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Kerala Syllabus Class 6 Mathematics - Unit 4 Arithmetic of Parts - Questions and Answers | Teaching Manual 

Questions and Answers for Class 6 Mathematics - Unit 4 Arithmetic of Parts - Study Notes | Text Books Solution STD 6 - Maths: Unit 4 Arithmetic of Parts - Questions and Answers | ഗണിതം - ഭാഗക്കണക്കുകൾ - ചോദ്യോത്തരങ്ങൾ 

ആറാം ക്ലാസ്സ്‌  Mathematics - Unit 4 Arithmetic of Parts എന്ന പാഠം ആസ്പദമാക്കി തയ്യാറാക്കിയ ചോദ്യോത്തരങ്ങള്‍. ഈ അധ്യായത്തിന്റെ Teachers Handbook, Teaching Manual എന്നിവ ഡൗൺലോഡ് ചെയ്യാനുള്ള ലിങ്ക് ചോദ്യോത്തരങ്ങളുടെ അവസാനം നൽകിയിട്ടുണ്ട്. പുതിയ അപ്‌ഡേറ്റുകൾക്കായി ഞങ്ങളുടെ Telegram Channel ൽ ജോയിൻ ചെയ്യുക.

Kerala Syllabus STD 6 Maths: Unit 4 Arithmetic of Parts - Textbook Solutions

♦ Textbook Activities (Textbook Page No: 50)

♦ In each of  the pictures below, write the parts of each colour and the total coloured parts as fractions. Write the sum of fractions got from each picture in lowest terms. 
Answer:
1. 🟦Colour: ⅜ Part
    🟩Colour: ⅛ Part
    Total ⁴⁄₈ Part
    ⅜ + ⅛ = ⁴⁄₈ =1 × 4 / 2 × 4 = ½

2. 🟨Colour: ⅒ Part
    🟥Colour: ³⁄₁₀ Part
    Total ⁴⁄₁₀ Part   
    ⅒ + ³⁄₁₀ = ⁴⁄₁₀ = 2 × 2 / 5 × 2 = ⅖

3. 🟧Colour: ⅜ Part
    🟨Colour: ⅜ Part
    Total ⁶⁄₈ Part
    ⅜ + ⅜ = ⁶⁄₈ = 3 × 2 / 4 × 2 = ¾

4. 🟨Colour: ⁵⁄₁₆ Part
    🟩Colour: ⁷⁄₁₆ Part
    Total ¹²⁄₁₆ Part
    ⁵⁄₁₆ + ⁷⁄₁₆ = 12 / 16 = 3 × 4 / 4 × 4 = ¾

♦ Textbook Activities (Textbook Page No: 56, 57)

(1) In each pair of  pictures below, find the fraction of  the circle we get by cutting up the coloured pieces of  both circles and putting them together: 
Answer:
  ⅓ + ⁵⁄₁₂ = ⁴⁄₁₂ + ⁵⁄₁₂ = ⁹⁄₁₂             

  ⅓ = 1 × 4 / 3 × 4 = ⁴⁄₁₂
  = 3 × 3 / 4 × 3 = ¾
If we cut out the coloured parts from the 2 circles and combine them, we will get ¾ of a circle.
Answer:
  ⅙ + ¾ = ²⁄₁₂ + ⁹⁄₁₂ = ¹¹⁄₁₂              

  ⅙ = 1 × 2 / 6 × 4 = ²⁄₁₂
  ¾ = 3 × 3 / 4 × 3 = ⁹⁄₁₂
If we cut out the coloured parts from the 2 circles and combine them, we will get ¹¹⁄₁₂ of a circle.
Answer:
  ⅗ + ⅜ = ²⁴⁄₄₀ + ¹⁵⁄₄₀ = ³⁹⁄₄₀              

  ⅗ = 3 × 8 / 5 × 8 = ²⁴⁄₄₀
  ⅜ = 3 × 5 / 8 × 5 = ¹⁵⁄₄₀
If we cut out the coloured parts from the 2 circles and combine them, we will get ³⁹⁄₄₀ of a circle.

2) Calculate the sums given below:
(i) ¼ + ⅛
    ¼ = 1 × 2 / 4 × 2 = ²⁄₈
    ∴ ¼ + ⅛ = ²⁄₈ + ⅛ = ⅜

(ii) ¾ + ⅙
     ¾ = 3 × 3 / 4 × 3 = ⁹⁄₁₂
     ⅙ = 1 × 2 / 6 × 2 = ²⁄₁₂
     ∴ ¾ + ⅙ = ⁹⁄₁₂ + ²⁄₁₂ = ¹¹⁄₁₂

(iii) ⅓ + ⅖
     ⅓ = 1 × 5 / 3 × 5 = ⁵⁄₁₅
     ⅖ = 2 × 3 / 5 × 3 = ⁶⁄₁₅
    ∴ ⅓ + ⅖ = ⁵⁄₁₅ + ⁶⁄₁₅ = ¹¹⁄₁₅

(iv) ½ + ⅖
     ½ = 1 × 5 / 2 × 5 = ⁵⁄₁₀
     ⅖ = 2 × 2 / 5 × 2 = ⁴⁄₁₀
    ∴ ½ + ⅖ = ⁵⁄₁₀ + ⁴⁄₁₀ = ⁹⁄₁₀

(v) ⅔ + ⅕
     ⅔ = 2 × 5 / 3 × 5 = ¹⁰⁄₁₅
     ⅕ = 1 × 3 / 5 × 3 = ³⁄₁₅
    ∴ ⅔ + ⅕ = ¹⁰⁄₁₅ + ³⁄₁₅ = ¹³⁄₁₅

(3) There are two taps to fill a tank with water. If the first tap alone is opened, the tank would fill up in 10 minutes. If the second tap alone is opened, it would take 15 minutes to fill up the tank. 
(i) If the first tap alone is opened, what fraction of the tank would be filled in one minute? 
If only the first tap is opened, the entire tank will be filled in 10 minutes. That is, the part filled in one minute = ⅒
(²⁄₁₀ parts in 2 minutes, ³⁄₁₀ parts in 3 minutes, ⁴⁄₁₀ parts in 4 minutes,.... ¹⁰⁄₁₀ parts in 10 minutes ¹⁰⁄₁₀ = 1 (Full))
 
(ii) If the second tap alone is opened, what fraction of the tank would be filled in one minute?
If only the second tap is opened, the entire tank will be filled in 15 minutes. That is, the part filled in one minute = ¹⁄₁₅

(iii) If both the taps are opened, what fraction of the tank would be filled in one minute? 
The part filled through the first tap in one minute = ⅒
The part filled through the second tap in one minute = ¹⁄₁₅
If both taps are opened, the part filled in one minute = ⅒ + ¹⁄₁₅
 = ³⁄₃₀ + ²⁄₃₀ = ⁵⁄₃₀
= 1 × 5 / 6 × 5 = ⅙ part

⅒ = 1 × 3 / 10 × 3 = ³⁄₃₀
¹⁄₁₅ = 1 × 2 / 15 × 2 = ²⁄₃₀

(iv) If both the taps are opened, how much time would it take for the tank to be filled up? 
⅙ part filled in 1 minute
²⁄₆ part filled in 2 minutes
-------------------------------
-------------------------------
In 6 minutes, ⁶⁄₆ = 1 (full) is filled.

Some other sums 
See these sums :
½ + ½ = 1 + 1 / 2 = ²⁄₂ = 1
⅓ + ⅔ = 1 + 2 / 3 = ³⁄₃ = 1
¼ + ¾ = 1 + 3 / 4 = ⁴⁄₄ = 1
⅕ + ⅘  = 1 + 4 / 5 = ⁵⁄₅ = 1
⅖ + ⅗  = 2 + 3 / 5 = ⁵⁄₅ = 1

Of the two fractions added, one is some parts of one divided into equal parts taken together; and the other is the remaining parts taken together. When both these are taken together, we get all the parts.

♦ For each fraction given below, can you mentally calculate the fraction to be added to make it 1? 
(i) ²⁄₇  (ii) ⁴⁄₇   (iii) ⅜  (iv) ³⁄₁₀
Answer:
(i) 7 − 2 / 7 = ⁵⁄₇
(ii) 7 − 4 / 7 = ³⁄₇
(iii) 8 − 3 / 8 = ⅝
(iv) 10 − 3 / 10 = ⁷⁄₁₀
♦ Textbook Activities (Textbook Page No: 60)
(1) Calculate the sums below :
2) One jar contains one and a half litres of milk, and another contains two and three-quarters litres of milk. How much milk is in both jars together?
The amount of milk in the first jar = 1½ litres
The amount of milk in the second jar = 2¾ litres
The total amount of milk = 1½ + 2¾
 = (1 + ½) + (2 + ¾) = (1 + 2) + (½ + ¾)
 = 3 + (½ + ½ + ¼) = 3 + (1 + ¼)
 = 3 + 1 + ¼ = 4 + ¼ = 4¼ litres

(3) Two strings of lengths one and a half metres are joined end to end. What is the total length?
The total length of the string = 1½ + 1½
  = 1 + ½ + 1 + ½      
  = (1 + 1) + (½ + ½)
  = 2 + 1 = 3 metrs          

(4) Ahirath bought one and a half kilograms of beans and three-quarters of a kilogram of yam. What is the total weight? 
Quantity of beans = 1½ kilogram
Quantity of yam = ¾ kilogram
Total weight = 1½ + ¾
  = ½ + ½ + ¼
  = 1 + (½ + ½) + ¼
  = 1 + 1 + ¼ = 2 + ¼
  = 2¼ kilogram

♦ Textbook Activities (Textbook Page No: 61)
♦ Removing parts 
(i) ½ − ⅛
    ½ − ⅛ = ⁴⁄₈ − ⅛ = 4 − 1 / 8 = ⅜

(ii) ¾ − ⅛
     ¾ − ⅛ = ⁶⁄₈ − ⅛ = 6 − 1 / 8 = ⅝

(iii) ⅓ − ⅕
     ⅓ − ⅕ = 1 × 5 / 3 × 5 − 1 × 3 / 5 × 3
     = ⁵⁄₁₅ − ³⁄₁₅ = 5 − 3 / 15 = ²⁄₁₅

(iv) ⅖ − ⅓
     ⅖ − ⅓ = 2 × 3 / 5 × 3 − 1 × 5 / 3 × 5
     = ⁶⁄₁₅ − ⁵⁄₁₅ = 6 − 5 / 15 = ¹⁄₁₅

(v) ⅔ − ⅕
     ⅔ − ⅕ = 2 × 5 / 3 × 5 − 1 × 3 / 5 × 3
     = ¹⁰⁄₁₅ − ³⁄₁₅ = 10 − 3 / 15 = ⁷⁄₁₅
♦ Textbook Activities (Textbook Page No: 63, 64)

♦ One and a half metres of cloth was bought for Anoop and two and a quarter metres for his father. How much more was bought for his father?
Half a metre added to one and a half metres makes two metres; another quarter of a metre added makes two and a quarter metres. Total added is half and a quarter, which is three-quarters. So, three-quarters of a metre more. 
That is, 2 ¼ − 1½ = ¾
Let's write down our reasoning:
• 2 = 1½ + 2½ 
• 2¼ = 2 + ¼ = 1½ + ½ + ¼
• 2¼ = 1½ + ¾
• 2¼ − 1½ = ¾
That is, Father bought ¾ metre of cloth more than Anoop.

(1) Natasha drew a circle and coloured ⁵⁄₁₂ of it. What fraction of  the circle 
remains to be coloured? 
The coloured part of the circle = ⁵⁄₁₂
The part to be coloured = 1 − ⁵⁄₁₂
  = ¹²⁄₁₂ − ⁵⁄₁₂ = 12 − 5 / 12 = ⁷⁄₁₂

(2) A bucket can hold 10 litres of water, and it contains 3¾ litres. How much more is needed to fill it?
If ¾ litre of water is added to 3¾ litres, it will become 4 litres of water added again, it will become 10 litres. That is, 6¼ litres are needed to fill the bucket.
This can also be done like this.
10 − 3¾ = 10 − (3 + ¾)
   = (10 − 3) − ¾
   = 7 − ¾ = 6¼ litres

3) From a string, one and three-quarters of a metre long, a piece half a metre long is cut off. What is the length of the remaining piece?
Total length of string = 1¾ metres
Length of the string to be cut = ½ metre
Length of the remaining string = 1¾ − ½ = 1 + ¾ − ½
 = 1 + ¾ − ²⁄₄ = 1 + 3 − 2 / 4
 = 1 + ¼ = 1¼ metres

(4) A panchayat constructed a new road, 14¾ kilometres long last year. This year, a road 16¼ kilometres long was constructed. How much more was constructed this year?
Length of the road constructed last year = 14¾ kilometres
Length of the road constructed this year = 16¼ km.
Length of the road constructed more this year than last year = 16¼ − 14¾ 
 = 15 + 1¼ − (14 + ¾) = 15 + 1¼ − 14 − ¾
 = (15 − 14) + (1¼ − ¾) = 1 + ½ = 1½ km.

(5) Ashadev bought 20 metres of string. He cut off a piece 5¾ metres long first, and then a piece 6½ metres long later. What is the length of string left?
Total length of the string = 20 metres
Total length of the string cut out = 5¾ + 6½
  = 5 + ¾ + 6 + ½
  = 5 + 6 + ¾ + ¼ = 11 + 1 + ¼ 
  = 12 + ¼ = 12¼  metres
Total length of the remaining string = 20 − 12¼
  = 20 − (12 + ¼) = 20 − 12 − ¼
  = 8 − ¼ = 7¾ metres

(6) The milk society got 75¼ litres in the morning and 55½ litres in the evening. Of this, 85¾ litres of milk was sold. How much milk is left?
  = 75¼ + 55½ = 75 + ¼ + 55 + ½
  = 75 + 55 + ¼ + ½
  = 130 + ¾ = 130¾  litres
Quantity of milk distributed = 85¾ litres
Quantity of milk left over = 130¾ − (85¾)
       = 130 + ¾ − (85¾)
       = 130 + ¾ − 85 − ¾
       = (130 − 85) + (¾ − ¾) = 45 litres

♦ Large and small 
Of two fractions with the same denominator, the one with larger numerator is the larger and the one with smaller numerator is the smaller.

Of two fractions with the same numerator, the one with larger denominator is smaller, and the one with smaller denominator is larger 
♦ Textbook Activities (Textbook Page No: 66)

(1) Find the larger and smaller of  each pair of  fractions below and write this using the < or > symbol:

(i) ⅖, ⅗
Here, the denominator of both fractions is 5. Therefore, the fraction with the larger denominator is smaller, and the fraction with the smaller denominator is greater.
Therefore, ⅖ < ⅗ or ⅗ > ⅖ 

(ii) ⅖, ⅔
Here, the numerator of both fractions is 2. Therefore, the fraction with the larger denominator is smaller, and the fraction with the smaller denominator is greater.
Therefore, ⅔ > ⅖ or ⅖ < ⅔ 

(iii) ⅖, ¾
Here, the denominator and numerator are different. To compare ⅖, and ¾, we cross‑multiply:
    ⅖ = 2 × 4 / 5 x 4 = ⁸⁄₂₀
    ¾ = 3 × 5 / 4 x 5 = ¹⁵⁄₂₀
    ∴ ⅖ < ¾ or ¾ > ⅖

(iv) ³⁄₇, ²⁄₉
    ³⁄₇ = 3 × 2 / 7 x 2 = ⁶⁄₁₄
    ²⁄₉ = 2 × 3 / 9 x 3 = ⁶⁄₂₇
    ∴ ³⁄₇ > ²⁄₉ or ²⁄₉ < ³⁄₇

(v) ²⁄₇, ⅜
    ²⁄₇ = 2 × 8 / 7 x 8 = ¹⁶⁄₅₆
    ⅜ = 3 × 7 / 8 x 7 = ²¹⁄₅₆
    ∴ ²⁄₇ < ⅜ or ⅜ > ²⁄₇ 

(vi) ⁴⁄₉, ⅜
    ⁴⁄₉ = 4 × 8 / 9 x 8 = ³²⁄₇₂
    ⅜ = 3 × 9 / 8 x 9 = ²⁷⁄₇₂
    ∴ ⅜ < ⁴⁄₉ or ⁴⁄₉ > ⅜

(2) Arrange each triple of  fractions below from the smallest to the largest and write it using the < symbol : 
(i) ⅖, ¾, ⅗
    ⅖ = 2 × 4 / 5 x 4 = ⁸⁄₂₀
    ¾ = 3 × 5 / 4 x 5 = ¹⁵⁄₂₀
    ⅗ = 3 × 4 / 5 x 4 = ¹²⁄₂₀
    ∴ ⅖ < ⅗ < ¾

(ii) ³⁄₇, ²⁄₉, ²⁄₇
    ³⁄₇ = 3 × 9 / 7 x 9 = ²⁷⁄₆₃
    ²⁄₉ = 2 × 7 / 9 x 7 = ¹⁴⁄₆₃
    ²⁄₇ = 2 × 9 / 7 x 9 = ¹⁸⁄₆₃
    ∴ ²⁄₉ < ²⁄₇ < ³⁄₇

(iii) ½, ⅓, ⅔
    ½ = 1 × 3 / 2 x 3 = ³⁄₆
    ⅓ = 1 × 2 / 3 x 2 = ²⁄₆
    ⅔ = 2 × 2 / 3 x 2 = ⁴⁄₆
    ∴ ⅓ < ½ < ⅔