GSEB Class 7 Maths Solutions Chapter 2 અપૂર્ણાંક અને દશાંશ સંખ્યાઓ Exercise 2.4

1. શોધોઃ

Question (i). \( 12 \div \frac {3}{4} \)
Answer: To divide 12 by the fraction \( \frac {3}{4} \), we first change the operation to multiplication and flip the second fraction.
\( 12 \div \frac {3}{4} \)
\( = 12 \times \frac {4}{3} \)
\( = \frac {4 \times 4}{1} \)
\( = 16 \)
The result of dividing 12 by \( \frac {3}{4} \) is 16.
In simple words: When dividing by a fraction, just multiply by its upside-down version. So, 12 divided by three-fourths becomes 12 multiplied by four-thirds, which gives 16.

Exam Tip: Remember that dividing by a fraction is the same as multiplying by its reciprocal. This is a crucial step in solving such problems.

Question (ii). \( 14 \div \frac {5}{6} \)
Answer: To find the value of 14 divided by \( \frac {5}{6} \), we need to multiply 14 by the inverse of the fraction \( \frac {5}{6} \).
\( 14 \div \frac {5}{6} \)
\( = 14 \times \frac {6}{5} \)
\( = \frac {14 \times 6}{5} \)
\( = \frac {84}{5} \)
The final answer, expressed as an improper fraction, is \( \frac {84}{5} \).
In simple words: Turn the division into multiplication by flipping the second fraction. Multiply 14 by six-fifths to get eighty-four over five.

Exam Tip: When the result is an improper fraction, you may also be asked to convert it to a mixed number. In this case, \( \frac{84}{5} \) would be \( 16\frac{4}{5} \).

Question (iii). \( 8 \div \frac {7}{3} \)
Answer: To compute 8 divided by \( \frac {7}{3} \), we will convert the division operation to multiplication and use the reciprocal of the divisor.
\( 8 \div \frac {7}{3} \)
\( = 8 \times \frac {3}{7} \)
\( = \frac {8 \times 3}{7} \)
\( = \frac {24}{7} \)
\( = 3\frac {3}{7} \)
The calculation shows that 8 divided by \( \frac {7}{3} \) gives \( 3\frac {3}{7} \).
In simple words: To divide 8 by seven-thirds, flip the fraction to three-sevenths and multiply by 8. This results in twenty-four over seven, which is three and three-sevenths.

Exam Tip: Always convert improper fractions to mixed numbers in your final answer if required by the question or standard practice.

Question (iv). \( 4 \div \frac {8}{3} \)
Answer: To solve 4 divided by \( \frac {8}{3} \), we replace the division with multiplication and use the inverted fraction.
\( 4 \div \frac {8}{3} \)
\( = 4 \times \frac {3}{8} \)
\( = \frac {1 \times 3}{2} \) (after simplifying 4 and 8)
\( = \frac {3}{2} \)
\( = 1\frac {1}{2} \)
Therefore, 4 divided by \( \frac {8}{3} \) equals \( 1\frac {1}{2} \).
In simple words: Change dividing by eight-thirds into multiplying by three-eighths. Simplify the numbers before multiplying, then convert the improper fraction to a mixed number.

Exam Tip: Simplify common factors between the numerator and denominator before multiplying to make calculations easier and reduce errors.

Question (v). \( 3 \div 2\frac {1}{3} \)
Answer: To compute 3 divided by \( 2\frac {1}{3} \), we must first transform the mixed number into an improper fraction.
\( 3 \div 2\frac {1}{3} \)
\( = 3 \div \frac {7}{3} \)
\( = 3 \times \frac {3}{7} \)
\( = \frac {9}{7} \)
\( = 1\frac {2}{7} \)
So, the outcome of 3 divided by \( 2\frac {1}{3} \) is \( 1\frac {2}{7} \).
In simple words: First, change the mixed number two and one-third into an improper fraction (seven-thirds). Then, flip that fraction and multiply by 3. Finally, convert your answer back into a mixed number.

Exam Tip: Always convert mixed numbers to improper fractions before performing multiplication or division operations.

Question (vi). \( 5 \div 3\frac {4}{7} \)
Answer: To find 5 divided by \( 3\frac {4}{7} \), we will begin by converting the mixed number to an improper fraction.
\( 5 \div 3\frac {4}{7} \)
\( = 5 \div \frac {25}{7} \)
\( = 5 \times \frac {7}{25} \)
\( = \frac {7}{5} \) (after simplifying 5 and 25)
\( = 1\frac {2}{5} \)
Thus, the result of 5 divided by \( 3\frac {4}{7} \) is \( 1\frac {2}{5} \).
In simple words: Change the mixed number to an improper fraction. Then, flip that improper fraction and multiply it by 5. Simplify the numbers if possible and write the final answer as a mixed number.

Exam Tip: Simplifying fractions before multiplying can save time and prevent errors in calculation. Always check for common factors in the numerators and denominators.

Question 2. નીચે આપેલ દરેક અપૂર્ણાકનો વ્યસ્ત શોધો. મેળવેલ વ્યસ્ત સંખ્યાઓનું શુદ્ધ અપૂર્ણાંક, અશુદ્ધ અપૂર્ણાંક અને પૂર્ણ સંખ્યામાં વર્ગીકરણ કરો:
(i) \( \frac {3}{7} \)
(ii) \( \frac {5}{8} \)
(iii) \( \frac {9}{7} \)
(iv) \( \frac {6}{5} \)
(v) \( \frac {12}{7} \)
(vi) \( \frac {1}{8} \)
(vii) \( \frac {1}{11} \)
Answer:
(i) \( \frac {3}{7} \) નો વ્યસ્ત \( \frac {7}{3} \). \( \frac {7}{3} \) એ અશુદ્ધ અપૂર્ણાંક છે.
(ii) \( \frac {5}{8} \) નો વ્યસ્ત \( \frac {8}{5} \). \( \frac {8}{5} \) એ અશુદ્ધ અપૂર્ણાંક છે.
(iii) \( \frac {9}{7} \) નો વ્યસ્ત \( \frac {7}{9} \). \( \frac {7}{9} \) એ શુદ્ધ અપૂર્ણાંક છે.
(iv) \( \frac {6}{5} \) નો વ્યસ્ત \( \frac {5}{6} \). \( \frac {5}{6} \) એ શુદ્ધ અપૂર્ણાંક છે.
(v) \( \frac {12}{7} \) નો વ્યસ્ત \( \frac {7}{12} \). \( \frac {7}{12} \) એ શુદ્ધ અપૂર્ણાંક છે.
(vi) \( \frac {1}{8} \) નો વ્યસ્ત 8. 8 એ પૂર્ણ સંખ્યા છે.
(vii) \( \frac {1}{11} \) નો વ્યસ્ત 11. 11 એ પૂર્ણ સંખ્યા છે.
In simple words: To find the reciprocal of a fraction, just flip the top and bottom numbers. If the top number is bigger, it's an improper fraction. If the bottom number is bigger, it's a proper fraction. If the fraction becomes a whole number, that's what we call a full number.

Exam Tip: Remember that a proper fraction has a numerator smaller than its denominator, an improper fraction has a numerator equal to or larger than its denominator, and a whole number has no fractional part.

3. શોધો:

Question (i). \( \frac {7}{3} \div 2 \)
Answer: To perform the division of \( \frac {7}{3} \) by 2, we will transform 2 into a fraction and then apply the reciprocal rule.
\( \frac {7}{3} \div 2 \)
\( = \frac {7}{3} \times \frac {1}{2} \)
\( = \frac {7 \times 1}{3 \times 2} \)
\( = \frac {7}{6} \)
\( = 1\frac {1}{6} \)
The final answer for \( \frac {7}{3} \) divided by 2 is \( 1\frac {1}{6} \).
In simple words: To divide a fraction by a whole number, flip the whole number into a fraction (like 2 becomes 1/2) and then multiply. Then, change the answer to a mixed number if needed.

Exam Tip: Any whole number can be written as a fraction by placing it over 1 (e.g., \( 2 = \frac{2}{1} \)). This makes it easier to find its reciprocal.

Question (ii). \( \frac {4}{9} \div 5 \)
Answer: To calculate \( \frac {4}{9} \) divided by 5, we will rewrite 5 as a fraction and then multiply by its inverse.
\( \frac {4}{9} \div 5 \)
\( = \frac {4}{9} \times \frac {1}{5} \)
\( = \frac {4 \times 1}{9 \times 5} \)
\( = \frac {4}{45} \)
The result of \( \frac {4}{9} \) divided by 5 is \( \frac {4}{45} \).
In simple words: Change dividing by 5 into multiplying by one-fifth. Then, just multiply the top numbers together and the bottom numbers together to get the answer.

Exam Tip: When multiplying fractions, multiply numerators together and denominators together. Simplify the resulting fraction if possible.

Question (iii). \( \frac {6}{13} \div 7 \)
Answer: To evaluate \( \frac {6}{13} \) divided by 7, we first express 7 as a fraction and then perform the multiplication with its reciprocal.
\( \frac {6}{13} \div 7 \)
\( = \frac {6}{13} \times \frac {1}{7} \)
\( = \frac {6 \times 1}{13 \times 7} \)
\( = \frac {6}{91} \)
Therefore, \( \frac {6}{13} \) divided by 7 equals \( \frac {6}{91} \).
In simple words: To divide a fraction by a whole number like 7, change the 7 to \( \frac{1}{7} \) and then multiply the two fractions. The result is six over ninety-one.

Exam Tip: Be careful not to cross-multiply in division; always find the reciprocal of the second fraction first.

Question (iv). \( 4\frac {1}{3} \div 3 \)
Answer: To solve \( 4\frac {1}{3} \) divided by 3, we start by converting the mixed number to an improper fraction.
\( 4\frac {1}{3} \div 3 \)
\( = \frac {13}{3} \times \frac {1}{3} \)
\( = \frac {13 \times 1}{3 \times 3} \)
\( = \frac {13}{9} \)
\( = 1\frac {4}{9} \)
The value of \( 4\frac {1}{3} \) divided by 3 is \( 1\frac {4}{9} \).
In simple words: First, change the mixed number four and one-third into an improper fraction (thirteen-thirds). Then, multiply this by the reciprocal of 3, which is one-third. Convert the answer back to a mixed number.

Exam Tip: Remember that \( 3 = \frac{3}{1} \), so its reciprocal is \( \frac{1}{3} \). This step is essential when dividing by whole numbers.

Question (v). \( 3\frac {1}{2} \div 4 \)
Answer: To determine the result of \( 3\frac {1}{2} \) divided by 4, we will first convert the mixed number to an improper fraction.
\( 3\frac {1}{2} \div 4 \)
\( = \frac {7}{2} \times \frac {1}{4} \)
\( = \frac {7 \times 1}{2 \times 4} \)
\( = \frac {7}{8} \)
Thus, \( 3\frac {1}{2} \) divided by 4 equals \( \frac {7}{8} \).
In simple words: Turn the mixed number three and one-half into seven-halves. Then, multiply this fraction by the flipped version of 4 (which is one-fourth). This gives you seven-eighths.

Exam Tip: When a mixed number is part of a division problem, converting it to an improper fraction is always the first action to take.

Question (vi). \( 4\frac {3}{7} \div 7 \)
Answer: To solve \( 4\frac {3}{7} \) divided by 7, we first change the mixed number into an improper fraction.
\( 4\frac {3}{7} \div 7 \)
\( = \frac {31}{7} \times \frac {1}{7} \)
\( = \frac {31 \times 1}{7 \times 7} \)
\( = \frac {31}{49} \)
The result of \( 4\frac {3}{7} \) divided by 7 is \( \frac {31}{49} \).
In simple words: Convert the mixed number (four and three-sevenths) into an improper fraction (thirty-one sevenths). Then, multiply this by the reciprocal of 7, which is one-seventh. The answer is thirty-one over forty-nine.

Exam Tip: Be careful with calculations involving larger numbers in the denominator. Double-check your multiplication steps to avoid errors.

4. શોધો :

Question (i). \( \frac {2}{5} \div \frac {1}{2} \)
Answer: To divide \( \frac {2}{5} \) by \( \frac {1}{2} \), we need to multiply the first fraction by the reciprocal of the second fraction.
\( \frac {2}{5} \div \frac {1}{2} \)
\( = \frac {2}{5} \times \frac {2}{1} \)
\( = \frac {2 \times 2}{5 \times 1} \)
\( = \frac {4}{5} \)
Thus, \( \frac {2}{5} \) divided by \( \frac {1}{2} \) equals \( \frac {4}{5} \).
In simple words: When dividing fractions, flip the second fraction over and then multiply. Two-fifths times two over one gives four-fifths.

Exam Tip: Always remember that "invert and multiply" is the key rule for dividing fractions.

Question (ii). \( \frac {4}{9} \div \frac {2}{3} \)
Answer: To compute \( \frac {4}{9} \) divided by \( \frac {2}{3} \), we will multiply \( \frac {4}{9} \) by the inverse of \( \frac {2}{3} \).
\( \frac {4}{9} \div \frac {2}{3} \)
\( = \frac {4}{9} \times \frac {3}{2} \)
\( = \frac {2 \times 1}{3 \times 1} \) (after simplifying 4 with 2 and 9 with 3)
\( = \frac {2}{3} \)
Therefore, \( \frac {4}{9} \) divided by \( \frac {2}{3} \) is \( \frac {2}{3} \).
In simple words: Flip the second fraction from two-thirds to three-halves, then multiply. You can cross-cancel numbers (like 4 and 2, or 3 and 9) before multiplying to make it easier.

Exam Tip: Simplifying diagonally before multiplying fractions can significantly simplify your final result and reduce calculation effort.

Question (iii). \( \frac {3}{7} \div \frac {8}{7} \)
Answer: To solve \( \frac {3}{7} \) divided by \( \frac {8}{7} \), we will multiply the first fraction by the flipped version of the second fraction.
\( \frac {3}{7} \div \frac {8}{7} \)
\( = \frac {3}{7} \times \frac {7}{8} \)
\( = \frac {3 \times 1}{1 \times 8} \) (after simplifying 7 and 7)
\( = \frac {3}{8} \)
Hence, \( \frac {3}{7} \) divided by \( \frac {8}{7} \) equals \( \frac {3}{8} \).
In simple words: Change the division to multiplication and flip the second fraction (eight-sevenths becomes seven-eighths). The 7s cancel out, leaving you with three-eighths.

Exam Tip: When the denominators are the same, they cancel out easily after finding the reciprocal, making the calculation faster.

Question (iv). \( 2\frac {1}{3} \div \frac {3}{5} \)
Answer: To calculate \( 2\frac {1}{3} \) divided by \( \frac {3}{5} \), we must first convert the mixed number to an improper fraction.
\( 2\frac {1}{3} \div \frac {3}{5} \)
\( = \frac {7}{3} \div \frac {3}{5} \)
\( = \frac {7}{3} \times \frac {5}{3} \)
\( = \frac {35}{9} \)
\( = 3\frac {8}{9} \)
The final answer for \( 2\frac {1}{3} \) divided by \( \frac {3}{5} \) is \( 3\frac {8}{9} \).
In simple words: First, turn two and one-third into an improper fraction (seven-thirds). Then, multiply this by the flipped version of three-fifths (which is five-thirds). Finally, change the improper fraction result into a mixed number.

Exam Tip: Always remember the order of operations: convert mixed numbers, then invert the divisor and multiply, and finally simplify.

Question (v). \( 3\frac {1}{2} \div \frac {8}{3} \)
Answer: To determine the value of \( 3\frac {1}{2} \) divided by \( \frac {8}{3} \), we will convert the mixed number into an improper fraction initially.
\( 3\frac {1}{2} \div \frac {8}{3} \)
\( = \frac {7}{2} \div \frac {8}{3} \)
\( = \frac {7}{2} \times \frac {3}{8} \)
\( = \frac {21}{16} \)
\( = 1\frac {5}{16} \)
Therefore, \( 3\frac {1}{2} \) divided by \( \frac {8}{3} \) equals \( 1\frac {5}{16} \).
In simple words: Convert three and one-half to seven-halves. Then, multiply this by the reciprocal of eight-thirds (which is three-eighths). The result is twenty-one over sixteen, which you then convert to a mixed number.

Exam Tip: Be cautious when converting mixed numbers and finding reciprocals. A small error in these initial steps can lead to an incorrect final answer.

Question (vi). \( \frac {2}{5} \div 1\frac {1}{2} \)
Answer: To compute \( \frac {2}{5} \) divided by \( 1\frac {1}{2} \), our first step is to transform the mixed number into an improper fraction.
\( \frac {2}{5} \div 1\frac {1}{2} \)
\( = \frac {2}{5} \div \frac {3}{2} \)
\( = \frac {2}{5} \times \frac {2}{3} \)
\( = \frac {4}{15} \)
So, \( \frac {2}{5} \) divided by \( 1\frac {1}{2} \) results in \( \frac {4}{15} \).
In simple words: Change one and one-half into three-halves. Then, flip three-halves to two-thirds and multiply it by two-fifths. The answer is four-fifteenths.

Exam Tip: Always make sure to convert any mixed numbers to improper fractions before attempting division or multiplication with other fractions.

Question (vii). \( 3\frac {1}{5} \div 1\frac {2}{3} \)
Answer: To solve \( 3\frac {1}{5} \) divided by \( 1\frac {2}{3} \), we will start by converting both mixed numbers into improper fractions.
\( 3\frac {1}{5} \div 1\frac {2}{3} \)
\( = \frac {16}{5} \div \frac {5}{3} \)
\( = \frac {16}{5} \times \frac {3}{5} \)
\( = \frac {48}{25} \)
\( = 1\frac {23}{25} \)
Thus, the calculation of \( 3\frac {1}{5} \) divided by \( 1\frac {2}{3} \) gives \( 1\frac {23}{25} \).
In simple words: Convert both mixed numbers into improper fractions. Then, flip the second fraction (five-thirds becomes three-fifths) and multiply the two improper fractions. Convert your final improper fraction back to a mixed number.

Exam Tip: When dividing two mixed numbers, both must be converted to improper fractions before applying the "invert and multiply" rule.

Question (viii). \( 2\frac {1}{5} \div 1\frac {1}{5} \)
Answer: To determine \( 2\frac {1}{5} \) divided by \( 1\frac {1}{5} \), we must convert both mixed numbers into improper fractions first.
\( 2\frac {1}{5} \div 1\frac {1}{5} \)
\( = \frac {11}{5} \div \frac {6}{5} \)
\( = \frac {11}{5} \times \frac {5}{6} \)
\( = \frac {11}{6} \) (after simplifying 5 and 5)
\( = 1\frac {5}{6} \)
Hence, \( 2\frac {1}{5} \) divided by \( 1\frac {1}{5} \) equals \( 1\frac {5}{6} \).
In simple words: Change both mixed numbers into improper fractions. Flip the second improper fraction and multiply. Simplify by canceling out common numbers, then change the result to a mixed number.

Exam Tip: Notice when denominators are the same in the improper fractions before multiplying; they often simplify nicely, as the 5s cancelled here.