RBSE Solutions Class 7 Maths Chapter 2 Fractions and Decimal Numbers Exercise 2.3

Question 1. Find:
(i) \( 12 \div \frac{2}{3} \)
(ii) \( 5 \div 3\frac{4}{7} \)
(iii) \( 3 \div 1\frac{1}{3} \)
(iv) \( 4 \div \frac{8}{3} \)
(v) \( 6 \div \frac{2}{3} \)
(vi) \( 15 \div \frac{5}{7} \)
Answer: For each part, we need to divide a whole number by a fraction or a mixed fraction. When dividing by a fraction, we multiply by its reciprocal (which means flipping the fraction). For mixed fractions, first change them into improper fractions. Understanding reciprocals is key to solving division problems with fractions easily.
(i) To divide 12 by \( \frac{2}{3} \), we multiply 12 by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). This gives \( 12 \times \frac{3}{2} = 6 \times 3 = 18 \).
(ii) First, change the mixed fraction \( 3\frac{4}{7} \) into an improper fraction: \( (3 \times 7 + 4) / 7 = 25/7 \). Then, 5 divided by \( \frac{25}{7} \) is 5 multiplied by \( \frac{7}{25} \). This gives \( \frac{5 \times 7}{25} = \frac{35}{25} \). We can simplify this to \( \frac{7}{5} \) by dividing both numbers by 5.
(iii) Change \( 1\frac{1}{3} \) to an improper fraction: \( (1 \times 3 + 1) / 3 = 4/3 \). Now, 3 divided by \( \frac{4}{3} \) is 3 multiplied by \( \frac{3}{4} \). This gives \( \frac{3 \times 3}{4} = \frac{9}{4} \).
(iv) To divide 4 by \( \frac{8}{3} \), we multiply 4 by the reciprocal of \( \frac{8}{3} \), which is \( \frac{3}{8} \). This gives \( \frac{4 \times 3}{8} = \frac{12}{8} \). We can simplify this to \( \frac{3}{2} \) by dividing both numbers by 4.
(v) To divide 6 by \( \frac{2}{3} \), we multiply 6 by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). This gives \( \frac{6 \times 3}{2} = \frac{18}{2} = 9 \).
(vi) To divide 15 by \( \frac{5}{7} \), we multiply 15 by the reciprocal of \( \frac{5}{7} \), which is \( \frac{7}{5} \). This gives \( \frac{15 \times 7}{5} = \frac{105}{5} = 21 \).
In simple words: When you divide a number by a fraction, you flip the fraction over and then multiply the numbers. Always convert mixed fractions to improper fractions first.

🎯 Exam Tip: Remember that division by a fraction is equivalent to multiplication by its reciprocal. Don't forget to simplify any resulting improper fractions.

Question 2. Find the reciprocal of the following:
(i) \( \frac{3}{7} \)
(ii) \( \frac{1}{8} \)
(iii) \( \frac{12}{7} \)
(iv) \( \frac{5}{8} \)
(v) \( \frac{9}{7} \)
Answer: To find the reciprocal of a fraction, simply flip the numerator and the denominator. The top number becomes the bottom, and the bottom number becomes the top. Reciprocals are important in mathematics for operations like division of fractions.
(i) The reciprocal of \( \frac{3}{7} \) is \( \frac{7}{3} \).
(ii) The reciprocal of \( \frac{1}{8} \) is \( \frac{8}{1} \) or \( 8 \).
(iii) The reciprocal of \( \frac{12}{7} \) is \( \frac{7}{12} \).
(iv) The reciprocal of \( \frac{5}{8} \) is \( \frac{8}{5} \).
(v) The reciprocal of \( \frac{9}{7} \) is \( \frac{7}{9} \).
In simple words: Just turn the fraction upside down to get its reciprocal. The number on top goes to the bottom, and the number on the bottom goes to the top.

🎯 Exam Tip: A whole number's reciprocal is 1 divided by that number (e.g., reciprocal of 5 is 1/5). A fraction's reciprocal simply swaps the numerator and denominator.

Question 3. Find:
(i) \( \frac{3}{7} \div 2 \)
(ii) \( 4\frac{3}{7} \div 7 \)
(iii) \( \frac{6}{13} \div 5 \)
(iv) \( 3\frac{1}{2} \div 4 \)
(v) \( \frac{6}{5} \div 3 \)
(vi) \( \frac{7}{3} \div 4 \)
Answer: For these problems, we need to divide fractions or mixed fractions by whole numbers. Remember that dividing by a whole number is the same as multiplying by its reciprocal (1 over that number). Always remember to convert mixed fractions into improper fractions before performing multiplication or division.
(i) To divide \( \frac{3}{7} \) by 2, multiply \( \frac{3}{7} \) by the reciprocal of 2, which is \( \frac{1}{2} \). This gives \( \frac{3 \times 1}{7 \times 2} = \frac{3}{14} \).
(ii) First, convert the mixed fraction \( 4\frac{3}{7} \) to an improper fraction: \( (4 \times 7 + 3) / 7 = 31/7 \). Then, divide \( \frac{31}{7} \) by 7. Multiply \( \frac{31}{7} \) by \( \frac{1}{7} \). This gives \( \frac{31 \times 1}{7 \times 7} = \frac{31}{49} \).
(iii) To divide \( \frac{6}{13} \) by 5, multiply \( \frac{6}{13} \) by the reciprocal of 5, which is \( \frac{1}{5} \). This gives \( \frac{6 \times 1}{13 \times 5} = \frac{6}{65} \).
(iv) Convert \( 3\frac{1}{2} \) to an improper fraction: \( (3 \times 2 + 1) / 2 = 7/2 \). Then, divide \( \frac{7}{2} \) by 4. Multiply \( \frac{7}{2} \) by \( \frac{1}{4} \). This gives \( \frac{7 \times 1}{2 \times 4} = \frac{7}{8} \).
(v) To divide \( \frac{6}{5} \) by 3, multiply \( \frac{6}{5} \) by the reciprocal of 3, which is \( \frac{1}{3} \). This gives \( \frac{6 \times 1}{5 \times 3} = \frac{6}{15} \). This can be simplified to \( \frac{2}{5} \) by dividing both numbers by 3.
(vi) To divide \( \frac{7}{3} \) by 4, multiply \( \frac{7}{3} \) by the reciprocal of 4, which is \( \frac{1}{4} \). This gives \( \frac{7 \times 1}{3 \times 4} = \frac{7}{12} \).
In simple words: When you divide a fraction by a regular number, just flip the regular number into a fraction (like 2 becomes 1/2) and then multiply. Change any mixed fractions to improper fractions first.

🎯 Exam Tip: Always represent whole numbers as fractions (e.g., 2 as 2/1) before finding their reciprocal or performing division with fractions.

Question 4. Find:
(i) \( \frac{7}{3} + \frac{8}{7} \)
(ii) \( 2\frac{1}{5} + \frac{3}{5} \)
(iii) \( 2\frac{1}{5} + 1\frac{1}{2} \)
(iv) \( 3\frac{1}{5} + 1\frac{3}{5} \)
(v) \( 3\frac{1}{3} + 2\frac{1}{3} \)
(vi) \( \frac{3}{5} + \frac{5}{7} \)
Answer: To add fractions, you need to make sure they have the same bottom number (denominator). If they are mixed fractions, first change them into improper fractions. Finding the least common multiple (LCM) of the denominators helps simplify the addition of fractions.
(i) To add \( \frac{7}{3} \) and \( \frac{8}{7} \), find a common denominator, which is 21. Convert both fractions: \( \frac{7}{3} = \frac{7 \times 7}{3 \times 7} = \frac{49}{21} \), and \( \frac{8}{7} = \frac{8 \times 3}{7 \times 3} = \frac{24}{21} \). Adding them gives \( \frac{49 + 24}{21} = \frac{73}{21} \).
(ii) Convert the mixed fraction \( 2\frac{1}{5} \) to an improper fraction: \( (2 \times 5 + 1) / 5 = 11/5 \). Since \( \frac{11}{5} \) and \( \frac{3}{5} \) already have the same denominator, add the top numbers: \( \frac{11 + 3}{5} = \frac{14}{5} \).
(iii) Convert the mixed fractions: \( 2\frac{1}{5} \) becomes \( \frac{11}{5} \) and \( 1\frac{1}{2} \) becomes \( \frac{3}{2} \). The common denominator for 5 and 2 is 10. Convert \( \frac{11}{5} \) to \( \frac{22}{10} \) and \( \frac{3}{2} \) to \( \frac{15}{10} \). Adding them gives \( \frac{22 + 15}{10} = \frac{37}{10} \).
(iv) Convert \( 3\frac{1}{5} \) to \( \frac{16}{5} \) and \( 1\frac{3}{5} \) to \( \frac{8}{5} \). Since the denominators are the same, add the top numbers: \( \frac{16 + 8}{5} = \frac{24}{5} \).
(v) Convert \( 3\frac{1}{3} \) to \( \frac{10}{3} \) and \( 2\frac{1}{3} \) to \( \frac{7}{3} \). Since the denominators are the same, add the top numbers: \( \frac{10 + 7}{3} = \frac{17}{3} \).
(vi) To add \( \frac{3}{5} \) and \( \frac{5}{7} \), find a common denominator, which is 35. Convert both fractions: \( \frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35} \), and \( \frac{5}{7} = \frac{5 \times 5}{7 \times 5} = \frac{25}{35} \). Adding them gives \( \frac{21 + 25}{35} = \frac{46}{35} \).
In simple words: When adding fractions, make sure the bottom numbers are the same. If they are different, change them to a common number before adding the top numbers. Always convert mixed fractions first.

🎯 Exam Tip: Always simplify fractions to their lowest terms after addition if possible. For mixed fractions, converting to improper fractions makes calculations easier.

Question 5. What will be number of \( \frac{1}{4} \) parts of bread if each of 6 breads is divided into equal parts of \( \frac{1}{4} \)?
Answer: We have 6 whole breads, and each bread needs to be cut into quarters (1/4 parts). To find out the total number of quarter-parts, we divide the total number of breads by the size of each part. This is similar to how you would cut a pizza into slices, where each slice is a fraction of the whole pizza.
So, total number of \( \frac{1}{4} \) parts = \( 6 \div \frac{1}{4} \).
When dividing by a fraction, we multiply by its reciprocal.
\( 6 \times \frac{4}{1} = 6 \times 4 = 24 \).
Therefore, there will be 24 parts of \( \frac{1}{4} \) from 6 breads.
In simple words: If you have 6 whole breads and cut each bread into 4 small pieces, you will get a total of 24 small pieces.

🎯 Exam Tip: Word problems involving division of fractions often require you to think about how many small parts are in a larger whole. Multiplying by the reciprocal is the key operation.

Question 6. How many \( \frac{1}{2} \) cm pieces can be cut of \( 11\frac{1}{2} \) cm long ribbon?
Answer: We have a ribbon that is \( 11\frac{1}{2} \) cm long. We want to cut it into smaller pieces, each \( \frac{1}{2} \) cm long. To find how many pieces we can get, we divide the total length of the ribbon by the length of each small piece. This is a practical example of how fractions are used in everyday measurements, like cutting fabric or wood.
First, convert the mixed fraction \( 11\frac{1}{2} \) to an improper fraction: \( (11 \times 2 + 1) / 2 = \frac{23}{2} \) cm.
Now, divide the total length by the length of one piece: \( \frac{23}{2} \div \frac{1}{2} \).
When dividing by a fraction, multiply by its reciprocal.
\( \frac{23}{2} \times \frac{2}{1} = \frac{23 \times 2}{2 \times 1} = \frac{46}{2} = 23 \).
So, 23 pieces of \( \frac{1}{2} \) cm length can be cut from the ribbon.
In simple words: You have a ribbon that is eleven and a half centimeters long. If you cut it into half-centimeter pieces, you will get 23 pieces in total.

🎯 Exam Tip: In word problems, converting mixed numbers to improper fractions before performing operations like division will help avoid errors and simplify calculations.