GSEB Class 7 Maths Solutions Chapter 1 પૂર્ણાંક સંખ્યાઓ Exercise 1.3

Question 1. નીચે આપેલ દરેકનો જવાબ લખો:
(a) \( 3 \times (-1) \)
Answer: \( 3 \times (-1) \)
\( = -(3 \times 1) \)
\( = (-3) \)
In simple words: When you multiply a positive number by a negative number, the result is always a negative number.

Exam Tip: Remember that multiplying any number by \( -1 \) simply changes its sign.

(b) \( (-1) \times 225 \)
Answer: \( (-1) \times 225 \)
\( = -(1 \times 225) \)
\( = (-225) \)
In simple words: When a negative one multiplies any positive integer, the outcome is the negative of that integer.

Exam Tip: A negative number multiplied by a positive number will always yield a negative result.

(c) \( (-21) \times (-30) \)
Answer: \( (-21) \times (-30) \)
\( = + (21 \times 30) \)
\( = 630 \)
In simple words: Multiplying two negative integers yields a positive product.

Exam Tip: The product of an even number of negative integers is always positive.

(d) \( (-316) \times (-1) \)
Answer: \( (-316) \times (-1) \)
\( = + (316 \times 1) \)
\( = 316 \)
In simple words: Multiplying a negative integer by negative one changes its sign to positive.

Exam Tip: Remember that \( (-1) \times (-1) = 1 \), so multiplying by \( -1 \) twice effectively reverts the sign, or makes a negative number positive.

(e) \( (-15) \times 0 \times (-18) \)
Answer: \( (-15) \times 0 \times (-18) \)
\( = [(-15) \times 0] \times (-18) \)
\( = 0 \times (-18) \)
\( = 0 \)
In simple words: Any number multiplied by zero will always give zero as the product, regardless of other factors.

Exam Tip: The "zero property of multiplication" states that anything multiplied by zero is zero. This simplifies calculations with zero significantly.

(f) \( (-12) \times (-11) \times 10 \)
Answer: \( (-12) \times (-11) \times 10 \)
\( = + (12 \times 11) \times 10 \)
\( = 132 \times 10 \)
\( = 1320 \)
In simple words: Multiply the first two negative numbers to get a positive result, then multiply that result by the third number.

Exam Tip: When multiplying multiple integers, count the number of negative signs. If even, the product is positive; if odd, the product is negative.

(g) \( 9 \times (-3) \times (-6) \)
Answer: \( 9 \times (-3) \times (-6) \)
\( = 9 \times [+ (3 \times 6)] \)
\( = 9 \times 18 \)
\( = 162 \)
In simple words: First, multiply the two negative numbers to get a positive value. Then, multiply this positive result by the remaining positive number.

Exam Tip: Multiplying two negative numbers always yields a positive number. Then, continue with standard multiplication.

(h) \( (-18) \times (-5) \times (-4) \)
Answer: \( (-18) \times (-5) \times (-4) \)
\( = [+ (18 \times 5)] \times (-4) \)
\( = + 90 \times (-4) \)
\( = (-360) \)
In simple words: Multiply the first two negative integers to get a positive product, then multiply this positive product by the last negative integer to get a final negative result.

Exam Tip: An odd number of negative factors (like three in this case) will always result in a negative product.

(i) \( (-1) \times (-2) \times (-3) \times 4 \)
Answer: \( (-1) \times (-2) \times (-3) \times 4 \)
\( = [+ (1 \times 2)] \times [-(3 \times 4)] \)
\( = 2 \times (-12) \)
\( = (-24) \)
In simple words: Multiply the negative numbers in pairs; two negatives give a positive. An odd number of negative factors will result in a negative product.

Exam Tip: Grouping negative factors can make the multiplication easier and help keep track of the sign. \( ( - ) \times ( - ) = ( + ) \).

(j) \( (-3) \times (-6) \times (-2) \times (-1) \)
Answer: \( (-3) \times (-6) \times (-2) \times (-1) \)
\( = [+(3 \times 6)] \times [+(2 \times 1)] \)
\( = 18 \times 2 \)
\( = 36 \)
In simple words: Multiply the negative numbers in pairs. Each pair of negative numbers results in a positive number. Since there is an even count of negative factors, the final product is positive.

Exam Tip: If there's an even count of negative signs in a multiplication, the final answer will be positive.

Question 2. નીચેનાને ચકાસોઃ
(a) \( 18 \times [7 + (-3)] = (18 \times 7) + [18 \times (-3)] \)
Answer:
ડાબા. \( = 18 \times [7 + (-3)] \)
\( = 18 \times (7-3) \)
\( = 18 \times 4 \)
\( = 72 \)
જ.બા. \( = (18 \times 7) + [18 \times (-3)] \)
\( = 126 + (-54) \)
\( = 72 \)
આમ, ડાબા. \( = \) જ.બા.
\( \implies 18 \times [7 + (-3)] = (18 \times 7) + [18 \times (-3)] \)
In simple words: To check this, calculate both sides separately. The left side uses addition inside the bracket before multiplication, while the right side multiplies first then adds. Both results match, showing the statement is true.

Exam Tip: This question verifies the distributive property of multiplication over addition. Always calculate both sides (LHS and RHS) separately to prove equality.

(b) \( (-21) \times [(-4) + (-6)] = [(-21) \times (-4)] + [(-21) \times (-6)] \)
Answer:
ડાબા. \( = (-21) \times [(-4) + (-6)] \)
\( = (-21) \times (-10) \)
\( = + (21 \times 10) \)
\( = 210 \)
જ.બા. \( = [(-21) \times (-4)] + [(-21) \times (-6)] \)
\( = [+ (21 \times 4)] + [+ (21 \times 6)] \)
\( = 84 + 126 \)
\( = 210 \)
આમ, ડાબા. \( = \) જ.બા.
\( \implies (-21) \times [(-4) + (-6)] = [(-21) \times (-4)] + [(-21) \times (-6)] \)
In simple words: Verify this by computing each side individually. The left expression combines numbers inside the bracket before multiplying by -21. The right expression multiplies -21 by each number inside the bracket first, then adds the results. Both sides give the same final value, proving the statement is correct.

Exam Tip: The distributive property holds true even with negative integers. Careful application of integer multiplication rules is essential for accuracy.

Question 3.
(i) કોઈ પણ પૂર્ણાંક સંખ્યા a માટે, \( (-1) \times a \) બરાબર શું થાય?
Answer: કોઈ પણ પૂર્ણાક a માટે \( (-1) \times a = (-a) \) થાય.
In simple words: For any integer 'a', multiplying it by negative one yields the opposite value, or negative 'a'.

Exam Tip: Multiplying by \( -1 \) is the same as finding the additive inverse of a number.

(ii) નીચેની પૂર્ણાંક સંખ્યાઓનો \( (-1) \) સાથેનો ગુણાકાર શું થશે?
(a) \( (-22) \)
(b) \( 37 \)
(c) \( 0 \)
Answer:
(a) \( (-1) \times (-22) = + (22) = 22 \)
(b) \( (-1) \times 37 = (-37) = -37 \)
(c) \( (-1) \times 0 = 0 = 0 \)
In simple words: When multiplying an integer by \( -1 \), its sign flips. A negative number becomes positive, a positive number becomes negative, and zero remains zero.

Exam Tip: Pay attention to the initial sign of the integer when multiplying by \( -1 \). A negative times a negative is positive, and a negative times a positive is negative.

Question 4. \( (-1) \times 5 \)થી શરૂ કરીને નિશ્ચિત પૅટર્ન વડે વિવિધ ગુણાકારો લઈને દર્શાવો કે \( (-1) \times (-1) = 1 \) થાય.
Answer:
\( (-1) \times 5 = (-5) \)
\( (-1) \times 4 = (-4) = (-5) + 1 \)
\( (-1) \times 3 = (-3) = (-4) + 1 \)
\( (-1) \times 2 = (-2) = (-3) + 1 \)
\( (-1) \times 1 = (-1) = (-2) + 1 \)
\( (-1) \times 0 = 0 = (-1) + 1 \)
\( (-1) \times (-1) = 1 = 0 + 1 \)
In simple words: Starting with \( (-1) \times 5 \), we decrease the positive multiplier by one each time. We notice that the product increases by 1 in each step. Continuing this pattern until the multiplier becomes \( -1 \), we can logically conclude that \( (-1) \times (-1) \) must equal 1 to maintain the observed increment.

Exam Tip: Demonstrating patterns like this helps in understanding the rules of integer multiplication, especially why a negative times a negative is positive.

Question 5. યોગ્ય ગુણધર્મોનો ઉપયોગ કરીને જવાબ શોધોઃ
(a) \( 26 \times (-48) + (-48) \times (-36) \)
Answer: \( 26 \times (-48) + (-48) \times (-36) \)
\( = (-48) \times [26 + (-36)] \) (વિભાજનનો ગુણધર્મ)
\( = (-48) \times (-10) \)
\( = 480 \)
In simple words: By using the distributive property, we can factor out \( (-48) \). Then, we add the remaining numbers inside the bracket and multiply by \( -48 \) to get the final positive outcome.

Exam Tip: Look for common factors to apply the distributive property \( a \times b + a \times c = a \times (b+c) \). This can greatly simplify calculations.

(b) \( 8 \times 53 \times (-125) \)
Answer: \( 8 \times 53 \times (-125) \)
\( = [8 \times (-125)] \times 53 \) (ગુણાકારમાં ક્રમનો નિયમ)
\( = (-1000) \times 53 \)
\( = (-53000) \)
In simple words: To simplify, we rearrange the numbers using the commutative property to group 8 and -125 together. Multiplying these first gives -1000, which makes it easy to multiply by 53.

Exam Tip: Use the commutative property to group numbers that are easy to multiply (e.g., \( 8 \times 125 = 1000 \)) to simplify complex multiplications.

(c) \( 15 \times (-25) \times (-4) \times (-10) \)
Answer: \( 15 \times (-25) \times (-4) \times (-10) \)
\( = [(-25) \times (-4)] \times [15 \times (-10)] \) (ગુણાકારમાં ક્રમનો નિયમ)
\( = 100 \times (-150) \)
\( = (-15000) \)
In simple words: To make calculation easier, we regroup the numbers. Multiplying the negative numbers \( (-25) \) and \( (-4) \) first yields a positive product. Then, multiply 15 by \( (-10) \) to get \( -150 \), and finally multiply the two results.

Exam Tip: Pairing numbers that multiply to powers of 10 (like \( 25 \times 4 = 100 \)) or easy multiples makes calculations much quicker and less error-prone.

(d) \( (-41) \times 102 \)
Answer: \( (-41) \times 102 \)
\( = (-41) \times (100 + 2) \) (વિભાજનનો ગુણધર્મ)
\( = (-41) \times 100 + (-41) \times 2 \)
\( = -4100 - 82 \)
\( = (-4182) \)
In simple words: Break down 102 into \( 100 + 2 \) and apply the distributive property. Multiply \( -41 \) by 100, then by 2, and finally sum these two products to get the total.

Exam Tip: Using the distributive property with numbers close to multiples of 10 or 100 (e.g., \( 102 = 100+2 \) or \( 98 = 100-2 \)) can simplify multiplication problems.

(e) \( 625 \times (-35) + (-625) \times 65 \)
Answer: \( 625 \times (-35) + (-625) \times 65 \)
\( = 625 \times (-35) + 625 \times (-65) \)
\( = 625 \times [(-35) + (-65)] \) (વિભાજનનો ગુણધર્મ)
\( = 625 \times (-100) \)
\( = (-62500) \)
In simple words: Recognize that \( (-625) \times 65 \) is the same as \( 625 \times (-65) \). Then, use the distributive property to factor out 625. Add the numbers inside the bracket and multiply by 625 to get the negative final answer.

Exam Tip: Always look for ways to rewrite terms (like \( -625 \) as \( 625 \times (-1) \)) to apply properties that simplify the expression.

(f) \( 7 \times (50 - 2) \)
Answer: \( 7 \times (50 - 2) \)
\( = 7 \times 50 - 7 \times 2 \) (વિભાજનનો ગુણધર્મ)
\( = 350 - 14 \)
\( = 336 \)
In simple words: Apply the distributive property by multiplying 7 with both 50 and 2 separately, then subtract the second product from the first to find the result.

Exam Tip: The distributive property works for both addition and subtraction. \( a \times (b-c) = a \times b - a \times c \).

(g) \( (-17) \times (-29) \)
Answer: \( (-17) \times (-29) \)
\( = + (17 \times 29) \)
\( = 17 \times (30 - 1) \)
\( = (17 \times 30) - (17 \times 1) \) (વિભાજનનો ગુણધર્મ)
\( = 510 - 17 \)
\( = 493 \)
In simple words: Multiplying two negative numbers gives a positive result. To simplify \( 17 \times 29 \), split 29 into \( 30 - 1 \) and use the distributive property to perform two simpler multiplications and then subtract.

Exam Tip: Breaking down one of the numbers into a sum or difference (e.g., \( 29 = 30-1 \)) is a clever application of the distributive property for mental math.

(h) \( (-57) \times (-19) + 57 \)
Answer: \( (-57) \times (-19) + 57 \)
\( = (-57) \times (-19) + (-57) \times (-1) \) (વિભાજનનો ગુણધર્મ)
\( = (-57) \times [(-19) + (-1)] \)
\( = (-57) \times (-20) \)
\( = 1140 \)
In simple words: Rewrite 57 as \( (-57) \times (-1) \). Then, apply the distributive property by factoring out \( (-57) \). Add the numbers inside the bracket and multiply by \( -57 \) to get the positive final product.

Exam Tip: Be creative in finding common factors. Recognizing that \( 57 = (-57) \times (-1) \) is key to applying the distributive property in this problem.

Question 6. ઠંડું કરવાની પ્રક્રિયા માટે ઓરડાના તાપમાનને 40 °Cથી શરૂ કરીને 5 °C પ્રતિ કલાકના દરે ઘટાડવું જરૂરી છે. પ્રક્રિયા ચાલુ કર્યાના 10 કલાક પછી ઓરડાનું તાપમાન કેટલું હશે?
Answer:
ઓરડાનું હાલનું તાપમાન \( = 40^\circ C \)
દર કલાકે બદલાતું તાપમાન \( = -5^\circ C \)
આ રીતે 10 કલાકમાં તાપમાનમાં થતો ધટાડો \( = 10 \times (-5^\circ C) = -50^\circ C \)
હવે, ઓરડાનું છેવટનું તાપમાન \( = [40 + (-50)]^\circ C = -10^\circ C \)
In simple words: The room starts at \( 40^\circ C \). Since the temperature drops by \( 5^\circ C \) every hour, after 10 hours, the total drop will be \( 10 \times 5 = 50^\circ C \). So, the final temperature will be \( 40^\circ C - 50^\circ C = -10^\circ C \).

Exam Tip: When dealing with changes, a decrease is represented by a negative value. Multiply the rate of change by the time period to find the total change.

Question 7. વર્ગ કસોટી પ્રશ્નપત્રમાં કુલ 10 પ્રશ્નો છે. દરેક સાચા જવાબના 5 ગુણ અને દરેક ખોટા જવાબના (-2) ગુણ છે અને પ્રશ્નનો જવાબ નહિ લખવાના 0 ગુણ આપવામાં આવે છે. નીચેના પ્રશ્નોના ગુણ કેટલા હશે?
(i) મોહનના 4 સાચા અને 6 ખોટા જવાબ છે, તો તેના ગુણ કેટલા હશે?
Answer:
મોહને મેળવેલા ગુણ \( = 4 \times 5 + 6 \times (-2) \)
\( = 20 + (-12) \)
\( = 8 \)
In simple words: Mohan got 4 correct answers, earning \( 4 \times 5 = 20 \) marks. He had 6 incorrect answers, losing \( 6 \times 2 = 12 \) marks. His total score is \( 20 - 12 = 8 \) marks.

Exam Tip: For scoring problems, clearly identify positive marks for correct answers and negative marks for incorrect ones. Sum them up for the final score.

(ii) રેશમાના 5 સાચા અને 5 ખોટા જવાબ છે, તો તેના ગુણ કેટલા હશે?
Answer:
રેશમાએ મેળવેલા ગુણ \( = 5 \times 5 + 5 \times (-2) \)
\( = 25 + (-10) \)
\( = 15 \)
In simple words: Reshma answered 5 questions correctly, getting \( 5 \times 5 = 25 \) marks. She also answered 5 questions incorrectly, losing \( 5 \times 2 = 10 \) marks. Her final score is \( 25 - 10 = 15 \) marks.

Exam Tip: Always make sure to count the number of correct, incorrect, and unattempted answers to apply the scoring rules accurately.

(iii) હીના 7 જવાબો લખે છે, જેમાંથી 2 સાચા અને 5 ખોટા જવાબો છે, તો તેના ગુણ કેટલા હશે?
Answer:
હીનાએ મેળવેલા ગુણ \( = 2 \times 5 + 5 \times (-2) + 3 \times 0 \)
\( = 10 + (-10) + 0 \)
\( = 10 - 10 \)
\( = 0 \)
In simple words: Heena gave 2 correct answers, earning \( 2 \times 5 = 10 \) marks. She gave 5 incorrect answers, losing \( 5 \times 2 = 10 \) marks. She did not attempt 3 questions, so no marks were added or subtracted. Her total score is \( 10 - 10 = 0 \) marks.

Exam Tip: Don't forget to account for questions not attempted, which usually carry zero marks, but confirm the specific rule if available.

Question 8. એક સિમેન્ટ કંપનીને સફેદ સિમેન્ટની એક ગૂણ વેચતાં Rs 8 નફો મળે છે અને રાખોડી સિમેન્ટની એક ગુણ વેચતાં Rs 5 ની ખોટ થાય છે. નીચેના પ્રશ્નોના જવાબ આપો:
(a) કંપનીએ એક મહિનામાં 3000 ગૂણ સફેદ સિમેન્ટની અને 5000 ગૂણ રાખોડી સિમેન્ટની વેચી છે, તો તે કંપનીને કેટલો નફો કે ખોટ થઈ હશે?
Answer:
સફેદ સિમેન્ટની ગુણ વેચતાં મળતો નફો \( = \) Rs 8
રાખોડી સિમેન્ટની ગૂણ વેચતાં થતી ખોટ \( = \) Rs 5
સફેદ સિમેન્ટની ગૂણનું વેચાણ \( = 3000 \) ગૂણ
રાખોડી સિમેન્ટની ગૂણનું વેચાણ \( = 5000 \) ગૂણ
સફેદ સિમેન્ટની ગૂણમાં નફો \( = \) Rs \( (3000 \times 8) = \) Rs \( 24,000 \)
રાખોડી સિમેન્ટની ગૂણમાં ખોટ \( = \) Rs \( (5000 \times 5) = \) Rs \( 25,000 \)
અહીં ખોટ \( > \) નફો. વેપારીને એકંદરે ખોટ જાય છે.
કુલ ખોટ \( = \) Rs \( (25,000 - 24,000) = \) Rs \( 1,000 \)
In simple words: The company sells 3000 bags of white cement, making a profit of \( 3000 \times \text{Rs } 8 = \text{Rs } 24,000 \). It also sells 5000 bags of gray cement, incurring a loss of \( 5000 \times \text{Rs } 5 = \text{Rs } 25,000 \). Since the total loss is greater than the total profit, the company experiences an overall loss of \( \text{Rs } 25,000 - \text{Rs } 24,000 = \text{Rs } 1,000 \).

Exam Tip: Calculate total profit and total loss separately. Compare them to determine whether there is an overall profit or loss, and by what amount.

(b) જો રાખોડી સિમેન્ટની 6400 ગૂણ વેચાઈ હોય, તો સફેદ સિમેન્ટની કેટલી ગૂણ વેચાય તો નફો પણ ન થાય અને ખોટ પણ ન જાય?
Answer:
રાખોડી સિમેન્ટની ગૂણનું વેચાણ \( = 6400 \) ગૂણ
રાખોડી સિમેન્ટની ગૂણમાં ખોટ \( = \) Rs \( (6400 \times 5) = \) Rs \( 32,000 \)
નફો પણ નહીં અને ખોટ પણ નહીં તે માટે સફેદ સિમેન્ટની ગૂણના વેચાણ દ્વારા Rs \( 32,000 \)નો નફો મળવો જોઈએ.
સફેદ સિમેન્ટની ગૂણ દીઠ Rs 8 નફો મળે છે.
સફેદ સિમેન્ટની વેચવાની ગૂણ \( = \frac{32,000}{8} = 4000 \)
આમ, વેપારીએ 4000 સફેદ સિમેન્ટની ગૂણ વેચવી જોઈએ. જેથી નફો પણ ન થાય અને ખોટ પણ ન જાય.
In simple words: If 6400 bags of gray cement are sold, the total loss will be \( 6400 \times \text{Rs } 5 = \text{Rs } 32,000 \). To break even, the company needs to make a profit of Rs 32,000 from white cement. Since each white cement bag brings a profit of Rs 8, the company must sell \( \text{Rs } 32,000 \div \text{Rs } 8 = 4000 \) bags of white cement to achieve neither profit nor loss.

Exam Tip: To find the break-even point, set the total profit equal to the total loss, or ensure the net profit/loss is zero.

Question 9. ખાલી જગ્યાને પૂર્ણાંક સંખ્યા વડે પૂરી સાચું વિધાન બનાવો:
(a) \( (-3) \times \underline{\hspace{1cm}} = 27 \)
Answer: \( (-3) \times (-9) = 27 \)
કારણઃ \( 3 \times 9 = 27 \) તથા બે ઋણ પૂર્ણાકોનો ગુણાકાર ધન મળે.
In simple words: The missing integer is -9. When you multiply two negative numbers, the outcome is always a positive number. Since \( 3 \times 9 = 27 \), then \( (-3) \times (-9) \) must be 27.

Exam Tip: Remember that negative times negative equals positive, and divide the product by the given factor to find the missing one.

(b) \( 5 \times \underline{\hspace{1cm}} = (-35) \)
Answer: \( 5 \times (-7) = (-35) \)
કારણ: \( 5 \times 7 = 35 \) તથા એક ધન અને બીજા ઋણ પૂર્ણાકનો ગુણાકાર ઋણ મળે.
In simple words: The missing integer is -7. When you multiply a positive number by a negative number, the result is always negative. Since \( 5 \times 7 = 35 \), then \( 5 \times (-7) \) gives \( -35 \).

Exam Tip: If the product is negative and one factor is positive, the other factor must be negative.

(c) \( \underline{\hspace{1cm}} \times (-8) = (-56) \)
Answer: \( 7 \times (-8) = (-56) \)
કારણ: \( 7 \times 8 = 56 \) તથા એક ધન અને બીજા ઋણ પૂર્ણાકનો ગુણાકાર ઋણ મળે.
In simple words: The missing integer is 7. When multiplying a positive number by a negative one, the outcome is negative. Given \( 7 \times 8 = 56 \), then \( 7 \times (-8) \) must equal \( -56 \).

Exam Tip: If one factor is negative and the product is negative, the other factor must be positive.

(d) \( \underline{\hspace{1cm}} \times (-12) = 132 \)
Answer: \( (-11) \times (-12) = 132 \)
કારણઃ \( 11 \times 12 = 132 \) તથા બે ઋણ પૂર્ણાકનો ગુણાકાર ધન મળે.
In simple words: The missing integer is -11. When you multiply two negative numbers, the outcome is positive. Since \( 11 \times 12 = 132 \), then \( (-11) \times (-12) \) will be 132.

Exam Tip: If the product is positive and one factor is negative, the other factor must also be negative.