GSEB Class 7 Maths Solutions Chapter 2 અપૂર્ણાંક અને દશાંશ સંખ્યાઓ Exercise 2.7

1. શોધોઃ

Question (i). \( 0.4 \div 2 \)
Answer:
\( 0.4 \div 2 = \frac {4}{10} \times \frac {1}{2} \)
\( = \frac{4 \times 1}{10 \times 2} \)
\( = \frac {2}{10} \)
\( = 0.2 \)
In simple words: To divide a decimal like 0.4 by 2, you can change 0.4 into a fraction (4/10), then multiply by the inverse of 2 (which is 1/2). Simplify the fraction to get 2/10, which converts back to 0.2.

Exam Tip: When dividing decimals, it often helps to convert them to fractions first for clearer simplification, especially when dividing by whole numbers.

Question (ii). \( 0.35 \div 5 \)
Answer:
\( 0.35 \div 5 \)
\( = \frac {35}{100} \times \frac {1}{5} \)
\( = \frac{35 \times 1}{100 \times 5} \)
\( = \frac {7}{100} \)
\( = 0.07 \)
In simple words: To divide 0.35 by 5, first write 0.35 as a fraction (35/100). Then, multiply it by 1/5. Simplify the fraction to find the result, which is 0.07.

Exam Tip: Remember that dividing by a whole number is the same as multiplying by its reciprocal. This technique simplifies decimal division problems effectively.

Question (iii). \( 2.48 \div 4 \)
Answer:
\( 2.48 \div 4 \)
\( = \frac {248}{100} \times \frac {1}{4} \)
\( = \frac{248 \times 1}{100 \times 4} \)
\( = \frac {62}{100} \)
\( = 0.62 \)
In simple words: To divide 2.48 by 4, change 2.48 into the fraction 248/100. Then, multiply this fraction by 1/4. After simplifying, the result is 62/100, which is 0.62 as a decimal.

Exam Tip: Always make sure to reduce the fraction to its simplest form before converting back to a decimal to avoid errors.

Question (iv). \( 65.4 \div 6 \)
Answer:
\( 65.4 \div 6 \)
\( = \frac {654}{10} \times \frac {1}{6} \)
\( = \frac{654 \times 1}{10 \times 6} \)
\( = \frac {109}{10} \)
\( = 10.9 \)
In simple words: To perform 65.4 divided by 6, first represent 65.4 as the fraction 654/10. Next, multiply it by the reciprocal of 6 (which is 1/6). After simplifying, the value becomes 109/10, which means 10.9.

Exam Tip: When simplifying fractions, look for common factors in the numerator and denominator to divide by, making the calculation easier.

Question (v). \( 651.2 \div 4 \)
Answer:
\( 651.2 \div 4 \)
\( = \frac {6512}{10} \times \frac {1}{4} \)
\( = \frac{6512 \times 1}{10 \times 4} \)
\( = \frac {1628}{10} \)
\( = 162.8 \)
In simple words: To divide 651.2 by 4, convert 651.2 to the fraction 6512/10. Then, multiply by 1/4. Simplify the resulting fraction, which gives you 1628/10, or 162.8.

Exam Tip: Always double-check your division, especially when dealing with larger numbers, to make sure no calculation errors occur.

Question (vi). \( 14.49 \div 7 \)
Answer:
\( 14.49 \div 7 \)
\( = \frac {1449}{100} \times \frac {1}{7} \)
\( = \frac{1449 \times 1}{100 \times 7} \)
\( = \frac {207}{100} \)
\( = 2.07 \)
In simple words: To divide 14.49 by 7, write 14.49 as the fraction 1449/100. Then, multiply this by 1/7. After simplification, the answer is 207/100, which is 2.07.

Exam Tip: Be careful with the number of decimal places when converting back from fractions, ensuring it matches the denominator (e.g., /100 means two decimal places).

Question (vii). \( 3.96 \div 4 \)
Answer:
\( 3.96 \div 4 \)
\( = \frac {396}{100} \times \frac {1}{4} \)
\( = \frac{396 \times 1}{100 \times 4} \)
\( = \frac {99}{100} \)
\( = 0.99 \)
In simple words: To divide 3.96 by 4, change 3.96 into the fraction 396/100. Next, multiply this fraction by 1/4. When simplified, you will get 99/100, which is 0.99 as a decimal.

Exam Tip: It is always useful to estimate the answer before calculating. For example, 3.96 is close to 4, so 4 divided by 4 is 1. Therefore, the answer should be slightly less than 1.

Question (viii). \( 0.80 \div 5 \)
Answer:
\( 0.80 \div 5 \)
\( = \frac {80}{100} \times \frac {1}{5} \)
\( = \frac{80 \times 1}{100 \times 5} \)
\( = \frac {16}{100} \)
\( = 0.16 \)
In simple words: To divide 0.80 by 5, first express 0.80 as a fraction (80/100). Then, multiply this fraction by 1/5. After simplifying the numbers, you will achieve 16/100, which translates to 0.16.

Exam Tip: Remember that trailing zeros after a decimal point (like in 0.80) do not change the value but can help in converting to fractions if you are thinking of places like hundredths.

2. શોધોઃ

Question (i). \( 4.8 \div 10 \)
Answer:
\( 4.8 \div 10 \)
\( = \frac {48}{10} \times \frac {1}{10} \)
\( = \frac {48}{100} \)
\( = 0.48 \)
In simple words: To divide 4.8 by 10, simply move the decimal point one place to the left. This changes 4.8 to 0.48.

Exam Tip: Dividing a decimal by 10, 100, or 1000 means moving the decimal point to the left by the number of zeros in the divisor.

Question (ii). \( 52.5 \div 10 \)
Answer:
\( 52.5 \div 10 \)
\( = \frac {525}{10} \times \frac {1}{10} \)
\( = \frac {525}{100} \)
\( = 5.25 \)
In simple words: When you divide 52.5 by 10, the decimal point shifts one place to the left. So, 52.5 becomes 5.25.

Exam Tip: This pattern is a quick mental math trick for dividing by powers of ten. Ensure you move the decimal in the correct direction.

Question (iii). \( 0.7 \div 10 \)
Answer:
\( 0.7 \div 10 \)
\( = \frac {7}{10} \times \frac {1}{10} \)
\( = \frac {7}{100} \)
\( = 0.07 \)
In simple words: To divide 0.7 by 10, move the decimal point one position to the left. This changes 0.7 to 0.07.

Exam Tip: When moving the decimal point to the left, add a zero as a placeholder if there are no digits in that position, as seen in 0.07.

Question (iv). \( 33.1 \div 10 \)
Answer:
\( 33.1 \div 10 \)
\( = \frac {331}{10} \times \frac {1}{10} \)
\( = \frac {331}{100} \)
\( = 3.31 \)
In simple words: Dividing 33.1 by 10 means shifting the decimal point one place to the left. Thus, 33.1 becomes 3.31.

Exam Tip: This method is quick and accurate for division by 10. Just visualize the decimal moving to the left.

Question (v). \( 272.23 \div 10 \)
Answer:
\( 272.23 \div 10 \)
\( = \frac {27223}{100} \times \frac {1}{10} \)
\( = \frac {27223}{1000} \)
\( = 27.223 \)
In simple words: When you divide 272.23 by 10, the decimal point moves one spot to the left. This turns 272.23 into 27.223.

Exam Tip: Practicing division by powers of ten improves your speed in calculations. Always count the zeros to know how many places to shift the decimal.

Question (vi). \( 0.56 \div 10 \)
Answer:
\( 0.56 \div 10 \)
\( = \frac {56}{100} \times \frac {1}{10} \)
\( = \frac {56}{1000} \)
\( = 0.056 \)
In simple words: To divide 0.56 by 10, move the decimal point one position to the left. This means 0.56 becomes 0.056.

Exam Tip: Be mindful of adding leading zeros if needed to correctly position the decimal point after moving it. For example, 0.56 becomes 0.056, not 0.560.

Question (vii). \( 3.97 \div 10 \)
Answer:
\( 3.97 \div 10 \)
\( = \frac {397}{100} \times \frac {1}{10} \)
\( = \frac {397}{1000} \)
\( = 0.397 \)
In simple words: Dividing 3.97 by 10 causes the decimal point to move one spot to the left. So, 3.97 changes into 0.397.

Exam Tip: Understanding place values is key to correctly moving decimal points. Each shift of one place to the left divides the number by ten.

3. શોધોઃ

Question (i). \( 2.7 \div 100 \)
Answer:
\( 2.7 \div 100 \)
\( = \frac {27}{10} \times \frac {1}{100} \)
\( = \frac {27}{1000} \)
\( = 0.027 \)
In simple words: To divide 2.7 by 100, shift the decimal point two places to the left. This transforms 2.7 into 0.027.

Exam Tip: Dividing by 100 means moving the decimal point two places to the left. Add zeros as placeholders if necessary to maintain correct placement.

Question (ii). \( 0.3 \div 100 \)
Answer:
\( 0.3 \div 100 \)
\( = \frac {3}{10} \times \frac {1}{100} \)
\( = \frac {3}{1000} \)
\( = 0.003 \)
In simple words: When dividing 0.3 by 100, the decimal point moves two spots to the left. You need to add extra zeros as placeholders, changing 0.3 to 0.003.

Exam Tip: Always count the number of zeros in the divisor (100 has two zeros) to determine how many places to move the decimal point.

Question (iii). \( 0.78 \div 100 \)
Answer:
\( 0.78 \div 100 \)
\( = \frac {78}{100} \times \frac {1}{100} \)
\( = \frac {78}{10000} \)
\( = 0.0078 \)
In simple words: To divide 0.78 by 100, move the decimal point two spots to the left. You will need to add two zeros as placeholders, resulting in 0.0078.

Exam Tip: Be careful with leading zeros. When the decimal moves past existing digits, fill the empty spots with zeros to maintain the correct value.

Question (iv). \( 432.6 \div 100 \)
Answer:
\( 432.6 \div 100 \)
\( = \frac {4326}{10} \times \frac {1}{100} \)
\( = \frac {4326}{1000} \)
\( = 4.326 \)
In simple words: Dividing 432.6 by 100 means moving the decimal point two places to the left. This transforms 432.6 into 4.326.

Exam Tip: Always remember that the decimal point shifts to the left when dividing by powers of 10. The number of places moved depends on the number of zeros in the divisor.

Question (v). \( 23.6 \div 100 \)
Answer:
\( 23.6 \div 100 \)
\( = \frac {236}{10} \times \frac {1}{100} \)
\( = \frac {236}{1000} \)
\( = 0.236 \)
In simple words: When you divide 23.6 by 100, you move the decimal point two positions to the left. So, 23.6 becomes 0.236.

Exam Tip: It helps to think of it as making the number smaller by a factor of 100. Moving the decimal point is a quick way to show this change.

Question (vi). \( 98.53 \div 100 \)
Answer:
\( 98.53 \div 100 \)
\( = \frac {9853}{100} \times \frac {1}{100} \)
\( = \frac {9853}{10000} \)
\( = 0.9853 \)
In simple words: To divide 98.53 by 100, the decimal point shifts two places to the left. This changes 98.53 to 0.9853.

Exam Tip: Pay close attention to how many places the decimal needs to move. For 100, it's two places to the left.

4. શોધોઃ

Question (i). \( 7.9 \div 1000 \)
Answer:
\( 7.9 \div 1000 \)
\( = \frac {79}{10} \times \frac {1}{1000} \)
\( = \frac {79}{10000} \)
\( = 0.0079 \)
In simple words: To divide 7.9 by 1000, move the decimal point three places to the left. You will need to add extra zeros as placeholders, making the result 0.0079.

Exam Tip: When dividing by 1000, remember to shift the decimal point three places to the left, adding placeholder zeros where necessary.

Question (ii). \( 26.3 \div 1000 \)
Answer:
\( 26.3 \div 1000 \)
\( = \frac {263}{10} \times \frac {1}{1000} \)
\( = \frac {263}{10000} \)
\( = 0.0263 \)
In simple words: Dividing 26.3 by 1000 means moving the decimal point three spots to the left. This transforms 26.3 into 0.0263.

Exam Tip: For divisions involving 1000, ensure you move the decimal exactly three places. Visualizing the decimal shift can help prevent mistakes.

Question (iii). \( 38.53 \div 1000 \)
Answer:
\( 38.53 \div 1000 \)
\( = \frac {3853}{100} \times \frac {1}{1000} \)
\( = \frac {3853}{100000} \)
\( = 0.03853 \)
In simple words: To divide 38.53 by 1000, shift the decimal point three places to the left. This changes 38.53 into 0.03853.

Exam Tip: When dividing by 1000, count the three zeros and move the decimal point that many places to the left. Add zeros if there aren't enough digits.

Question (iv). \( 128.9 \div 1000 \)
Answer:
\( 128.9 \div 1000 \)
\( = \frac {1289}{10} \times \frac {1}{1000} \)
\( = \frac {1289}{10000} \)
\( = 0.1289 \)
In simple words: To divide 128.9 by 1000, move the decimal point three positions to the left. This results in 0.1289.

Exam Tip: This simple rule of shifting decimal points makes dividing by powers of ten much quicker than long division.

Question (v). \( 0.5 \div 1000 \)
Answer:
\( 0.5 \div 1000 \)
\( = \frac {5}{10} \times \frac {1}{1000} \)
\( = \frac {5}{10000} \)
\( = 0.0005 \)
In simple words: To divide 0.5 by 1000, move the decimal point three places to the left. You will need to add three zeros as placeholders, making the result 0.0005.

Exam Tip: Always verify that the number of placeholder zeros matches the power of ten you are dividing by. Three zeros for 1000, so three decimal shifts.

5. શોધોઃ

Question (i). \( 7 \div 3.5 \)
Answer:
\( 7 \div 3.5 \)
\( = 7 \div \frac {35}{10} \)
\( = \frac {7}{1} \times \frac {10}{35} \)
\( = \frac {1}{1} \times \frac {10}{5} \)
\( = \frac{1 \times 2}{1 \times 1} \)
\( = 2 \)
In simple words: To divide 7 by 3.5, convert 3.5 into the fraction 35/10. Then, multiply 7 by the reciprocal of 35/10, which is 10/35. Simplify the expression to get the answer, which is 2.

Exam Tip: When dividing by a decimal, it is often easiest to convert the divisor to a fraction and then multiply by its reciprocal.

Question (ii). \( 36 \div 0.2 \)
Answer:
\( 36 \div 0.2 \)
\( = 36 \div \frac {2}{10} \)
\( = \frac {36}{1} \times \frac {10}{2} \)
\( = \frac{18 \times 10}{1 \times 1} \)
\( = 180 \)
In simple words: To divide 36 by 0.2, change 0.2 to the fraction 2/10. Then, multiply 36 by the inverse of 2/10, which is 10/2. Simplify the numbers to achieve 180.

Exam Tip: A useful trick for dividing by a decimal is to make the divisor a whole number by multiplying both dividend and divisor by a power of 10. For example, multiply both by 10: \( 360 \div 2 = 180 \).

Question (iii). \( 3.25 \div 0.5 \)
Answer:
\( 3.25 \div 0.5 \)
\( = \frac {325}{100} \div \frac {5}{10} \)
\( = \frac {325}{100} \times \frac {10}{5} \)
\( = \frac{65 \times 10}{100 \times 1} \)
\( = \frac{65 \times 1}{10 \times 1} \)
\( = \frac {65}{10} \)
\( = 6.5 \)
In simple words: To divide 3.25 by 0.5, convert both to fractions: 325/100 and 5/10. Then, multiply 325/100 by the reciprocal of 5/10 (which is 10/5). Simplify the resulting fractions to get 65/10, which equals 6.5.

Exam Tip: When dividing by a decimal, you can also move the decimal point in both numbers until the divisor is a whole number: \( 3.25 \div 0.5 \) becomes \( 32.5 \div 5 \), which is simpler to calculate.

Question (iv). \( 30.94 \div 0.7 \)
Answer:
\( 30.94 \div 0.7 \)
\( = \frac {3094}{100} \div \frac {7}{10} \)
\( = \frac {3094}{100} \times \frac {10}{7} \)
\( = \frac{442 \times 1}{10 \times 1} \)
\( = \frac {442}{10} \)
\( = 44.2 \)
In simple words: To perform 30.94 divided by 0.7, change them into fractions (3094/100 and 7/10). Then, multiply the first fraction by the inverse of the second (10/7). Simplify the terms to arrive at 442/10, or 44.2.

Exam Tip: Always make sure to simplify the fractions as much as possible before performing the final multiplication to keep numbers manageable.

Question (v). \( 0.5 \div 0.25 \)
Answer:
\( 0.5 \div 0.25 \)
\( = \frac {5}{10} \div \frac {25}{100} \)
\( = \frac {5}{10} \times \frac {100}{25} \)
\( = \frac{1 \times 2}{1 \times 1} \)
\( = 2 \)
In simple words: To divide 0.5 by 0.25, convert both decimals to fractions (5/10 and 25/100). Then, multiply 5/10 by the reciprocal of 25/100 (which is 100/25). Simplify the fractions to get 2.

Exam Tip: It is easier to make the divisor a whole number by multiplying both numbers by 100: \( 0.5 \times 100 = 50 \) and \( 0.25 \times 100 = 25 \), so \( 50 \div 25 = 2 \).

Question (vi). \( 7.75 \div 0.25 \)
Answer:
\( 7.75 \div 0.25 \)
\( = \frac {775}{100} \div \frac {25}{100} \)
\( = \frac {775}{100} \times \frac {100}{25} \)
\( = \frac{31 \times 1}{1 \times 1} \)
\( = 31 \)
In simple words: To divide 7.75 by 0.25, change both to fractions (775/100 and 25/100). Then, multiply 775/100 by the reciprocal of 25/100 (which is 100/25). After simplifying, the answer is 31.

Exam Tip: When both the dividend and divisor have the same number of decimal places, you can often just divide the whole numbers. For example, \( 775 \div 25 = 31 \).

Question (vii). \( 76.5 \div 0.15 \)
Answer:
\( 76.5 \div 0.15 \)
\( = \frac {765}{10} \div \frac {15}{100} \)
\( = \frac {765}{10} \times \frac {100}{15} \)
\( = \frac{51 \times 10}{1 \times 1} \)
\( = 510 \)
In simple words: To divide 76.5 by 0.15, convert both to fractions (765/10 and 15/100). Then, multiply 765/10 by the reciprocal of 15/100 (which is 100/15). Simplify the numbers to get 510.

Exam Tip: To avoid fractions, multiply both numbers by 100 to remove decimals: \( 76.5 \times 100 = 7650 \) and \( 0.15 \times 100 = 15 \). Then calculate \( 7650 \div 15 \).

Question (viii). \( 37.8 \div 1.4 \)
Answer:
\( 37.8 \div 1.4 \)
\( = \frac {378}{10} \div \frac {14}{10} \)
\( = \frac {378}{10} \times \frac {10}{14} \)
\( = \frac{27 \times 1}{1 \times 1} \)
\( = 27 \)
In simple words: To divide 37.8 by 1.4, convert both into fractions (378/10 and 14/10). Then, multiply 378/10 by the reciprocal of 14/10 (which is 10/14). Simplify to find the answer, which is 27.

Exam Tip: When both numbers have one decimal place, you can simply divide the whole numbers directly: \( 378 \div 14 \).

Question (ix). \( 2.73 \div 1.3 \)
Answer:
\( 2.73 \div 1.3 \)
\( = \frac {273}{100} \div \frac {13}{10} \)
\( = \frac {273}{100} \times \frac {10}{13} \)
\( = \frac{21 \times 1}{10 \times 1} \)
\( = \frac {21}{10} \)
\( = 2.1 \)
In simple words: To divide 2.73 by 1.3, express both as fractions (273/100 and 13/10). Then, multiply 273/100 by the inverse of 13/10 (which is 10/13). Simplify the result to 21/10, which gives 2.1.

Exam Tip: To make the division simpler, multiply both numbers by 100 to remove decimals: \( 2.73 \times 100 = 273 \) and \( 1.3 \times 100 = 130 \). Then calculate \( 273 \div 130 \).

Question 6. એક વાહન 2.4 લિટર પેટ્રોલમાં 43.2 કિમીનું અંતર કાપે છે, તો 1 લિટર પેટ્રોલમાં વાહન દ્વારા કેટલું અંતર કપાયું હશે?
Answer:
2.4 લિટર પેટ્રોલથી કપાતું અંતર \( = 43.2 \) કિમી
1 લિટર પેટ્રોલથી કપાતું અંતર \( = \frac{43.2}{2.4} \) કિમી
\( = 18 \) કિમી
\( = \frac {43.2}{2.4} = \frac {432}{10} \div \frac {24}{10} \)
\( = \frac {432}{10} \times \frac {10}{24} \)
\( = 18 \)
1 લિટર પેટ્રોલથી 18 કિમી અંતર કપાયું હશે.
In simple words: If a vehicle goes 43.2 km on 2.4 liters of petrol, to find out how far it goes on 1 liter, you divide the total distance by the total petrol used. This calculation gives you 18 km per liter.

Exam Tip: This is a unit rate problem. Always divide the total quantity (distance) by the total units (liters of petrol) to find the amount per single unit.