 8 gives 10/3.

(vi) For 84/56: Factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. Factors of 56 are 1, 2, 4, 7, 8, 14, 28, 56. Common factors are 1, 2, 4, 7, 14, and 28. The HCF is 28. Dividing both by 28 gives 3/2.

Exam Tip: Always find the HCF of the numerator and denominator, then divide both by this number - this gives the fraction in its simplest form in one step.

 

Question 2. Simplify each of the following to its lowest form:
(i) 75/80
(ii) 52/76
(iii) 84/98
(iv) 68/17
(v) 150/50
(vi) 162/108
Answer:
(i) 75/80
The divisors of 75 are 1, 3, 5, 15, 25, and 75.
The divisors of 80 are 1, 2, 4, 5, 8, 10, 12, 16, 20, 40, and 80.
The common divisors of 75 and 80 are 1 and 5.
HCF of 75 and 80 is 5.
When we divide the top and bottom by 5, we get:
\( \frac{75}{80} \div \frac{5}{5} = \frac{15}{16} \)
Therefore, the simplest form is \( \frac{75}{80} = \frac{15}{16} \)

(ii) 52/76
The divisors of 52 are 1, 2, 4, 13, 26, and 52.
The divisors of 76 are 1, 2, 4, 19, 38, and 76.
The common divisors of 52 and 76 are 1, 2, and 4.
HCF of 52 and 76 is 4.
When we divide the top and bottom by 4, we get:
\( \frac{52}{76} \div \frac{4}{4} = \frac{13}{19} \)
Therefore, the simplest form is \( \frac{52}{76} = \frac{13}{19} \)

(iii) 84/98
The divisors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.
The divisors of 98 are 1, 2, 7, 14, 49, and 98.
The common divisors of 84 and 98 are 1, 2, 7, and 14.
HCF of 84 and 98 is 14.
When we divide the top and bottom by 14, we get:
\( \frac{84}{98} \div \frac{14}{14} = \frac{6}{7} \)
Therefore, the simplest form is \( \frac{84}{98} = \frac{6}{7} \)

(iv) 68/17
The divisors of 68 are 1, 2, 4, 17, 34, and 68.
The divisors of 17 are 1 and 17.
The common divisor of 68 and 17 is 17.
HCF of 68 and 17 is 17.
When we divide the top and bottom by 17, we get:
\( \frac{68}{17} \div \frac{17}{17} = \frac{4}{1} \)
Therefore, the simplest form is \( \frac{68}{17} = \frac{4}{1} \)

(v) 150/50
The divisors of 150 are 1, 2, 3, 5, 6, 10, 15, 25, 50, and 150.
The divisors of 50 are 1, 2, 5, 10, 25, and 50.
The common divisor of 150 and 50 is 50.
HCF of 150 and 50 is 50.
When we divide the top and bottom by 50, we get:
\( \frac{150}{50} \div \frac{50}{50} = \frac{3}{1} \)
Therefore, the simplest form is \( \frac{150}{50} = \frac{3}{1} \)

(vi) 162/108
The divisors of 162 are 1, 2, 3, 6, 9, 18, 27, 54, 81, and 162.
The divisors of 108 are 108, 1, 2, 3, 4, 6, 9, 12, 18, 27, and 54.
The common divisors of 162 and 108 are 1, 2, 3, 6, 9, 18, 27, and 54.
HCF of 162 and 108 is 54.
When we divide the top and bottom by 54, we get:
\( \frac{162}{108} \div \frac{54}{54} = \frac{3}{2} \)
Therefore, the simplest form is \( \frac{162}{108} = \frac{3}{2} \)
In simple words: To reduce a fraction to its simplest form, find the highest common factor of the top and bottom numbers. Then split both by that factor. Keep dividing until you cannot divide any further.

Exam Tip: Always verify your final answer by checking that the numerator and denominator share no common factors other than 1. This confirms you have reached the simplest form.

 

Exercise 6.7

 

Question 1. Write each fraction. Arrange them in ascending and descending order using correct sign ` < ', ' = ' > ' between the fractions:
(i) [Pie chart visual representations showing fractions 0/8, 3/8, 4/8, 6/8]
(ii) [Grid visual representations showing fractions 3/9, 4/9, 6/9, 8/9]
(iii) [Circle visual representations showing fractions 1/6, 2/6, 3/6, 4/6, 5/6, 6/6]
Answer:
(i) Ascending order: \( \frac{0}{8} < \frac{3}{8} < \frac{4}{8} < \frac{6}{8} \)
Descending order: \( \frac{6}{8} > \frac{4}{8} > \frac{3}{8} > \frac{0}{8} \)

(ii) Ascending order: \( \frac{3}{9} < \frac{4}{9} < \frac{6}{9} < \frac{8}{9} \)
Descending order: \( \frac{8}{9} > \frac{6}{9} > \frac{4}{9} > \frac{3}{9} \)

(iii) Ascending order: \( \frac{1}{6} < \frac{2}{6} < \frac{3}{6} < \frac{4}{6} < \frac{5}{6} < \frac{6}{6} \)
Descending order: \( \frac{6}{6} > \frac{5}{6} > \frac{4}{6} > \frac{3}{6} > \frac{2}{6} > \frac{1}{6} \)
In simple words: When fractions have the same bottom number, just compare the top numbers. The one with the larger top is the larger fraction. Arrange from smallest to biggest for ascending, and biggest to smallest for descending.

Exam Tip: Always ensure that when denominators are equal, you only need to compare numerators. For ascending order, place the smallest numerator first; for descending, place the largest first.

 

Question 2. Mark 2/6, 4/6, 8/6, 6/6 on the number line and put appropriate signs between fractions given below:
(i) \( \frac{5}{6} \) _____ \( \frac{2}{6} \)
(ii) \( \frac{3}{6} \) _____ \( \frac{0}{6} \)
(iii) \( \frac{1}{6} \) _____ \( \frac{6}{6} \)
(iv) \( \frac{8}{6} \) _____ \( \frac{5}{6} \)
Answer:

(i) \( \frac{5}{6} \) > \( \frac{2}{6} \) because 5 > 2 and the denominator is the same.
(ii) \( \frac{3}{6} \) > \( \frac{0}{6} \) because 3 > 0 and the denominator is the same.
(iii) \( \frac{1}{6} \) < \( \frac{6}{6} \) because 1 < 6 and the denominator is the same.
(iv) \( \frac{8}{6} \) > \( \frac{5}{6} \) because 8 > 5 and the denominator is the same.
In simple words: When the bottom numbers match, examine the top numbers. The fraction with the bigger top is larger. Mark each fraction's position on the line by counting out the divisions.

Exam Tip: Use a number line to visualize fraction positions - this makes comparison much clearer and helps you place them correctly on the line.

 

Question 3. Compare the following fractions and put an appropriate sign:
(i) \( \frac{3}{6} \) _____ \( \frac{5}{6} \)
(ii) \( \frac{4}{5} \) _____ \( \frac{0}{5} \)
(iii) \( \frac{3}{20} \) _____ \( \frac{4}{20} \)
(iv) \( \frac{1}{7} \) _____ \( \frac{1}{4} \)
Answer:
(i) \( \frac{3}{6} < \frac{5}{6} \) because 3 < 5 and the denominator is the same.
(ii) \( \frac{4}{5} > \frac{0}{5} \) because 4 > 0 and the denominator is the same.
(iii) \( \frac{3}{20} < \frac{4}{20} \) because 3 < 4 and the denominator is the same.
(iv) \( \frac{1}{7} < \frac{1}{4} \) because 7 > 4; when the numerator is the same, then the fraction that has smaller denominator is greater.
In simple words: If both fractions have matching denominators, look at the top numbers - the bigger top means a bigger fraction. If the top numbers are the same, look at the bottom - the smaller bottom gives a bigger fraction.

Exam Tip: Remember the key rule: when numerators are equal, the fraction with the smaller denominator is always larger. This is because you are dividing by a smaller number.

 

Question 4. Compare the following fractions using the symbol > or <:
(i) 6/7 and 6/11
(ii) 3/7 and 5/7
(iii) 2/3 and 8/12
(iv) 1/5 and 4/15
(v) 8/3 and 8/13
(vi) 4/9 and 15/8
Answer:
(i) \( \frac{6}{7} > \frac{6}{11} \) because when the numerator is the same, then the fraction with smaller denominator is greater.

(ii) \( \frac{3}{7} < \frac{5}{7} \) because 3 < 5 and the denominator is the same.

(iii) \( \frac{8}{12} = \frac{2 \times 2 \times 2}{2 \times 2 \times 3} = \frac{2}{3} \)

Therefore, \( \frac{2}{3} = \frac{8}{12} \)

(iv) \( \frac{1}{5} = \frac{1}{5} \times \frac{3}{3} = \frac{3}{15} \)

Therefore, \( \frac{3}{15} < \frac{4}{15} \)

(Because 3 < 4 and the denominator is the same. Therefore, 1/15 < 4/15)

(v) \( \frac{8}{3} < \frac{8}{13} \) Because when the numerator is the same, then the fraction with smaller denominator is greater.

(vi) \( \frac{4}{9} = \frac{4}{9} \times \frac{8}{8} = \frac{32}{72} \)

\( \frac{15}{8} = \frac{15}{8} \times \frac{9}{9} = \frac{135}{72} \)

(Because 135 > 32 and the denominator is the same)

Therefore, \( \frac{4}{9} < \frac{15}{8} \)
In simple words: Compare fractions by examining their numerators and denominators. If both have the same bottom number, pick the one with the bigger top. If both have the same top, the one with the smaller bottom is larger. When nothing matches, convert to the same denominator first.

Exam Tip: For fractions with different denominators, always find a common denominator before comparing - this removes any confusion and makes the comparison direct.

 

Question 5. The following fractions represent just three different numbers. Separate them in to three groups of equal fractions by changing each one to its simplest form:
(i) 2/12
(ii) 3/15
(iii) 8/50
(iv) 16/100
(v) 10/60
(vi) 15/75
(vii) 12/60
(viii) 16/96
(ix) 12/75
(x) 12/72
(xi) 3/18
(xii) 4/25
Answer:
(i) 2/12
HCF of 2 & 12 is 2.
Divide both the numerator & denominator by the HCF of 2 &12
\( \frac{2}{12} \div \frac{2}{2} = \frac{1}{6} \)

(ii) 3/15
HCF of 3 & 15 is 3.
Divide both the numerator & denominator by the HCF of 3 &15.
\( \frac{3}{15} \div \frac{3}{3} = \frac{1}{5} \)

(iii) 8/50
HCF of 8 & 50 is 2.
Divide both the numerator & denominator by the HCF of 8 & 50.
\( \frac{8}{50} \div \frac{2}{2} = \frac{4}{25} \)

(iv) 16/100
HCF of 16 & 100 is 4.
Divide both the numerator & denominator by the HCF of 16 & 100.
\( \frac{16}{100} \div \frac{4}{4} = \frac{4}{25} \)

(v) 10/60
HCF of 10 & 60 is 10. (appears to be HCF - interpreting source)
Divide both the numerator & denominator by the HCF of 10 & 60.
\( \frac{10}{60} \div \frac{10}{10} = \frac{1}{6} \)

(vi) 15/75
HCF of 15 & 75 is 15.
Divide both the numerator & denominator by the HCF of 15 & 75.
\( \frac{15}{75} \div \frac{15}{15} = \frac{1}{5} \)

(vii) 12/60
HCF of 12 & 60 is 12.
Divide both the numerator & denominator by the HCF of 12 & 60.
\( \frac{12}{60} \div \frac{12}{12} = \frac{1}{5} \)

(viii) 16/96
HCF of 16 & 96 is 16.
Divide both the numerator & denominator by the HCF of 16 & 96.
\( \frac{16}{96} \div \frac{16}{16} = \frac{1}{6} \)

(ix) 12/75
HCF of 12 & 75 is 3.
Divide both the numerator & denominator by the HCF of 12 & 75.
\( \frac{12}{75} \div \frac{3}{3} = \frac{4}{25} \)

(x) 12/72
HCF of 12 & 72 is 12.
Divide both the numerator & denominator by the HCF of 12 & 72.
\( \frac{12}{72} \div \frac{12}{12} = \frac{1}{6} \)

(xi) 3/18
HCF of 3 & 18 is 3.
Divide both the numerator & denominator by the HCF of 3 & 18.
\( \frac{3}{18} \div \frac{3}{3} = \frac{1}{6} \)

(xii) 4/25
HCF of 4 & 25 is 1 (already in simplest form).
\( \frac{4}{25} \) is already in its simplest form.

Group 1 (equal to 1/6): 2/12, 10/60, 16/96, 12/72, 3/18
Group 2 (equal to 1/5): 3/15, 15/75, 12/60
Group 3 (equal to 4/25): 8/50, 16/100, 12/75, 4/25
In simple words: Reduce each fraction to its simplest form by dividing the top and bottom by their highest common factor. Then group all fractions that simplify to the same result together.

Exam Tip: Always double-check your HCF calculation before dividing - an incorrect HCF will lead to wrong groupings. Verify by checking that the final simplified numerator and denominator have no common factors.

 

Question 6. Isha read 25 pages of a book containing 100 pages. Nagma read 1/2 of the same book. Who read less?
Answer: Isha's reading amount equals 25 pages out of 100 total pages, which simplifies to 1/4 of the book (by dividing numerator and denominator by their HCF of 25). Nagma read 1/2 of the book. To compare, use 4 as the common denominator: 1/4 becomes 1/4, and 1/2 becomes 2/4. Since 1/4 is smaller than 2/4, Isha read a smaller portion of the book.
In simple words: Isha read 1/4 of the book while Nagma read 1/2. One quarter is less than one half, so Isha read less.

Exam Tip: Always reduce fractions to their simplest form before comparing, and convert to a common denominator when the denominators differ.

 

Question 7. Arrange the following fractions in the ascending order:
(i) \( \frac{2}{9}, \frac{7}{9}, \frac{3}{9}, \frac{4}{9}, \frac{1}{9}, \frac{6}{9}, \frac{5}{9} \)
(ii) \( \frac{7}{8}, \frac{7}{25}, \frac{7}{11}, \frac{7}{18}, \frac{7}{10} \)
(iii) \( \frac{37}{47}, \frac{37}{50}, \frac{37}{100}, \frac{37}{100}, \frac{37}{85}, \frac{37}{41} \)
(iv) \( \frac{3}{5}, \frac{1}{5}, \frac{4}{5}, \frac{2}{5} \)
(v) \( \frac{2}{5}, \frac{3}{4}, \frac{1}{2}, \frac{3}{5} \)
(vi) \( \frac{3}{8}, \frac{3}{12}, \frac{3}{6}, \frac{3}{4} \)
(vii) \( \frac{4}{6}, \frac{3}{8}, \frac{6}{12}, \frac{5}{16} \)
Answer:
(i) When denominators stay the same and numerators vary, the fraction having the larger numerator has a greater value. Therefore: \( \frac{1}{9}, \frac{2}{9}, \frac{3}{9}, \frac{4}{9}, \frac{5}{9}, \frac{6}{9}, \frac{7}{9} \)
(ii) When numerators stay the same and denominators differ, the fraction having the larger denominator has a smaller value. Therefore: \( \frac{7}{25}, \frac{7}{18}, \frac{7}{11}, \frac{7}{10}, \frac{7}{8} \)
(iii) When numerators are the same and denominators have different values, the fraction having the smaller denominator has a greater value. Therefore: \( \frac{37}{100}, \frac{37}{100}, \frac{37}{85}, \frac{37}{50}, \frac{37}{47}, \frac{37}{41} \)
(iv) When denominators remain constant and numerators are different, the fraction with the larger numerator has a greater value. Therefore: \( \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} \)
(v) LCM of 5, 4, and 2 is 20. Converting each fraction into an equivalent fraction with 20 as its denominator: \( \frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20} \), \( \frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20} \), \( \frac{2}{2} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20} \). Therefore: \( \frac{2}{5}, \frac{1}{2}, \frac{3}{5}, \frac{3}{4} \)
(vi) \( \frac{3}{12}, \frac{3}{8}, \frac{3}{6}, \frac{3}{4} \)
(vii) \( \frac{5}{16}, \frac{3}{8}, \frac{6}{12}, \frac{4}{6} \)
In simple words: Fractions with identical denominators arrange by numerator size. Fractions with the same numerator arrange by denominator - larger denominators mean smaller values. For different numerators and denominators, find a common denominator first.

Exam Tip: Always check whether denominators or numerators match before arranging - this determines which rule applies and saves calculation time.

 

Question 8. Arrange in descending order in each of the following using symbols >:
(i) 8/17, 8/9, 8/5, 8/13
(ii) 5/9, 3/12, 1/3, 4/15
Answer:
(i) \( \frac{8}{5} > \frac{8}{9} > \frac{8}{13} > \frac{8}{17} \)
(ii) \( \frac{5}{9} > \frac{1}{3} > \frac{3}{12} > \frac{4}{15} \)
In simple words: When numerators are the same, larger denominators make smaller fractions - so arrange by putting the smallest denominator first. When numerators differ, convert to a common denominator to compare accurately.

Exam Tip: For descending order with identical numerators, the fraction with the smallest denominator comes first.

 

Question 9. Find answers to the following. Write and indicate how you solved them.
(i) Is 5/9 equal to 4/5?
(ii) Is 9/16 equal to 5/9?
(iii) Is 4/5 equal to 16/20?
(iv) Is 1/15 equal to 4/30?
Answer:
(i) No. When you multiply the numerator and denominator across: 5 × 5 = 25 and 9 × 4 = 36. Since 25 ≠ 36, these fractions are not equal.
(ii) No. Cross multiplying gives: 9 × 9 = 81 and 16 × 5 = 80. Since 81 ≠ 80, they are not equal.
(iii) Yes. Cross multiplying: 4 × 20 = 80 and 5 × 16 = 80. Since both products equal 80, the fractions are equal.
(iv) No. Cross multiplying: 1 × 30 = 30 and 15 × 4 = 60. Since 30 ≠ 60, they are not equal.
In simple words: To check if two fractions are equal, cross multiply: multiply the first numerator by the second denominator, then multiply the first denominator by the second numerator. If both products match, the fractions are equal.

Exam Tip: Cross multiplication is the fastest method to test fraction equality - no need to find common denominators or reduce to simplest form.

 

Exercise 6.8

 

Question 1. Write these fractions appropriately as additions or subtraction:
Answer:
(i) \( \frac{1}{5} + \frac{2}{5} = \frac{3}{5} \) (or equivalently: \( \frac{1+2}{5} = \frac{3}{5} \))
(ii) \( \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \)
In simple words: When fractions share the same denominator, add the numerators together and keep the denominator unchanged. For subtraction, subtract the numerators while keeping the denominator the same.

Exam Tip: Always ensure denominators match before adding or subtracting fractions - if they don't, find a common denominator first.

 

Question 2. Solve:
(i) \( \frac{5}{12} + \frac{1}{12} \)
(ii) \( \frac{3}{15} + \frac{7}{15} \)
(iii) \( \frac{3}{22} + \frac{7}{22} \)
(iv) \( \frac{1}{4} + \frac{0}{4} \)
(v) \( \frac{4}{13} + \frac{2}{13} + \frac{1}{13} \)
(vi) \( \frac{0}{15} + \frac{2}{15} + \frac{1}{15} \)
(vii) \( \frac{7}{31} - \frac{4}{31} + \frac{9}{31} \)
(viii) \( 3\frac{2}{7} + \frac{1}{7} - 2\frac{3}{7} \)
(ix) \( 2\frac{1}{3} - 1\frac{2}{3} - 4\frac{1}{3} \)
(x) \( \frac{1}{3} - \frac{2}{3} + \frac{7}{3} \)
(xi) \( \frac{16}{7} - \frac{5}{7} + \frac{9}{7} \)
Answer:
(i) The given fractions are: \( \frac{5}{12} + \frac{1}{12} = \frac{5+1}{12} = \frac{6}{12} = \frac{1}{2} \). Hence the answer is 1/2.
(ii) The given fractions are: \( \frac{3}{6} + \frac{2}{6} = \frac{3+2}{6} = \frac{8}{6} = \frac{4}{3} \). Hence the answer is 4/3.
(iii) The given fractions are: \( \frac{3}{22} + \frac{7}{22} = \frac{3+7}{22} = \frac{10}{22} = \frac{5}{11} \). Hence the answer is 5/11.
(iv) The given fractions are: \( \frac{1}{4} + \frac{0}{4} = \frac{1+0}{4} = \frac{1}{4} \). Hence the answer is 1/4.
(v) The given fractions are: \( \frac{4}{13} + \frac{2}{13} + \frac{1}{13} = \frac{4+2+1}{13} = \frac{7}{13} \). Hence the answer is 7/13.
(vi) The given fractions are: \( \frac{0}{15} + \frac{2}{15} + \frac{1}{15} = \frac{0+2+1}{15} = \frac{3}{15} = \frac{1}{5} \). Hence the answer is 1/5.
(vii) The given fractions are: \( \frac{7}{31} - \frac{4}{31} + \frac{9}{31} = \frac{7-4+9}{31} = \frac{12}{31} \). Hence the answer is 12/31.
(viii) The given fractions are: \( 3\frac{2}{7} + \frac{1}{7} - 2\frac{3}{7} = \frac{23+1-17}{7} = \frac{7}{7} = \frac{1}{1} = 1 \). Hence the answer is 1/1 = 1.
(ix) The given fractions are: \( 3\frac{2}{7} + \frac{1}{7} - 2\frac{3}{7} = \frac{23+1-17}{7} = \frac{35}{7} = \frac{5}{1} = 5 \). Hence the answer is 5/1 = 5.
(x) The given fractions are: \( \frac{1}{3} - \frac{2}{3} + \frac{7}{3} = \frac{3-2+7}{3} = \frac{8}{3} \). Hence the answer is 8/3.
(xi) The given fractions are: \( \frac{16}{7} - \frac{5}{7} + \frac{9}{7} = \frac{16-5+9}{7} = \frac{20}{7} \). Hence the answer is 20/7.
In simple words: When adding or subtracting fractions with the same denominator, combine all numerators while keeping the denominator fixed, then simplify the result if possible.

Exam Tip: Always simplify your final answer to its lowest terms by dividing both numerator and denominator by their HCF.

 

Question 3. Shikha painted 1/5 of the wall space in her room. Her brother Ravish helped and painted 3/5 of the wall space. How much did they paint together? How much the room is left unpainted?
Answer: Shikha covered 1/5 of the wall, and Ravish covered 3/5 of the wall. The combined portion they painted is \( \frac{1}{5} + \frac{3}{5} = \frac{1+3}{5} = \frac{4}{5} \) of the wall. The unpainted section remaining is \( \frac{1-4}{5} = \frac{5-4}{5} = \frac{1}{5} \) of the wall.
In simple words: Shikha and Ravish together painted 4/5 of the wall. This leaves 1/5 of the wall still unpainted.

Exam Tip: Always verify that the painted and unpainted portions add up to 1 (or the whole) as a final check.

 

Question 4. Ramesh bought 2\(\frac{1}{2}\) kg sugar whereas rohit bought 3\(\frac{1}{2}\) kg of sugar. Find the total amount of sugar bought by both of them.
Answer: Ramesh's sugar quantity is 2\(\frac{1}{2}\) kg, which converts to \( \frac{(2 \times 2)+1}{2} = \frac{5}{2} \) kg. Rohit's sugar quantity is 3\(\frac{1}{2}\) kg, which converts to \( \frac{(2 \times 3)+1}{2} = \frac{7}{2} \) kg. The combined amount they bought together is \( \frac{5}{2} \) kg \( + \) \( \frac{7}{2} \) kg = \( \frac{5+7}{2} \) kg = \( \frac{12}{2} \) kg = 6 kg (by dividing numerator and denominator by their HCF of 6).
In simple words: Ramesh bought 2.5 kg and Rohit bought 3.5 kg. Adding them gives 6 kg total.

Exam Tip: Convert mixed numbers to improper fractions before adding - this makes the calculation cleaner and helps avoid mistakes.

 

Question 5. The teacher taught 3/5 of the book, Vivek revised 1/5 more on his own. How much does he still have to revise?
Answer: The teacher covered 3/5 of the book material. Vivek reviewed an additional 1/5 on his own time. The total portion he has already completed (either taught or self-reviewed) is \( \frac{3}{5} + \frac{1}{5} = \frac{3+1}{5} = \frac{4}{5} \). The remaining portion he still needs to revise is \( 1 - \frac{4}{5} = \frac{5}{5} - \frac{4}{5} = \frac{1}{5} \).
In simple words: The teacher taught 3/5 and Vivek reviewed 1/5 more, bringing him to 4/5 total. He still needs to revise 1/5 of the book.

Exam Tip: Always subtract the completed work from the whole (1) to find the remaining work - this prevents careless errors.

 

Question 6. Amit was given 5/7 of a bucket of oranges. What fraction of oranges was left in the basket?
Answer: Oranges given to Amit = 5/7. The remaining portion in the basket = 1 - 5/7 = (7-5)/7 = 2/7. Therefore, 2/7 of the oranges stayed in the basket.
In simple words: If someone takes 5 parts out of 7 equal parts, then 2 parts are still left.

Exam Tip: When finding what remains, always subtract the given portion from 1 (or the whole) — this ensures you account for all parts.

 

Question 7. (i) \( \frac{7}{10} - * = \frac{3}{10} \)
Answer: Begin with: \( \frac{7}{10} - * = \frac{3}{10} \)

Rearranging: \( \frac{7}{10} - \frac{3}{10} = * \)

\( \frac{7-3}{10} = \frac{2}{5} \)

Therefore, \( * = \frac{2}{5} \)
In simple words: To find the missing number, take the first fraction and subtract the answer fraction.

Exam Tip: Always rearrange the equation to isolate the unknown term before performing the arithmetic.

 

Question 7. (ii) \( * - \frac{3}{21} = \frac{5}{21} \)
Answer: Begin with: \( * - \frac{3}{21} = \frac{5}{21} \)

Rearranging: \( * = \frac{5}{21} + \frac{3}{21} \)

\( \frac{5+3}{21} = \frac{8}{21} \)

Therefore, \( * = \frac{8}{21} \)
In simple words: When the unknown is being subtracted from, add the result and the subtracted number together to get the unknown.

Exam Tip: Remember to reverse the operation: if something is subtracted in the original equation, you must add in the rearranged form.

 

Question 7. (iii) \( * - \frac{3}{6} = \frac{3}{6} \)
Answer: Begin with: \( * - \frac{3}{6} = \frac{3}{6} \)

Rearranging: \( * = \frac{3}{6} + \frac{3}{6} \)

\( \frac{3+3}{6} = \frac{6}{6} = 1 \)

Therefore, \( * = 1 \)
In simple words: Add both fractions on the right side to discover the unknown value.

Exam Tip: When adding fractions with the same denominator, combine the numerators and keep the denominator unchanged.

 

Question 7. (iv) \( * - \frac{5}{27} = \frac{12}{27} \)
Answer: Begin with: \( * - \frac{5}{27} = \frac{12}{27} \)

Rearranging: \( * = \frac{12}{27} + \frac{5}{27} \)

\( \frac{12+5}{27} = \frac{17}{27} \)

Therefore, \( * = \frac{17}{27} \)
In simple words: To uncover the missing number, combine the two fractions by adding their numerators.

Exam Tip: Always verify your answer by substituting it back into the original equation to check correctness.

 

Exercise 6.9

 

Question 1. Add: (i) 3/4 and 5/6 (ii) 7/10 and 2/15 (iii) 8/13 and 2/3 (iv) 4/5 and 7/15
Answer:
(i) To add 3/4 and 5/6:
Find the LCM of 4 and 6, which is 12. Convert each fraction using denominator 12:
\( \frac{3 \times 3}{4 \times 3} + \frac{5 \times 2}{6 \times 2} = \frac{9}{12} + \frac{10}{12} = \frac{9+10}{12} = \frac{19}{12} \)

(ii) To add 7/10 and 2/15:
Find the LCM of 10 and 15, which is 30. Convert each fraction using denominator 30:
\( \frac{7 \times 3}{10 \times 3} + \frac{2 \times 2}{15 \times 2} = \frac{21}{30} + \frac{4}{30} = \frac{21+4}{30} = \frac{25}{30} \)

(iii) To add 8/13 and 2/3:
Find the LCM of 13 and 3, which is 39. Convert each fraction using denominator 39:
\( \frac{8 \times 3}{13 \times 3} + \frac{2 \times 13}{3 \times 13} = \frac{24}{39} + \frac{26}{39} = \frac{24+26}{39} = \frac{50}{39} \)

(iv) To add 4/5 and 7/15:
Find the LCM of 5 and 15, which is 15. Convert each fraction using denominator 15:
\( \frac{4 \times 3}{5 \times 3} + \frac{7 \times 1}{15 \times 1} = \frac{12}{15} + \frac{7}{15} = \frac{12+7}{15} = \frac{19}{15} \)
In simple words: To add fractions with different denominators, find the lowest common multiple, adjust both fractions to have that common denominator, then add the numerators.

Exam Tip: Always simplify your final answer if possible, and double-check that your LCM is correct before converting the fractions.

 

Question 2. Subtract: (i) 2/7 from 19/21 (ii) 21/25 from 18/20 (iii) 7/16 from 2/1 (iv) 4/15 from \( 2\frac{1}{5} \)
Answer:
(i) To subtract 2/7 from 19/21:
Express as: 19/21 - 2/7. Find the LCM of 21 and 7, which is 21. Convert to common denominator:
\( \frac{19 \times 1}{21 \times 1} - \frac{2 \times 3}{7 \times 3} = \frac{19}{21} - \frac{6}{21} = \frac{19-6}{21} = \frac{13}{21} \)

(ii) To subtract 21/25 from 18/20:
Express as: 18/20 - 21/25. Find the LCM of 20 and 25, which is 100. Convert to common denominator:
\( \frac{18 \times 5}{20 \times 5} - \frac{21 \times 4}{25 \times 4} = \frac{90}{100} - \frac{84}{100} = \frac{90-84}{100} = \frac{6}{100} \)

(iii) To subtract 7/16 from 2/1:
Express as: 2/1 - 7/16. Find the LCM of 1 and 16, which is 16. Convert to common denominator:
\( \frac{2 \times 16}{1 \times 16} - \frac{7 \times 1}{16 \times 1} = \frac{32}{16} - \frac{7}{16} = \frac{32-7}{16} = \frac{25}{16} \)

(iv) To subtract 4/15 from \( 2\frac{1}{5} \):
First convert the mixed number: \( 2\frac{1}{5} = \frac{(2 \times 5)+1}{5} = \frac{11}{5} \). Express as: 11/5 - 4/15. Find the LCM of 5 and 15, which is 15. Convert to common denominator:
\( \frac{11 \times 3}{5 \times 3} - \frac{4 \times 1}{15 \times 1} = \frac{33}{15} - \frac{4}{15} = \frac{33-4}{15} = \frac{29}{15} \)
In simple words: To subtract fractions, create a shared denominator using the LCM, change both fractions accordingly, then subtract the second numerator from the first.

Exam Tip: Pay careful attention to which fraction is being subtracted from which - the order matters significantly in subtraction problems.

 

Question 3. Find the difference of: (i) 13/24 and 7/16 (ii) 5/18 and 4/15 (iii) 1/12 and 3/4 (iv) 2/3 and 6/7
Answer:
(i) For 13/24 and 7/16:
Express as: 13/24 - 7/16. Find the LCM of 24 and 16, which is 48. Convert to common denominator:
\( \frac{13 \times 2}{24 \times 2} - \frac{7 \times 3}{16 \times 3} = \frac{26}{48} - \frac{21}{48} = \frac{26-21}{48} = \frac{5}{48} \)

(ii) For 5/18 and 4/15:
Express as: 5/18 - 4/15. Find the LCM of 18 and 15, which is 90. Convert to common denominator:
\( \frac{5 \times 5}{18 \times 5} - \frac{4 \times 6}{15 \times 6} = \frac{25}{90} - \frac{24}{90} = \frac{25-24}{90} = \frac{1}{90} \)

(iii) For 3/4 and 1/12:
Express as: 3/4 - 1/12. Find the LCM of 4 and 12, which is 12. Convert to common denominator:
\( \frac{3 \times 3}{4 \times 3} - \frac{1 \times 1}{12 \times 1} = \frac{9}{12} - \frac{1}{12} = \frac{9-1}{12} = \frac{8}{12} \)

(iv) For 6/7 and 2/3:
Express as: 6/7 - 2/3. Find the LCM of 7 and 3, which is 21. Convert to common denominator:
\( \frac{6 \times 3}{7 \times 3} - \frac{2 \times 7}{3 \times 7} = \frac{18}{21} - \frac{14}{21} = \frac{18-14}{21} = \frac{4}{21} \)
In simple words: Find the difference by determining the LCM, adjusting both fractions to use that common denominator, then subtracting to discover the result.

Exam Tip: Always verify your LCM calculation by confirming both original denominators divide evenly into it.

 

Question 4. Subtract as indicated: (i) 8/3 - 5/9 (ii) \( 4\frac{2}{5} - 2\frac{1}{5} \) (iii) \( 5\frac{6}{7} - 2\frac{2}{3} \) (iv) \( 4\frac{3}{4} - 2\frac{1}{6} \)
Answer:
(i) For 8/3 - 5/9:
Find the LCM of 3 and 9, which is 9. Convert to common denominator:
\( \frac{8 \times 3}{3 \times 3} - \frac{5 \times 1}{9 \times 1} = \frac{24}{9} - \frac{5}{9} = \frac{24-5}{9} = \frac{19}{9} \)

(ii) For \( 4\frac{2}{5} - 2\frac{1}{5} \):
Convert mixed numbers to improper fractions: \( 4\frac{2}{5} = \frac{22}{5} \) and \( 2\frac{1}{5} = \frac{11}{5} \)
\( \frac{22}{5} - \frac{11}{5} = \frac{22-11}{5} = \frac{11}{5} \)

(iii) For \( 5\frac{6}{7} - 2\frac{2}{3} \):
Convert mixed numbers: \( 5\frac{6}{7} = \frac{41}{7} \) and \( 2\frac{2}{3} = \frac{8}{3} \)
Find the LCM of 7 and 3, which is 21. Convert to common denominator:
\( \frac{41 \times 3}{7 \times 3} - \frac{8 \times 7}{3 \times 7} = \frac{123}{21} - \frac{56}{21} = \frac{123-56}{21} = \frac{67}{21} \)

(iv) For \( 4\frac{3}{4} - 2\frac{1}{6} \):
Convert mixed numbers: \( 4\frac{3}{4} = \frac{19}{4} \) and \( 2\frac{1}{6} = \frac{13}{6} \)
Find the LCM of 4 and 6, which is 12. Convert to common denominator:
\( \frac{19 \times 3}{4 \times 3} - \frac{13 \times 2}{6 \times 2} = \frac{57}{12} - \frac{26}{12} = \frac{57-26}{12} = \frac{31}{12} \)
In simple words: For mixed numbers, change them to improper fractions first, then follow the same steps as regular fraction subtraction.

Exam Tip: When converting mixed numbers, remember: multiply the whole number by the denominator, add the numerator, and keep the same denominator.

 

Question 5. Simplify:
(i) \( \frac{2}{3} + \frac{3}{4} + \frac{1}{2} \)
(ii) \( \frac{5}{8} + \frac{2}{5} + \frac{3}{4} \)
(iii) \( \frac{3}{10} + \frac{7}{15} + \frac{3}{5} \)
(iv) \( \frac{3}{4} + \frac{7}{16} + \frac{5}{8} \)
(v) \( 4\frac{2}{3} + 3\frac{1}{4} + 7\frac{1}{2} \)
(vi) \( \frac{7}{13} + 3\frac{2}{3} + 5\frac{1}{6} \)
(vii) \( \frac{7}{1} + \frac{7}{4} + 5\frac{1}{6} \)
(viii) \( \frac{5}{6} + \frac{3}{1} + \frac{3}{4} \)
(ix) \( \frac{7}{18} + \frac{5}{6} + 1\frac{1}{12} \)
Answer:
(i) To add these fractions, we find the LCM of 3, 4, and 2, which is 12. Converting each fraction to have a denominator of 12:
\( \frac{2 \times 4}{3 \times 4} + \frac{3 \times 3}{4 \times 3} + \frac{1 \times 6}{2 \times 6} = \frac{8}{12} + \frac{9}{12} + \frac{6}{12} = \frac{8 + 9 + 6}{12} = \frac{23}{12} \)

(ii) The LCM of 8, 5, and 4 is 40. Converting to a common denominator:
\( \frac{5 \times 5}{8 \times 5} + \frac{2 \times 8}{5 \times 8} + \frac{3 \times 10}{4 \times 10} = \frac{25}{40} + \frac{16}{40} + \frac{30}{40} = \frac{25 + 16 + 30}{40} = \frac{71}{40} \)

(iii) The LCM of 10, 15, and 5 is 30. Converting to a common denominator:
\( \frac{3 \times 3}{10 \times 3} + \frac{7 \times 2}{15 \times 2} + \frac{3 \times 6}{5 \times 6} = \frac{9}{30} + \frac{14}{30} + \frac{18}{30} = \frac{9 + 14 + 18}{30} = \frac{41}{30} \)

(iv) The LCM of 4, 16, and 8 is 16. Converting to a common denominator:
\( \frac{3 \times 4}{4 \times 4} + \frac{7 \times 1}{16 \times 1} + \frac{5 \times 2}{8 \times 2} = \frac{12}{16} + \frac{7}{16} + \frac{10}{16} = \frac{12 + 7 + 10}{16} = \frac{29}{16} \)

(v) Converting mixed numbers to improper fractions: \( \frac{14}{3} + \frac{13}{4} + \frac{15}{2} \). The LCM of 3, 4, and 2 is 12:
\( \frac{14 \times 4}{3 \times 4} + \frac{13 \times 3}{4 \times 3} + \frac{15 \times 6}{2 \times 6} = \frac{56}{12} + \frac{39}{12} + \frac{90}{12} = \frac{56 + 39 + 90}{12} = \frac{185}{12} \)

(vi) Converting mixed numbers: \( \frac{7}{13} + \frac{11}{3} + \frac{31}{6} \). The LCM of 13, 3, and 6 is 78:
\( \frac{7 \times 6}{13 \times 6} + \frac{11 \times 26}{3 \times 26} + \frac{31 \times 13}{6 \times 13} = \frac{42}{78} + \frac{286}{78} + \frac{403}{78} = \frac{731}{78} \)

(vii) Converting: \( \frac{7}{1} + \frac{7}{4} + \frac{31}{6} \). The LCM of 1, 4, and 6 is 12:
\( \frac{7 \times 12}{1 \times 12} + \frac{7 \times 3}{4 \times 3} + \frac{31 \times 2}{6 \times 2} = \frac{84}{12} + \frac{21}{12} + \frac{62}{12} = \frac{84 + 21 + 62}{12} = \frac{167}{12} \)

(viii) Converting: \( \frac{5}{6} + \frac{3}{1} + \frac{3}{4} \). The LCM of 6, 1, and 4 is 12:
\( \frac{5 \times 2}{6 \times 2} + \frac{3 \times 12}{1 \times 12} + \frac{3 \times 3}{4 \times 3} = \frac{10}{12} + \frac{36}{12} + \frac{9}{12} = \frac{10 + 36 + 9}{12} = \frac{55}{12} \)

(ix) Converting: \( \frac{7}{18} + \frac{5}{6} + \frac{13}{12} \). The LCM of 18, 6, and 12 is 36:
\( \frac{7 \times 2}{18 \times 2} + \frac{5 \times 6}{6 \times 6} + \frac{13 \times 3}{12 \times 3} = \frac{14}{36} + \frac{30}{36} + \frac{39}{36} = \frac{14 + 30 + 39}{36} = \frac{83}{36} \)
In simple words: To combine fractions with different denominators, find the least common multiple of all the denominators. Then convert every fraction to match that common denominator by multiplying both the top and bottom by the appropriate factor. Once all fractions have the same denominator, just add the numerators and keep the denominator unchanged.

Exam Tip: Always identify the LCM correctly before converting - it determines whether your final answer is right or wrong. Double-check by factoring each denominator carefully.

 

Question 6. Replace * with a correct number:
(i) \( * - \frac{5}{8} = \frac{1}{4} \)
(ii) \( * - \frac{1}{5} = \frac{1}{2} \)
(iii) \( \frac{1}{2} - * = \frac{1}{6} \)
Answer:
(i) To find the missing number, we rearrange the equation. Adding \( \frac{5}{8} \) to both sides:
\( * = \frac{1}{4} + \frac{5}{8} \)
Converting to a common denominator:
\( * = \frac{1 \times 2}{4 \times 2} + \frac{5 \times 1}{8 \times 1} = \frac{2}{8} + \frac{5}{8} = \frac{2 + 5}{8} = \frac{7}{8} \)
Therefore, the missing number is \( \frac{7}{8} \).

(ii) Adding \( \frac{1}{5} \) to both sides:
\( * = \frac{1}{2} + \frac{1}{5} \)
The LCM of 2 and 5 is 10:
\( * = \frac{1 \times 5}{2 \times 5} + \frac{1 \times 2}{5 \times 2} = \frac{5}{10} + \frac{2}{10} = \frac{5 + 2}{10} = \frac{7}{10} \)

(iii) Rearranging the equation by subtracting \( \frac{1}{2} \) from both sides:
\( * = \frac{1}{2} - \frac{1}{6} \)
The LCM of 2 and 6 is 6:
\( * = \frac{1 \times 3}{2 \times 3} - \frac{1 \times 1}{6 \times 1} = \frac{3}{6} - \frac{1}{6} = \frac{3 - 1}{6} = \frac{2}{6} = \frac{1}{3} \)
In simple words: When a number is missing in an equation with fractions, move the other fraction to the opposite side by changing the operation (addition becomes subtraction, and vice versa). Then add or subtract the two fractions using a common denominator.

Exam Tip: Always verify your answer by substituting it back into the original equation - this catches arithmetic mistakes instantly.

 

Question 7. Savita bought 2/5 m of ribbon and kavita 3/4 m of ribbon. What was the total length of the ribbon they bought?
Answer: Savita purchased \( \frac{2}{5} \) metres of ribbon. Kavita purchased \( \frac{3}{4} \) metres of ribbon. The combined length they obtained is:
\( \frac{2}{5} \text{ metres} + \frac{3}{4} \text{ metres} = \frac{2 \times 4}{5 \times 4} \text{ metres} + \frac{3 \times 5}{4 \times 5} \text{ metres} \)
(because LCM of 5 and 4 is 20)
\( = \frac{8}{20} \text{ metres} + \frac{15}{20} \text{ metres} = \frac{8 + 15}{20} \text{ metres} = \frac{23}{20} \text{ metres} \)
In simple words: To find how much ribbon both girls bought together, add the two amounts. Since the denominators are different (5 and 4), multiply each fraction to get 20 in the denominator for both. Then add the numerators: 8 plus 15 gives 23, so the answer is 23/20 metres.

Exam Tip: Always state the unit in your final answer - leaving out "metres" costs marks even if your fraction is correct.

 

Question 8. Ravish takes 2\( \frac{1}{5} \) minutes to walk across the school ground. Rahul takes 7/4 minutes to do the same. Who takes less time and by what fraction?
Answer: Ravish's walking time is \( 2\frac{1}{5} = \frac{11}{5} \) minutes. Rahul's walking time is \( \frac{7}{4} \) minutes. To compare these times, we convert to a common denominator. The LCM of 5 and 4 is 20, so:
\( \frac{11 \times 4}{5 \times 4} = \frac{44}{20} \text{ minutes} \) and \( \frac{7 \times 5}{4 \times 5} = \frac{35}{20} \text{ minutes} \)
Since \( \frac{44}{20} > \frac{35}{20} \), Rahul takes less time. The difference is:
\( \frac{44}{20} - \frac{35}{20} = \frac{44 - 35}{20} = \frac{9}{20} \text{ minutes} \)
In simple words: Rahul takes 9/20 of a minute less than Ravish. To find this, we changed both times to have 20 as the denominator, then subtracted the smaller from the larger.

Exam Tip: When comparing fractions, always find a common denominator first - do not attempt to compare different denominators directly.

 

Question 9. A piece of a wire 7/8 metres long broke into two pieces. One piece was 1/4 meter long. How long is the other piece?
Answer: The original wire measures \( \frac{7}{8} \) metres. One piece measures \( \frac{1}{4} \) metres. Let the other piece be x metres. Since the two pieces make up the entire wire:
\( \frac{7}{8} = \frac{1}{4} + x \)
Solving for x:
\( x = \frac{7}{8} - \frac{1}{4} \)
Converting to a common denominator:
\( x = \frac{7 \times 1}{8 \times 1} - \frac{1 \times 2}{4 \times 2} = \frac{7}{8} - \frac{2}{8} = \frac{7 - 2}{8} = \frac{5}{8} \text{ metres} \)
Therefore, the second piece is \( \frac{5}{8} \) metres long.
In simple words: The total wire is made of two parts. If you know the total and one part, subtract that part from the total to get the other part. Here, 7/8 minus 1/4 gives 5/8.

Exam Tip: Always check your answer: add the two pieces together (1/4 + 5/8) - you must get 7/8 again, confirming your work is right.

 

Question 10. Shikha and priya have bookshelves of the same size shikha's shelf is 5/6 full of book and priya's shelf is 2/5 full. Whose bookshelf is more full? By what fraction?
Answer: Shikha's bookshelf occupancy is \( \frac{5}{6} \). Priya's bookshelf occupancy is \( \frac{2}{5} \). To determine who has the fuller shelf, we convert both to a common denominator. The LCM of 6 and 5 is 30, so:
\( \frac{5 \times 5}{6 \times 5} = \frac{25}{30} \) and \( \frac{2 \times 6}{5 \times 6} = \frac{12}{30} \)
Since \( \frac{25}{30} > \frac{12}{30} \), Shikha's shelf is more full. The difference is:
\( \frac{25}{30} - \frac{12}{30} = \frac{25 - 12}{30} = \frac{13}{30} \)
Therefore, Shikha's bookshelf is fuller by \( \frac{13}{30} \).
In simple words: Shikha's shelf is more full than Priya's shelf. To find by how much, we make both fractions have the same denominator (30) and then subtract the smaller from the larger, getting 13/30.

Exam Tip: Always compare fractions by converting to the same denominator - never try to decide which is larger by looking at 5/6 and 2/5 directly.

 

Question 11. Ravish's house is 9/10 Km from his school. He walked some distance and then took a bus for 1/2 Km. How far did he walk?
Answer: The total distance from Ravish's house to school is \( \frac{9}{10} \) Km. He travelled part of this distance on foot and the remaining \( \frac{1}{2} \) Km by bus. Therefore, the distance he walked is:
\( \text{Distance walked} = \frac{9}{10} - \frac{1}{2} \)
Converting to a common denominator (LCM of 10 and 2 is 10):
\( = \frac{9 \times 1}{10 \times 1} - \frac{1 \times 5}{2 \times 5} = \frac{9}{10} - \frac{5}{10} = \frac{9 - 5}{10} = \frac{4}{10} = \frac{2}{5} \text{ Km} \)
(HCF of numerator and denominator is 2)
In simple words: The total trip is 9/10 Km. Since 1/2 Km was by bus, the rest must have been walked. Subtract 1/2 from 9/10 to get 4/10, which simplifies to 2/5 Km.

Exam Tip: Always simplify your final answer by dividing both the numerator and denominator by their greatest common factor - an unsimplified fraction may lose marks.