GSEB Class 6 Maths Solutions Chapter 2 પૂર્ણ સંખ્યાઓ InText Questions

પ્રયત્ન કરો [પાન નંબર 28]

 

Question 1. પછીની સંખ્યાઓની પહેલાં અને પછીની સંખ્યા લખો: 1; 19; 1997; 12,000; 49; 1,00,000
Answer:
For 1: The number before it is \( 1 - 1 = 0 \); the number after it is \( 1 + 1 = 2 \).
For 19: The number before it is \( 19 - 1 = 18 \); the number after it is \( 19 + 1 = 20 \).
For 1997: The number before it is \( 1997 - 1 = 1996 \); the number after it is \( 1997 + 1 = 1998 \).
For 12,000: The number before it is \( 12,000 - 1 = 11,999 \); the number after it is \( 12,000 + 1 = 12,001 \).
For 49: The number before it is \( 49 - 1 = 48 \); the number after it is \( 49 + 1 = 50 \).
For 1,00,000: The number before it is \( 1,00,000 - 1 = 99,999 \); the number after it is \( 1,00,000 + 1 = 1,00,001 \).
Note: If we consider 1 as a natural number, then there is no number before it. If we consider 1 as a whole number, then the number before it is 0.
In simple words: To find the number before, subtract 1. To find the number after, add 1. Natural numbers start from 1, so 1 has no natural number before it. Whole numbers start from 0, so 0 is the number before 1 if 1 is a whole number.

Exam Tip: Remember that "predecessor" means the number before (subtract 1) and "successor" means the number after (add 1). Pay attention to whether you are dealing with natural numbers or whole numbers, as the rules for 0 and 1 differ between them.

 

Question 2. કઈ પ્રાકૃતિક સંખ્યા પાસે તેના પહેલાં આવતી સંખ્યા નથી?
Answer: The smallest natural number, 1, is the only number that does not have a preceding natural number. This is because \( 1 - 1 = 0 \), but 0 itself is not a natural number (it is a whole number).
In simple words: The number 1 is the first natural number. Because natural numbers begin at 1, there is no natural number that comes right before it. If you subtract 1 from 1, you get 0, which is a whole number, not a natural number.

Exam Tip: Clearly distinguish between natural numbers (1, 2, 3...) and whole numbers (0, 1, 2, 3...) to avoid confusion regarding predecessors of 1 and 0.

 

Question 3. કઈ પ્રાકૃતિક સંખ્યા પાસે તેના પછીની સંખ્યા નથી? શું તે સૌથી છેલ્લી આવતી પ્રાકૃતિક સંખ્યા છે?
Answer: There is no natural number that does not have a succeeding number. For instance, \( 6 + 1 = 7 \), \( 7 + 1 = 8 \), \( 8 + 1 = 9 \), and so on. There is no last natural number because natural numbers are infinite. Natural numbers begin from 1 and continue indefinitely.
In simple words: Natural numbers never end, so you can always find a number after any given natural number just by adding 1. This means there is no "last" natural number.

Exam Tip: Understand the concept of infinite sets, such as natural numbers, where there is no largest or last element.

પ્રયત્ન કરો [પાન નંબર 29]

 

Question 1. શું દરેક પ્રાકૃતિક સંખ્યા પૂર્ણ સંખ્યા હોય છે?
Answer: Yes, every natural number is also a whole number.
In simple words: Since natural numbers start from 1 and go up, and whole numbers start from 0 and go up, all natural numbers are also included in the set of whole numbers.

Exam Tip: Remember that the set of natural numbers is a subset of the set of whole numbers.

 

Question 2. શું દરેક પૂર્ણ સંખ્યા પ્રાકૃતિક સંખ્યા હોય છે?
Answer: No, not every whole number is a natural number. This is because 0 is a whole number, but it is not a natural number.
In simple words: No, because 0 is a whole number, but it is not a natural number. This means not all whole numbers are natural numbers.

Exam Tip: The key difference between whole numbers and natural numbers is the inclusion of 0 in the set of whole numbers.

 

Question 3. સૌથી નાની પૂર્ણ સંખ્યા કઈ છે?
Answer: The smallest whole number is 0.
In simple words: The first and smallest number in the set of whole numbers is 0.

Exam Tip: Memorize the definition of whole numbers, which begin with 0.

 

Question 4. સૌથી મોટી પૂર્ણ સંખ્યા કઈ છે?
Answer: There is no largest whole number. This is because after every whole number, there is always another one. Also, the set of whole numbers is infinite.
In simple words: Whole numbers never end; you can always add 1 to any whole number to get a bigger one. So, there isn't a "biggest" whole number.

Exam Tip: Understand that the set of whole numbers is infinite, meaning there is no maximum element.

પ્રયત્ન કરો [પાન નંબર 30]

સંખ્યારેખાનો ઉપયોગ કરીને

 

Question 1. સંખ્યારેખાનો ઉપયોગ કરીને (1) 4 + 5 (2) 2 + 6 (3) 3 + 5 અને (4) 1 + 6નો સરવાળો મેળવો.
Answer:
(1) To add 5 to 4, we start at 4 and move 5 steps to the right, one unit at a time. After taking 5 steps, we reach 9.
\( \implies 4 + 5 = 9 \)
(2) To add 6 to 2, we start at 2 and move 6 steps to the right, one unit at a time. After taking 6 steps, we arrive at 8.
\( \implies 2 + 6 = 8 \)
(3) To add 5 to 3, we begin at 3 and move 5 steps to the right, one unit at a time. After taking 5 steps, we reach 8.
\( \implies 3 + 5 = 8 \)
(4) To add 6 to 1, we start at 1 and move 6 steps to the right, one unit at a time. After taking 6 steps, we arrive at 7.
\( \implies 1 + 6 = 7 \)
In simple words: To add numbers on a number line, start at the first number and move to the right by the value of the second number. Each jump represents one unit. The point you land on is the sum.

Exam Tip: Always start at the first number in the sum and move to the right for addition. Ensure each jump corresponds to a single unit on the number line for accuracy.

 

પ્રયત્ન કરો [પાન નંબર 30]

* સંખ્યારેખાના ઉપયોગથી

 

Question 1. સંખ્યારેખાના ઉપયોગથી (1) 8-3 (2) 6-2 (3) 9 - 6ની બાદબાકી મેળવો.
Answer:
(1) To subtract 3 from 8, we start at 8 and move 3 steps to the left, one unit at a time. After taking 3 steps, we reach 5.
\( \implies 8 - 3 = 5 \)
(2) To subtract 2 from 6, we begin at 6 and move 2 steps to the left, one unit at a time. After taking 2 steps, we arrive at 4.
\( \implies 6 - 2 = 4 \)
(3) To subtract 6 from 9, we start at 9 and move 6 steps to the left, one unit at a time. After taking 6 steps, we reach 3.
\( \implies 9 - 6 = 3 \)
In simple words: To subtract numbers using a number line, start at the first number and move to the left by the value of the second number. Each jump moves one unit. The point you land on is the difference.

Exam Tip: For subtraction on a number line, always move to the left from the starting number. The number of steps corresponds to the subtrahend (the number being subtracted).

પ્રયત્ન કરો [પાન નંબર 31]

* સંખ્યારેખાના ઉપયોગથી 2 × 6, 3 × 3, 4 × 2 મેળવો.

 

Question 1. સંખ્યારેખાના ઉપયોગથી (1) 2 × 6 (2) 3 × 3 અને (3) 4 × 2 મેળવો.
Answer:
(1) To multiply 2 by 6, we take 2 units, 6 times. Starting from 0, we move 6 steps of 2 units each to the right. This way, we reach 12.
\( \implies 2 \times 6 = 12 \)
(2) To multiply 3 by 3, we take 3 units, 3 times. Starting from 0, we move 3 steps of 3 units each to the right. This way, we reach 9.
\( \implies 3 \times 3 = 9 \)
(3) To multiply 4 by 2, we take 4 units, 2 times. Starting from 0, we move 2 steps of 4 units each to the right. This way, we reach 8.
\( \implies 4 \times 2 = 8 \)
In simple words: To multiply numbers on a number line, start at 0 and make equal-sized jumps to the right. The size of each jump is one of the numbers, and the count of jumps is the other number. The point where you finish jumping is the product.

Exam Tip: For multiplication on a number line, ensure you consistently use jumps of the same size. The final position on the line represents the product.

વિચારો, ચર્ચા કરો અને લખો [પાન નંબર 32-33]

 

Question 1. પૂર્ણ સંખ્યાઓ બાદબાકી માટે સંવૃત્ત નથી. શા માટે? તમારી બાદબાકી આ પ્રમાણે હોઈ શકે છે? તમે કોઈ પણ પાંચ ઉદાહરણ લઈ જાતે પ્રયત્ન કરો.
Answer: Whole numbers are not closed under subtraction because if you subtract a larger whole number from a smaller whole number, the result might not be a whole number. For example, if you subtract 9 from 3, the result is -6, which is not a whole number. Also, if you subtract 8 from 7, the result is -1, which is also not a whole number. This is why subtraction does not always produce a whole number when working with whole numbers.

Here are five examples:
1. \( 7 - 2 = 5 \), which is a whole number.
2. \( 10 - 15 = -5 \), which is not a whole number.
3. \( 20 - 22 = -2 \), which is not a whole number.
4. \( 14 - 4 = 10 \), which is a whole number.
5. \( 8 - 9 = -1 \), which is not a whole number.
In simple words: Whole numbers are not "closed" for subtraction, meaning when you subtract two whole numbers, you don't always get another whole number. For example, if you take a smaller whole number and subtract a larger one from it, you'll get a negative number, which isn't a whole number. You can try many examples to see this.

Exam Tip: To demonstrate that a set is not closed under an operation, a single counterexample (like \( 3 - 9 = -6 \)) is sufficient.

 

Question 2. શું પૂર્ણ સંખ્યાઓ ભાગાકાર માટે સંવૃત્ત છે? ના, નીચે આપેલું કોષ્ટક જુઓઃ
Answer: No, whole numbers are not closed under division. When you divide one whole number by another, the result is not always a whole number. Consider the following table:

Here are some examples:
1. \( 9 \div 3 = 3 \), which is a whole number.
2. \( 10 \div 7 = \frac{10}{7} \), which is not a whole number.
3. \( 11 \div 4 = \frac{11}{4} \), which is not a whole number.
4. \( 15 \div 5 = 3 \), which is a whole number.
5. \( 8 \div 3 = \frac{8}{3} \), which is not a whole number.
In simple words: Whole numbers are not "closed" for division because when you divide one whole number by another, the answer isn't always a whole number. For example, dividing 5 by 7 gives a fraction, not a whole number. Try more examples to confirm this.

Exam Tip: Remember that closure under an operation means that performing the operation on any two elements of the set always yields an element within the same set. Division often results in fractions, demonstrating that whole numbers are not closed under division.

પ્રયત્ન કરો [પાન નંબર 37]

 

Question 1. શોધોઃ (1) 7 + 18 + 13 અને (2) 16 + 10 + 4
Answer:
(1) This sum can be calculated as follows:
\( 7 + 18 + 13 \)
\( = 7 + 13 + 18 \) (Using the commutative property of addition)
\( = (7 + 13) + 18 \) (Using the associative property of addition)
\( = 20 + 18 \)
\( = 38 \)
(2) This sum can be calculated as follows:
\( 16 + 10 + 4 \)
\( = 16 + 4 + 10 \) (Using the commutative property of addition)
\( = (16 + 4) + 10 \) (Using the associative property of addition)
\( = 20 + 10 \)
\( = 30 \)
In simple words: When adding several numbers, you can change their order (commutative property) or group them differently (associative property) to make the calculation easier, without changing the final sum.

Exam Tip: Applying the commutative (reordering) and associative (regrouping) properties can simplify complex additions by allowing you to sum numbers that easily combine, such as those that add up to 10 or 20.

પ્રયત્ન કરો [પાન નંબર 39]

 

Question 1. શોધો : (1) 25 × 8358 × 4 અને (2) 625 × 3759 × 8
Answer:
(1) This product can be calculated efficiently:
\( 25 \times 8358 \times 4 \)
\( = 25 \times 4 \times 8358 \) (Using the commutative property of multiplication)
\( = (25 \times 4) \times 8358 \) (Using the associative property of multiplication)
\( = 100 \times 8358 \)
\( = 8,35,800 \)
(2) This product can be calculated efficiently:
\( 625 \times 3759 \times 8 \)
\( = 625 \times 8 \times 3759 \) (Using the commutative property of multiplication)
\( = (625 \times 8) \times 3759 \) (Using the associative property of multiplication)
\( = 5000 \times 3759 \)
\( = 1,87,95,000 \)
In simple words: When multiplying three or more numbers, you can change the order or group them in different ways to make the multiplication easier. For example, pairing 25 with 4 or 625 with 8 makes them easy multiples of 100 or 1000.

Exam Tip: Look for pairs of numbers that multiply to 10, 100, 1000, etc., as these can greatly simplify multiplication problems using the commutative and associative properties.

 

Question 2. વિભાજનના ગુણધર્મનો ઉપયોગ કરી (1) 15 × 68 (2) 7 × 23 અને (3) 69 × 78 + 22 × 69 શોધો.
Answer:
(1) Using the distributive property, we can calculate:
\( 15 \times 68 \)
\( = 15 \times (60 + 8) \) (Applying the distributive property)
\( = (15 \times 60) + (15 \times 8) \)
\( = 900 + 120 \)
\( = 1020 \)
(2) Using the distributive property, we can calculate:
\( 7 \times 23 \)
\( = 7 \times (20 + 3) \) (Applying the distributive property)
\( = (7 \times 20) + (7 \times 3) \)
\( = 140 + 21 \)
\( = 161 \)
(3) Using the distributive property (taking 69 as a common factor), we can calculate:
\( 69 \times 78 + 22 \times 69 \)
\( = 69 \times (78 + 22) \) (Taking 69 as the common factor)
\( = 69 \times 100 \)
\( = 6900 \)
In simple words: The distributive property helps to multiply numbers by breaking one number into a sum (like 68 into 60 + 8) and then multiplying each part. It also helps to factor out a common number in an addition problem.

Exam Tip: The distributive property \( a \times (b + c) = (a \times b) + (a \times c) \) is useful for mental math and simplifying expressions. Always look for ways to break down numbers into easier parts for calculation.

પ્રયત્ન કરો [પાન નંબર 42]

 

Question 1. કઈ સંખ્યાઓ કેવળ રેખાના રૂપમાં દર્શાવી શકાય છે?
Answer: All whole numbers can be represented as a line of dots. For example, 4 can be shown like this:
In simple words: Every number can be visually shown as a row of dots. For example, the number 4 can be drawn as four dots in a straight line.

Exam Tip: Remember that any natural number can be represented by a line of dots, demonstrating its sequential nature.

 

Question 2. કઈ સંખ્યાઓ ચોરસના રૂપમાં દર્શાવી શકાય છે?
Answer: All numbers that are perfect squares can be represented in the form of squares using dots. Such numbers include 4, 9, 16, 25, 36, and so on.
In simple words: Numbers that can be arranged into a perfect square shape with dots are called square numbers. These are numbers like 4 (2x2), 9 (3x3), 16 (4x4), and so on.

Exam Tip: Square numbers are obtained by multiplying an integer by itself. Their dot patterns always form a square.

 

Question 3. કઈ સંખ્યાઓ લંબચોરસના રૂપમાં દર્શાવી શકાય છે?
Answer: Numbers that are multiples of 2, such as 6, 8, 10, and 12, can be represented in the form of rectangles using dots.
In simple words: Numbers that can be arranged into a rectangular shape with dots are called rectangular numbers. Many composite numbers (numbers with more than two factors), like 6, 8, and 10, can form rectangles.

Exam Tip: A number is rectangular if it can be expressed as a product of two unequal integers greater than 1, or as an integer multiplied by itself (square numbers are also a type of rectangle).

 

Question 4. પ્રથમ સાત ત્રિકોણાકાર સંખ્યાઓ લખો. (એટલે તે સંખ્યાઓ જેને ત્રિકોણના રૂપમાં ગોઠવી શકાય છે.) દા. ત., 3, 6, .....
Answer: The first seven triangular numbers, which can form a triangular shape with dots, are 1, 3, 6, 10, 15, 21, and 28. Dot patterns for 3 and 6 are provided below. You can imagine and draw the patterns for others yourself.
In simple words: Triangular numbers are numbers that can form a triangle when you arrange them as dots. You get them by adding consecutive natural numbers: 1, then \( 1+2=3 \), then \( 1+2+3=6 \), and so on.

Exam Tip: Triangular numbers are found by summing consecutive natural numbers starting from 1. The formula for the nth triangular number is \( n(n+1)/2 \).

 

Question 5. કેટલીક સંખ્યાઓને જુદાં જુદાં બે લંબચોરસના રૂપમાં દર્શાવી શકાય છે. ઉદાહરણ તરીકે, 12 ને 3 × 4 અથવા 2 × 6. આ પ્રકારનાં ઓછામાં ઓછાં પાંચ ઉદાહરણો આપો.
Answer: Some numbers can be shown in the shape of two different rectangles. For instance, 12 can be represented as \( 3 \times 4 \) or \( 2 \times 6 \). Here are at least five more examples of this kind of number:
(i) 6 can be represented as \( 3 \times 2 \) or \( 2 \times 3 \).
or
(ii) 10 can be represented as \( 2 \times 5 \) or \( 5 \times 2 \).
or
(iii) 15 can be represented as \( 3 \times 5 \) or \( 5 \times 3 \).
or
(iv) 14 can be represented as \( 2 \times 7 \) or \( 7 \times 2 \).
or
(v) 20 can be represented as \( 5 \times 4 \) or \( 4 \times 5 \).
or
In simple words: Some numbers, like 6, 10, 12, 14, 15, and 20, can be shown as dots forming different rectangles because they have more than one pair of factors. You can turn the rectangle on its side (like 2x5 or 5x2) to get a different shape.

Exam Tip: Numbers that can form multiple distinct rectangles (excluding rotations as distinct) are composite numbers with more than two unique factors, often called highly composite numbers. Factors dictate the dimensions of the rectangles.

HOTs પ્રકારના પ્રશ્નોત્તર

 

Question 1. .............. પ્રાકૃતિક સંખ્યા નથી.
(a) 1000
(b) 100
(c) 1
(d) 0
Answer: (d) 0
In simple words: Natural numbers start from 1 (\(1, 2, 3, \dots \)), so 0 is not considered a natural number.

Exam Tip: Remember that natural numbers are positive integers (1, 2, 3, ...) while whole numbers include 0 (0, 1, 2, 3, ...).

 

Question 2. \( 10 \times (6 + 7) = 10 \times 6 + .............. \times 7 \)
(a) 10
(b) 6
(c) 7
(d) 13
Answer: (a) 10
In simple words: This equation shows the distributive property of multiplication over addition, where the number outside the bracket (10) multiplies both numbers inside the bracket.

Exam Tip: The distributive property states that \( a \times (b + c) = (a \times b) + (a \times c) \). The number outside the parenthesis (a) is distributed to each term inside.

 

Question 3. \( 8 \times (5 \times 4) = (8 \times 5) \times 4 \) ગુણાકારમાં ............. નો નિયમ સૂચવે છે.
(a) ક્રમ
(b) જૂથ
(c) વિભાજન
(d) સંવૃત્તતા
Answer: (b) જૂથ
In simple words: This property means that when you multiply three or more numbers, you can group them in any order without changing the final product.

Exam Tip: The associative property of multiplication means that the way numbers are grouped in a multiplication problem does not affect the product, i.e., \( (a \times b) \times c = a \times (b \times c) \).

 

Question 4. ................ સિવાયની દરેક સંખ્યાને ડૉટ્સ સ્વરૂપે રેખામાં દર્શાવી શકાય.
(a) 5
(b) 7
(c) 11
(d) 1
Answer: (d) 1
In simple words: While all numbers can be represented by dots, the number 1 is often considered a single point rather than forming a line in the same way other numbers can.

Exam Tip: This question tests a nuanced understanding of geometric representation. While 1 is a line of one dot, it's often conceptually differentiated from lines of multiple dots in visual patterns.

 

Question 5. ............... ને ચોરસ સ્વરૂપે ડૉટ્સ દ્વારા દર્શાવી શકાય.
(a) 6
(b) 8
(c) 4
(d) 10
Answer: (c) 4
In simple words: A number can be represented as a square of dots only if it is a perfect square, like 4 (which is \( 2 \times 2 \)).

Exam Tip: Identify perfect squares (numbers whose square root is an integer) to determine which numbers can form a square dot pattern.

 

Question 6. 10ને ડૉટ્સ દ્વારા ................ સ્વરૂપે દર્શાવી શકાય.
(a) ચોરસ
(b) ષટ્કોણ
(c) ત્રિકોણ
(d) વર્તુળ
Answer: (c) ત્રિકોણ
In simple words: The number 10 is a triangular number because you can arrange 10 dots to form a perfect triangle.

Exam Tip: Recall the sequence of triangular numbers (1, 3, 6, 10, 15, ...) to quickly identify which numbers can form a triangular dot pattern.