RBSE Solutions Class 7 Maths Chapter 2 Fractions and Decimal Numbers Exercise 2.2

Question 1. Match appropriate product with line diagrams:
(i)
(ii)
(iii)
(a) \( \frac{3}{4} \times 3 \)
(b) \( \frac{1}{4} \times 2 \)
(c) \( \frac{3}{5} \times 3 \)
Answer:
(i) (b) (This diagram shows 2 groups, each with 1 out of 4 parts shaded, so it matches \( \frac{1}{4} \times 2 \).)
(ii) (c) (This diagram shows 3 groups, each with 3 out of 5 parts shaded, so it matches \( \frac{3}{5} \times 3 \).)
(iii) (a) (This diagram shows 3 groups, each with 3 out of 4 parts shaded, so it matches \( \frac{3}{4} \times 3 \).)
In simple words: Look at each picture to see how many equal parts are shaded and how many groups there are. Then match it to the correct multiplication problem that shows the same thing.

🎯 Exam Tip: Pay close attention to both the fraction represented in each individual figure and the number of times that figure is repeated to correctly identify the multiplication expression.

Question 2. Show the following figures in terms of multiplication (repeated addition) :
(i)
(ii)
(iii)
Answer:
(i) The figure shows two groups, and in each group, 1 out of 3 parts is shaded. This means we have \( \frac{1}{3} \) added two times. So, the multiplication is \( \frac{1}{3} \times 2 \).
(ii) The figure shows three groups, and in each group, 2 out of 5 parts are shaded. This means we have \( \frac{2}{5} \) added three times. So, the multiplication is \( \frac{2}{5} \times 3 \).
(iii) The figure shows three groups, and in each group, 2 out of 4 parts are shaded. This means we have \( \frac{2}{4} \) added three times. So, the multiplication is \( \frac{2}{4} \times 3 \).
In simple words: Count how many groups you see. Then, in one group, count the shaded parts out of the total parts to get a fraction. Multiply the fraction by the number of groups to show repeated addition as multiplication.

🎯 Exam Tip: When writing repeated addition as multiplication, the number of groups always comes after the fraction. For example, if you have three groups of \( \frac{1}{2} \), it's \( \frac{1}{2} \times 3 \), not \( 3 \times \frac{1}{2} \).

Question 3. Multiply and express in the simplest form:
(i) \( 8 \times \frac{3}{5} \)
(ii) \( \frac{2}{3} \times 4 \)
(iii) \( \frac{5}{2} \times 6 \)
(iv) \( 15 \times \frac{3}{5} \)
(v) \( 20 \times \frac{2}{3} \)
(vi) \( 18 \times \frac{1}{9} \)
(vii) \( 2\frac{2}{3} \times \frac{6}{7} \)
(viii) \( 12 \times \frac{5}{3} \)
(ix) \( \frac{3}{8} \times \frac{6}{4} \)
(x) \( \frac{4}{5} \times \frac{12}{7} \)
Answer:
(i) We multiply the whole number by the numerator and keep the denominator the same:
\( 8 \times \frac{3}{5} = \frac{8 \times 3}{5} = \frac{24}{5} \)
(ii) We multiply the numerator by the whole number:
\( \frac{2}{3} \times 4 = \frac{2 \times 4}{3} = \frac{8}{3} \)
(iii) First, multiply the numerator by the whole number, then simplify:
\( \frac{5}{2} \times 6 = \frac{5 \times 6}{2} = \frac{30}{2} = 15 \)
(iv) Multiply the whole number by the numerator and simplify:
\( 15 \times \frac{3}{5} = \frac{15 \times 3}{5} = \frac{45}{5} = 9 \)
(v) Multiply the whole number by the numerator, the fraction stays improper:
\( 20 \times \frac{2}{3} = \frac{20 \times 2}{3} = \frac{40}{3} \)
(vi) Multiply and then simplify the fraction:
\( 18 \times \frac{1}{9} = \frac{18 \times 1}{9} = \frac{18}{9} = 2 \)
(vii) First, change the mixed fraction to an improper fraction, then multiply:
\( 2\frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{8}{3} \)
Now, multiply:
\( \frac{8}{3} \times \frac{6}{7} = \frac{8 \times 6}{3 \times 7} = \frac{48}{21} \)
Now, simplify by dividing both by 3:
\( \frac{48 \div 3}{21 \div 3} = \frac{16}{7} \)
(viii) Multiply the whole number by the numerator and simplify:
\( 12 \times \frac{5}{3} = \frac{12 \times 5}{3} = \frac{60}{3} = 20 \)
(ix) Multiply the numerators and the denominators, then simplify:
\( \frac{3}{8} \times \frac{6}{4} = \frac{3 \times 6}{8 \times 4} = \frac{18}{32} \)
Simplify by dividing both by 2:
\( \frac{18 \div 2}{32 \div 2} = \frac{9}{16} \)
(x) Multiply the numerators and the denominators:
\( \frac{4}{5} \times \frac{12}{7} = \frac{4 \times 12}{5 \times 7} = \frac{48}{35} \)
In simple words: To multiply fractions, multiply the top numbers together and the bottom numbers together. If there's a whole number, think of it as a fraction over 1. Always simplify your answer to the smallest possible numbers at the end. If you have a mixed fraction, change it into a top-heavy fraction first.

🎯 Exam Tip: Always look for common factors in the numerator and denominator *before* multiplying to simplify calculations and avoid larger numbers. This is called cross-cancellation.

Question 5. Find the following:
(i) \( \frac{1}{3} \) of 27
(ii) \( \frac{1}{3} \) of 18
(iii) \( \frac{1}{5} \) of 50
(iv) \( \frac{3}{4} \) of 24
(v) \( \frac{5}{4} \) of 32
(vi) \( \frac{3}{7} \) of 28
Answer:
(i) To find a fraction "of" a number, we multiply the fraction by the number:
\( \frac{1}{3} \text{ of } 27 = 27 \times \frac{1}{3} = \frac{27 \times 1}{3} = \frac{27}{3} = 9 \)
(ii) We multiply the fraction by 18:
\( \frac{1}{3} \text{ of } 18 = 18 \times \frac{1}{3} = \frac{18 \times 1}{3} = \frac{18}{3} = 6 \)
(iii) We multiply the fraction by 50:
\( \frac{1}{5} \text{ of } 50 = 50 \times \frac{1}{5} = \frac{50 \times 1}{5} = \frac{50}{5} = 10 \)
(iv) We multiply the fraction by 24:
\( \frac{3}{4} \text{ of } 24 = 24 \times \frac{3}{4} = \frac{24 \times 3}{4} = \frac{72}{4} = 18 \)
(v) We multiply the fraction by 32:
\( \frac{5}{4} \text{ of } 32 = 32 \times \frac{5}{4} = \frac{32 \times 5}{4} = \frac{160}{4} = 40 \)
(vi) We multiply the fraction by 28:
\( \frac{3}{7} \text{ of } 28 = 28 \times \frac{3}{7} = \frac{28 \times 3}{7} = \frac{84}{7} = 12 \)
In simple words: When you see "of" between a fraction and a number, it means multiply. You just multiply the fraction by that number to get your answer. It's like finding a part of a whole.

🎯 Exam Tip: Always remember that "of" means multiplication in mathematics. It's an easy way to understand what calculation to perform.

Question 6. Find the following:
(i) \( \frac{1}{5} \) of 4
(ii) \( \frac{2}{5} \) of \( 5\frac{1}{5} \)
(iii) \( \frac{8}{5} \) of \( 3\frac{2}{5} \)
(iv) \( \frac{2}{3} \) of \( 9\frac{2}{3} \)
(v) \( \frac{1}{5} \) of \( \frac{3}{5} \)
(vi) \( \frac{1}{7} \) of \( \frac{3}{10} \)
Answer:
(i) We multiply the fraction by the whole number:
\( \frac{1}{5} \text{ of } 4 = 4 \times \frac{1}{5} = \frac{4 \times 1}{5} = \frac{4}{5} \)
(ii) First, convert the mixed fraction to an improper fraction: \( 5\frac{1}{5} = \frac{26}{5} \). Then multiply:
\( \frac{2}{5} \text{ of } 5\frac{1}{5} = \frac{2}{5} \times \frac{26}{5} = \frac{2 \times 26}{5 \times 5} = \frac{52}{25} \)
(iii) Convert the mixed fraction to an improper fraction: \( 3\frac{2}{5} = \frac{17}{5} \). Then multiply:
\( \frac{8}{5} \text{ of } 3\frac{2}{5} = \frac{8}{5} \times \frac{17}{5} = \frac{8 \times 17}{5 \times 5} = \frac{136}{25} \)
(iv) Convert the mixed fraction to an improper fraction: \( 9\frac{2}{3} = \frac{29}{3} \). Then multiply:
\( \frac{2}{3} \text{ of } 9\frac{2}{3} = \frac{2}{3} \times \frac{29}{3} = \frac{2 \times 29}{3 \times 3} = \frac{58}{9} \)
(v) Multiply the two fractions:
\( \frac{1}{5} \text{ of } \frac{3}{5} = \frac{1}{5} \times \frac{3}{5} = \frac{1 \times 3}{5 \times 5} = \frac{3}{25} \)
(vi) Multiply the two fractions:
\( \frac{1}{7} \text{ of } \frac{3}{10} = \frac{1}{7} \times \frac{3}{10} = \frac{1 \times 3}{7 \times 10} = \frac{3}{70} \)
In simple words: When you need to find a fraction "of" another fraction or a mixed number, always change mixed numbers into top-heavy fractions first. Then, multiply the top numbers together and the bottom numbers together.

🎯 Exam Tip: Always convert mixed fractions to improper (top-heavy) fractions before performing any multiplication or division operations. This helps avoid errors.

Question 7. Multiply the following fractions
(i) \( 3\frac{2}{5} \times 4 \)
(ii) \( \frac{3}{2} \times 6\frac{2}{5} \)
(iii) \( 3\frac{4}{7} \times \frac{4}{5} \)
(iv) \( 3\frac{2}{5} \times 4\frac{3}{8} \)
Answer:
(i) First, convert the mixed fraction to an improper fraction: \( 3\frac{2}{5} = \frac{(3 \times 5) + 2}{5} = \frac{17}{5} \). Then multiply by the whole number:
\( \frac{17}{5} \times 4 = \frac{17 \times 4}{5} = \frac{68}{5} \)
(ii) First, convert the mixed fraction to an improper fraction: \( 6\frac{2}{5} = \frac{(6 \times 5) + 2}{5} = \frac{32}{5} \). Then multiply:
\( \frac{3}{2} \times \frac{32}{5} = \frac{3 \times 32}{2 \times 5} = \frac{96}{10} \)
Now, simplify by dividing both by 2:
\( \frac{96 \div 2}{10 \div 2} = \frac{48}{5} \)
(iii) First, convert the mixed fraction to an improper fraction: \( 3\frac{4}{7} = \frac{(3 \times 7) + 4}{7} = \frac{25}{7} \). Then multiply:
\( \frac{25}{7} \times \frac{4}{5} = \frac{25 \times 4}{7 \times 5} = \frac{100}{35} \)
Now, simplify by dividing both by 5:
\( \frac{100 \div 5}{35 \div 5} = \frac{20}{7} \)
(iv) First, convert both mixed fractions to improper fractions:
\( 3\frac{2}{5} = \frac{(3 \times 5) + 2}{5} = \frac{17}{5} \)
\( 4\frac{3}{8} = \frac{(4 \times 8) + 3}{8} = \frac{35}{8} \)
Then, multiply:
\( \frac{17}{5} \times \frac{35}{8} = \frac{17 \times 35}{5 \times 8} = \frac{17 \times 7}{1 \times 8} \)
(We cancelled 35 and 5 by dividing both by 5)
\( \frac{119}{8} \)
In simple words: When multiplying fractions, always change mixed numbers into improper fractions first. Then, multiply the top numbers and the bottom numbers. Remember to simplify your final answer if you can, by dividing both the top and bottom by a common number.

🎯 Exam Tip: Always simplify fractions by cross-cancellation before multiplying to make the numbers smaller and easier to work with, leading to fewer errors in the final simplification step.

Question 8. Which is greater?
(i) \( \frac{2}{7} \) of \( \frac{3}{4} \) or \( \frac{3}{5} \) of \( \frac{5}{8} \)
(ii) \( \frac{1}{2} \) of \( \frac{6}{7} \) or \( \frac{2}{3} \) of \( \frac{3}{7} \)
Answer:
(i) First, calculate the value of each expression:
\( \frac{2}{7} \text{ of } \frac{3}{4} = \frac{2}{7} \times \frac{3}{4} = \frac{2 \times 3}{7 \times 4} = \frac{6}{28} \)
We can simplify this to \( \frac{3}{14} \).
Now, for the second expression:
\( \frac{3}{5} \text{ of } \frac{5}{8} = \frac{3}{5} \times \frac{5}{8} = \frac{3 \times 5}{5 \times 8} = \frac{15}{40} \)
We can simplify this to \( \frac{3}{8} \).
Now we compare \( \frac{3}{14} \) and \( \frac{3}{8} \). When numerators are the same, the fraction with the smaller denominator is greater.
Here, 8 is smaller than 14, so \( \frac{3}{8} \) is greater than \( \frac{3}{14} \).
Therefore, \( \frac{3}{5} \text{ of } \frac{5}{8} \) is greater.
(ii) First, calculate the value of each expression:
\( \frac{1}{2} \text{ of } \frac{6}{7} = \frac{1}{2} \times \frac{6}{7} = \frac{1 \times 6}{2 \times 7} = \frac{6}{14} \)
We can simplify this to \( \frac{3}{7} \).
Now, for the second expression:
\( \frac{2}{3} \text{ of } \frac{3}{7} = \frac{2}{3} \times \frac{3}{7} = \frac{2 \times 3}{3 \times 7} = \frac{6}{21} \)
We can simplify this to \( \frac{2}{7} \).
Now we compare \( \frac{3}{7} \) and \( \frac{2}{7} \). When denominators are the same, the fraction with the larger numerator is greater.
Here, 3 is larger than 2, so \( \frac{3}{7} \) is greater than \( \frac{2}{7} \).
Therefore, \( \frac{1}{2} \text{ of } \frac{6}{7} \) is greater.
In simple words: First, work out the answer for both sides by multiplying the fractions. Then, compare the two final fractions. If their top numbers are the same, the one with the smaller bottom number is bigger. If their bottom numbers are the same, the one with the bigger top number is larger.

🎯 Exam Tip: To compare fractions with different denominators, always find a common denominator or simplify them first. If the numerators are the same, the fraction with the smaller denominator is the greater one.

Question 9. A milk vendor has 15 L of milk. She sold \( \frac{2}{5} \) part to Kanchan's house and \( \frac{1}{5} \) part to Bhavna's house. The remaining milk was sold to a hotel.
(i) How much milk was given to Kanchan's house?
(ii) How much milk was given to Bhavna's house?
(iii) How much milk was sold to the hotel?
Answer:
Total milk = 15 L
(i) Quantity of milk given to Kanchan's house:
\( = 15 \times \frac{2}{5} = \frac{15 \times 2}{5} = \frac{30}{5} = 6 \text{ L} \)
(ii) Quantity of milk given to Bhavna's house:
\( = 15 \times \frac{1}{5} = \frac{15 \times 1}{5} = \frac{15}{5} = 3 \text{ L} \)
(iii) Total milk sold to Kanchan and Bhavna:
\( = 6 \text{ L} + 3 \text{ L} = 9 \text{ L} \)
Quantity of milk sold to the hotel (remaining milk):
\( = \text{Total milk} - \text{Milk sold to Kanchan and Bhavna} \)
\( = 15 \text{ L} - 9 \text{ L} = 6 \text{ L} \)
In simple words: First, find out how much milk each person bought by multiplying the total milk by the fraction they bought. Then, add those amounts to find the total milk sold to individuals. Finally, subtract that total from the initial amount to see how much milk was left for the hotel.

🎯 Exam Tip: When solving word problems involving fractions, always clearly identify the total amount first, then calculate each fractional part, and finally use addition or subtraction to find the remaining quantities.

Question 10. 7 boys were placed each \( \frac{3}{4} \) meter apart from the other for PT demonstration on Independence Day. What is the distance between the first and the last boy?
Answer:
Let the 7 children be A, B, C, D, E, F, and G, standing in a line. The distance between each boy is \( \frac{3}{4} \) meter.
A --- \( \frac{3}{4} \) m --- B --- \( \frac{3}{4} \) m --- C --- \( \frac{3}{4} \) m --- D --- \( \frac{3}{4} \) m --- E --- \( \frac{3}{4} \) m --- F --- \( \frac{3}{4} \) m --- G
To find the distance between the first boy (A) and the last boy (G), we count the number of gaps between them. For 7 boys, there are 6 gaps.
Distance between A and G = Number of gaps \( \times \) Distance of one gap
\( = 6 \times \frac{3}{4} \) m
\( = \frac{18}{4} \) m
\( = \frac{9}{2} \) m
So, the distance between the first and the last boy is \( \frac{9}{2} \) meters, or 4.5 meters. The number of intervals is always one less than the number of items. This is a common pattern in spacing problems.
In simple words: If you have 7 boys in a row, there are 6 spaces between them. Since each space is \( \frac{3}{4} \) meter, you just multiply 6 by \( \frac{3}{4} \) to find the total distance from the first to the last boy.

🎯 Exam Tip: For problems involving items arranged in a line, the number of intervals or gaps is always one less than the number of items. For 'n' items, there are 'n-1' intervals.

Question 11. Rahul works on a painting \( 2\frac{3}{4} \) hours daily. If he takes 8 days to complete this painting then calculate the number of hours he worked.
Answer:
Rahul's daily work time = \( 2\frac{3}{4} \) hours.
First, convert the mixed fraction to an improper fraction:
\( 2\frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4} \) hours.
Number of days taken to complete the painting = 8 days.
Total hours worked = Daily work time \( \times \) Number of days
\( = \frac{11}{4} \times 8 \)
\( = \frac{11 \times 8}{4} \)
\( = \frac{88}{4} \)
\( = 22 \) hours.
So, Rahul worked for a total of 22 hours to complete the painting. This problem involves basic multiplication of fractions to find a total over a period.
In simple words: Rahul works for \( 2\frac{3}{4} \) hours each day. To find out how many hours he worked in 8 days, first change \( 2\frac{3}{4} \) into a simple fraction, then multiply that fraction by 8.

🎯 Exam Tip: Always convert mixed numbers to improper fractions before performing multiplication or division to ensure accurate calculations.

Question 12. Apex travels \( \frac{1}{5} \) km using 1 litre of petrol. What distance will it travel using \( 2\frac{4}{5} \) litres of petrol?
Answer:
Distance travelled with 1 litre of petrol = \( \frac{1}{5} \) km.
Quantity of petrol = \( 2\frac{4}{5} \) litres.
First, convert the mixed fraction to an improper fraction:
\( 2\frac{4}{5} = \frac{(2 \times 5) + 4}{5} = \frac{10 + 4}{5} = \frac{14}{5} \) litres.
Total distance travelled = Distance per litre \( \times \) Total litres of petrol
\( = \frac{1}{5} \times \frac{14}{5} \)
\( = \frac{1 \times 14}{5 \times 5} \)
\( = \frac{14}{25} \) km.
So, Apex will travel \( \frac{14}{25} \) km using \( 2\frac{4}{5} \) litres of petrol. This is a straightforward application of multiplying fractions to scale a given rate. For example, if you travel 2 km per liter, then 3 liters means 2x3=6 km.
In simple words: The car goes \( \frac{1}{5} \) km for every 1 litre of petrol. To find out how far it goes with \( 2\frac{4}{5} \) litres, change \( 2\frac{4}{5} \) into a simple fraction, then multiply that by \( \frac{1}{5} \).

🎯 Exam Tip: For "per unit" problems, multiply the rate by the total units to find the total quantity. Ensure all numbers are in fractional form before multiplying.

Question 13.
(i) Write the number in the box so that \( \frac{3}{4} \times \boxed{\phantom{X}} = \frac{6}{40} \)
(ii) The simplest form of number in the box is.......
Answer:
(i) Let the number in the box be \( x \).
So, the equation is \( \frac{3}{4} \times x = \frac{6}{40} \)
To find \( x \), we divide \( \frac{6}{40} \) by \( \frac{3}{4} \). When dividing by a fraction, we multiply by its reciprocal:
\( x = \frac{6}{40} \div \frac{3}{4} \)
\( x = \frac{6}{40} \times \frac{4}{3} \)
Now, we multiply the fractions:
\( x = \frac{6 \times 4}{40 \times 3} = \frac{24}{120} \)
To simplify, we can divide both the numerator and the denominator by their greatest common divisor, which is 24:
\( x = \frac{24 \div 24}{120 \div 24} = \frac{1}{5} \)
So, the number in the box is \( \frac{1}{5} \). This calculation demonstrates how to solve for an unknown factor in a fractional multiplication problem.
(ii) The simplest form of the number in the box is \( \frac{1}{5} \).
In simple words: To find the missing number in the box, you need to do the opposite of multiplying, which is dividing. You divide the answer fraction by the first fraction. Remember to flip the second fraction and multiply to divide fractions. Then, make sure your answer is in its simplest form.

🎯 Exam Tip: When solving for an unknown factor in a multiplication of fractions, divide the product by the known factor. Always simplify the final fraction to its lowest terms.

Question 14.
(i) Write the number in the box so that \( \frac{5}{8} \times \boxed{\phantom{X}} = \frac{10}{24} \)
(ii) The simplest form of number In the box of (i) is ......
Answer:
(i) Let the number in the box be \( x \).
So, the equation is \( \frac{5}{8} \times x = \frac{10}{24} \)
To find \( x \), we divide \( \frac{10}{24} \) by \( \frac{5}{8} \). When dividing by a fraction, we multiply by its reciprocal:
\( x = \frac{10}{24} \div \frac{5}{8} \)
\( x = \frac{10}{24} \times \frac{8}{5} \)
Now, we multiply the fractions and simplify by cross-cancellation:
\( x = \frac{10 \div 5}{24 \div 8} \times \frac{8 \div 8}{5 \div 5} = \frac{2}{3} \times \frac{1}{1} = \frac{2}{3} \)
So, the number in the box is \( \frac{2}{3} \). This problem involves finding a missing fraction, a common algebra skill applied to fractions.
(ii) The simplest form of the number in the box is \( \frac{2}{3} \).
In simple words: To figure out the hidden number, divide the answer fraction by the first fraction. When you divide, flip the second fraction upside down and then multiply. Always reduce your final answer to the smallest possible fraction.

🎯 Exam Tip: Always remember that division by a fraction is the same as multiplication by its reciprocal. Cross-cancellation can save time and reduce errors in complex fraction calculations.