RBSE Solutions Class 7 Maths Chapter 2 Fractions and Decimal Numbers More Ques

 

Question. (Page 16)

(i) You know that area of a rectangle = length x breadth. But if length and breadth is given in fraction, then how will you find the area of rectangle?
(ii) Are you agree with that to find the area of a rectangle, we must know how to multiply two or more than fractional numbers?
Answer:
(i) If the length and breadth of a rectangle are given as fractions, you multiply the numerators (top numbers) together and the denominators (bottom numbers) together. This gives you a new fraction, which is the area of the rectangle. This method is consistent for all types of numbers.
For example:
Length \( = \frac{4}{5} \) unit
Breadth \( = \frac{3}{7} \) unit
Area of rectangle \( = \text{length} \times \text{breadth} \)
\( = \frac{4}{5} \times \frac{3}{7} \)
\( = \frac{4 \times 3}{5 \times 7} \)
\( = \frac{12}{35} \) sq unit.
(ii) Yes, we agree. To find the area of a rectangle when its sides are fractional numbers, we absolutely need to know how to multiply fractions. Understanding this fundamental operation is key to solving such problems.
In simple words: (i) To find the area of a rectangle when sides are fractions, multiply the top numbers together and the bottom numbers together. (ii) Yes, you need to know how to multiply fractions to find the area of a rectangle with fractional sides.

🎯 Exam Tip: Remember that "area" is always measured in square units. When multiplying fractions, ensure you multiply numerator by numerator and denominator by denominator.

 

Question. (Page 19) Complete the table:

Answer: Yes, we agree that the value of the product of two proper fractions is always less than the value of each individual fraction. This happens because a proper fraction is always less than 1, so multiplying it by another proper fraction makes it even smaller.
In simple words: When you multiply two fractions that are each smaller than 1, their answer will always be smaller than both of the original fractions.

🎯 Exam Tip: To compare fractions, find a common denominator. For example, to compare \( \frac{2}{15} \) and \( \frac{1}{3} \), convert \( \frac{1}{3} \) to \( \frac{5}{15} \), then it's clear that \( \frac{2}{15} < \frac{5}{15} \).

 

Question. (Page 20) Complete the table:

Answer: On completing the table, we can see that the product of two improper fractions is greater than each individual fraction. This is because an improper fraction is always greater than 1, so multiplying it by another improper fraction makes the product even larger. This is the opposite of multiplying proper fractions.
In simple words: When you multiply two fractions that are both bigger than 1, their answer will always be bigger than both of the original fractions.

🎯 Exam Tip: An improper fraction has a numerator that is greater than or equal to its denominator. Always reduce fractions to their simplest form if asked, but for comparisons, finding a common denominator is often enough.

 

Question. (Page 24)
Interchange the numerator and denominator with each of the function \( \frac{1}{5} \) and \( \frac{2}{3} \).
(ii) Fill in the blanks:
(a) \( \frac{1}{5} \times \text{_______} = \text{.......} \)
(b) \( \frac{2}{3} \times \frac{3}{2} = \text{.......} \)
(c) \( 2 + 2 = 2 \times \text{.......} = 2 \times (\text{Inverse of } \frac{3}{4}) \)
(d) \( 2 \frac{1}{3} + \frac{5}{4} = \text{.......} = \text{.......} \times (\text{Inverse of } \frac{5}{4}) \)
Answer:
(i) After changing the numerator and denominator of \( \frac{1}{5} \), the required fraction is \( \frac{5}{1} \).
After changing the numerator and denominator of \( \frac{2}{3} \), the required fraction is \( \frac{3}{2} \).
(ii) Fill in the blanks:
(a) \( \frac{1}{5} \times \frac{5}{1} = 1 \)
(b) \( \frac{2}{3} \times \frac{3}{2} = 1 \)
(c) \( 2 \div \frac{3}{4} = 2 \times \frac{4}{3} = 2 \times (\text{Inverse of } \frac{3}{4}) \)
(d) \( 2 \frac{1}{3} \div \frac{5}{4} = \frac{7}{3} \div \frac{5}{4} = \frac{7}{3} \times (\text{Inverse of } \frac{5}{4}) \)
In simple words: (i) When you swap the top and bottom numbers of a fraction, you get its inverse. (ii) When dividing by a fraction, you can change it to multiplying by its inverse (the flipped fraction).

🎯 Exam Tip: The term "inverse" for a fraction means its reciprocal. The product of a fraction and its reciprocal is always 1.

 

Question. (Page 25)
How can you read those number
(i) decimal two
(ii) 2.04 = Two decimal zero four
Answer:
(i) If "decimal two" refers to the number \( 0.2 \), then we read it as "zero point two".
(ii) \( 2.04 \) is read as "Two point zero four". The digits after the decimal point are read individually, not as a whole number. This is important for clarity in scientific and mathematical contexts.
In simple words: When you say numbers after a decimal point, you say each digit alone. For example, 2.04 is "two point zero four", not "two point four".

🎯 Exam Tip: Always read digits after the decimal point one by one to avoid confusion, especially when there are zeros. For example, 0.25 is "zero point two five", not "zero point twenty-five".

 

How do we write \( \frac{1}{8} \) as a decimal? To convert \( \frac{1}{8} \) to a decimal, divide the numerator by the denominator.
\( \frac{1}{8} = 0.125 \). This is done by dividing 1 by 8.

 

Question. (Page 26)
See the following table and fill up the blanks:

(ii) We can also write these numbers in their extended form
\( 421.258 = 4 \times 100 + 2 \times 10 + 1 \times 1 + 2 \times \frac{1}{10} + 5 \times \frac{1}{100} + 8 \times \frac{1}{1000} \)
Similarly write the remaining numbers from above table.
(iii) Fill in the blanks : \( 120 \text{ m} = \text{.......} \text{ km} \).
Answer:
(i) The completed table is shown below, with each digit placed in its correct column according to its place value:

(ii) The extended forms of the numbers are:
\( 608.507 = 6 \times 100 + 0 \times 10 + 8 \times 1 + 5 \times \frac{1}{10} + 0 \times \frac{1}{100} + 7 \times \frac{1}{1000} \)
\( 303.210 = 3 \times 100 + 0 \times 10 + 3 \times 1 + 2 \times \frac{1}{10} + 1 \times \frac{1}{100} + 0 \times \frac{1}{1000} \)
\( 876.170 = 8 \times 100 + 7 \times 10 + 6 \times 1 + 1 \times \frac{1}{10} + 7 \times \frac{1}{100} + 0 \times \frac{1}{1000} \)
\( 784.035 = 7 \times 100 + 8 \times 10 + 4 \times 1 + 0 \times \frac{1}{10} + 3 \times \frac{1}{100} + 5 \times \frac{1}{1000} \)
\( 012.345 = 0 \times 100 + 1 \times 10 + 2 \times 1 + 3 \times \frac{1}{10} + 4 \times \frac{1}{100} + 5 \times \frac{1}{1000} \)
(iii) To convert meters to kilometers, we divide by 1000, since 1 km = 1000 m.
\( 120 \text{ m} = \frac{120}{1000} \text{ km} = 0.12 \text{ km} \).
In simple words: (i) The table shows how numbers are made up of hundreds, tens, ones, tenths, hundredths, and thousandths. (ii) Writing a number in extended form means showing the value of each digit by multiplying it by its place value. (iii) To change meters to kilometers, divide the number of meters by 1000.

🎯 Exam Tip: When writing numbers in extended form, remember to include terms for zero digits as \( 0 \times \text{place value} \), or simply omit them for brevity after understanding the concept. Be careful with place values after the decimal point.

 

Question. (Page 30)
Put decimal in -
\( 1.52 \times 1000 = \text{.......} \)
Answer:
When you multiply a decimal number by 1000, you move the decimal point three places to the right. This is because 1000 has three zeros.
\( 1.52 \times 1000 = 1520.00 \)
In simple words: To multiply a number with a decimal by 1000, just move the decimal point three spots to the right.

🎯 Exam Tip: Multiplying by powers of 10 (10, 100, 1000, etc.) involves shifting the decimal point to the right by the number of zeros in the power of 10.

 

Question. (Page 34)
(i) There are 8 black and 7 white strips in a zebra crossing. So tell what part of total strips is the number of white strips?
(ii) On a day 100 people crossed the road by zebra crossing out of which 20 are men, 30 women, 10 children and 40 students. Show all these data's In decimal.
Answer:
(i) First, find the total number of strips on the zebra crossing.
Total strips \( = 8 \text{ (black)} + 7 \text{ (white)} = 15 \text{ strips} \).
Now, find the part of white strips compared to the total strips.
Part of white strips \( = \frac{\text{Number of white strips}}{\text{Total strips}} = \frac{7}{15} \).
(ii) The total number of people who crossed the road is 100. We need to express each group as a decimal part of the total. A decimal represents a fraction where the denominator is a power of ten.
For men: \( \frac{20}{100} = 0.2 \)
For women: \( \frac{30}{100} = 0.3 \)
For children: \( \frac{10}{100} = 0.1 \)
For students: \( \frac{40}{100} = 0.4 \)
Here is the data in a table form:

In simple words: (i) There are 15 strips in total, and 7 of them are white, so the white part is \( \frac{7}{15} \). (ii) For every group of people, you divide their number by the total (100) to get a decimal. So, 20 men out of 100 is 0.2, 30 women is 0.3, and so on.

🎯 Exam Tip: When expressing a "part" or "portion", always write it as a fraction (part/whole) and simplify it if possible. To convert a fraction to a decimal, divide the numerator by the denominator.

 

Question 1. (Page 14) Find five equivalent fractions of \( \frac{4}{7} \).
Answer:
To find equivalent fractions, you multiply both the numerator (top number) and the denominator (bottom number) by the same non-zero whole number. This keeps the value of the fraction the same, just in a different form. Here are five equivalent fractions for \( \frac{4}{7} \):
\( \frac{4}{7} = \frac{4 \times 2}{7 \times 2} = \frac{8}{14} \)
\( \frac{4}{7} = \frac{4 \times 3}{7 \times 3} = \frac{12}{21} \)
\( \frac{4}{7} = \frac{4 \times 4}{7 \times 4} = \frac{16}{28} \)
\( \frac{4}{7} = \frac{4 \times 5}{7 \times 5} = \frac{20}{35} \)
\( \frac{4}{7} = \frac{4 \times 6}{7 \times 6} = \frac{24}{42} \)
Therefore, the equivalent fractions of \( \frac{4}{7} \) are \( \frac{8}{14}, \frac{12}{21}, \frac{16}{28}, \frac{20}{35}, \text{ and } \frac{24}{42} \).
In simple words: To get fractions that are the same as \( \frac{4}{7} \), multiply the top number (4) and the bottom number (7) by the same counting number. This gives new fractions that have the same value.

🎯 Exam Tip: You can multiply by any whole number (except zero) to find equivalent fractions. The goal is to show that the new fraction still represents the same proportion as the original one.

 

Question. (Page 18) Solve
(i) \( 3 \times \frac{8}{7} \)
(ii) \( \frac{9}{7} \times 6 \)
(iii) \( 4 \times \frac{7}{5} \)
(iv) \( 4 \times \frac{4}{9} \)
Answer:
To multiply a whole number by a fraction, you can think of the whole number as a fraction with a denominator of 1. Then, multiply the numerators together and the denominators together.
(i) \( 3 \times \frac{8}{7} = \frac{3}{1} \times \frac{8}{7} = \frac{3 \times 8}{1 \times 7} = \frac{24}{7} \)
(ii) \( \frac{9}{7} \times 6 = \frac{9}{7} \times \frac{6}{1} = \frac{9 \times 6}{7 \times 1} = \frac{54}{7} \)
(iii) \( 4 \times \frac{7}{5} = \frac{4}{1} \times \frac{7}{5} = \frac{4 \times 7}{1 \times 5} = \frac{28}{5} \)
(iv) \( 4 \times \frac{4}{9} = \frac{4}{1} \times \frac{4}{9} = \frac{4 \times 4}{1 \times 9} = \frac{16}{9} \)
In simple words: When you multiply a whole number by a fraction, you multiply the whole number only by the top number of the fraction. The bottom number stays the same.

🎯 Exam Tip: Always remember that a whole number can be written as a fraction by putting it over 1 (e.g., \( 3 = \frac{3}{1} \)). This helps in visualizing the multiplication of fractions.

 

Question. (Page 18) Solve
(i) \( 5 \times \frac{1}{2} = \text{?} \)
(ii) \( 1\frac{4}{9} \times 6 = \text{?} \)
Answer:
(i) To solve \( 5 \times \frac{1}{2} \), multiply the whole number 5 by the numerator 1, and keep the denominator 2.
\( 5 \times \frac{1}{2} = \frac{5 \times 1}{2} = \frac{5}{2} \)
(ii) First, convert the mixed fraction \( 1\frac{4}{9} \) into an improper fraction. To do this, multiply the whole number (1) by the denominator (9) and add the numerator (4), keeping the original denominator.
\( 1\frac{4}{9} = \frac{(1 \times 9) + 4}{9} = \frac{9 + 4}{9} = \frac{13}{9} \)
Now, multiply this improper fraction by 6:
\( \frac{13}{9} \times 6 = \frac{13}{9} \times \frac{6}{1} = \frac{13 \times 6}{9 \times 1} = \frac{78}{9} \)
This fraction can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 3.
\( \frac{78}{9} = \frac{78 \div 3}{9 \div 3} = \frac{26}{3} \)
In simple words: (i) \( 5 \times \frac{1}{2} \) means five halves, which is \( \frac{5}{2} \). (ii) First, change the mixed number \( 1\frac{4}{9} \) into a simple fraction \( \frac{13}{9} \). Then multiply \( \frac{13}{9} \) by 6 to get \( \frac{78}{9} \), which can be made simpler to \( \frac{26}{3} \).

🎯 Exam Tip: Always convert mixed fractions to improper fractions before performing multiplication or division. Also, simplify your final answer to its lowest terms if possible.

 

Question. (Page 18) Solve
(i) \( \frac{1}{2} \) of 5
(ii) \( \frac{1}{4} \) of 16
(iii) \( \frac{2}{5} \) of 25
Answer:
The word "of" in mathematics often means to multiply. So, to find a fraction "of" a number, you multiply the fraction by that number.
(i) \( \frac{1}{2} \text{ of } 5 = \frac{1}{2} \times 5 = \frac{1 \times 5}{2} = \frac{5}{2} \)
(ii) \( \frac{1}{4} \text{ of } 16 = \frac{1}{4} \times 16 = \frac{1 \times 16}{4} = \frac{16}{4} = 4 \)
(iii) \( \frac{2}{5} \text{ of } 25 = \frac{2}{5} \times 25 = \frac{2 \times 25}{5} = \frac{50}{5} = 10 \)
In simple words: "Of" means to multiply. So, for example, \( \frac{1}{2} \) of 5 means \( \frac{1}{2} \times 5 \).

🎯 Exam Tip: When calculating "fraction of a number," always multiply the fraction by the number. You can simplify by canceling common factors before multiplying for easier calculation.

 

Question. (Page 10) Solve
(i) \( \frac{1}{3} \times \frac{1}{7} \)
(ii) \( \frac{3}{2} \times \frac{4}{7} \)
(iii) \( \frac{1}{7} \times \frac{1}{5} \)
(iv) \( \frac{3}{5} \times \frac{2}{3} \)
Answer:
When multiplying fractions, multiply the numerators (top numbers) together and the denominators (bottom numbers) together.
(i) \( \frac{1}{3} \times \frac{1}{7} = \frac{1 \times 1}{3 \times 7} = \frac{1}{21} \)
(ii) \( \frac{3}{2} \times \frac{4}{7} = \frac{3 \times 4}{2 \times 7} = \frac{12}{14} \)
This fraction can be simplified by dividing both the numerator and denominator by 2.
\( \frac{12}{14} = \frac{12 \div 2}{14 \div 2} = \frac{6}{7} \)
(iii) \( \frac{1}{7} \times \frac{1}{5} = \frac{1 \times 1}{7 \times 5} = \frac{1}{35} \)
(iv) \( \frac{3}{5} \times \frac{2}{3} = \frac{3 \times 2}{5 \times 3} = \frac{6}{15} \)
This fraction can be simplified by dividing both the numerator and denominator by 3.
\( \frac{6}{15} = \frac{6 \div 3}{15 \div 3} = \frac{2}{5} \)
In simple words: To multiply fractions, just multiply the top numbers together and the bottom numbers together. Then, make the answer simpler if you can.

🎯 Exam Tip: Always check if the resulting fraction can be simplified by finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it.

 

Question. (Page 20) Find the product of one proper and one improper fraction and prepare the table showing the result.

Answer:
When you multiply a proper fraction (less than 1) by an improper fraction (greater than 1), the product will always be greater than the proper fraction but smaller than the improper fraction. This makes sense because you are essentially scaling the proper fraction up and the improper fraction down towards a middle value. For example:
Let's take proper fraction \( \frac{1}{5} \) and improper fraction \( \frac{6}{5} \).
Product \( = \frac{1}{5} \times \frac{6}{5} = \frac{6}{25} \).
Comparing the product with the proper fraction: \( \frac{6}{25} \) vs \( \frac{1}{5} \). Convert \( \frac{1}{5} \) to \( \frac{5}{25} \). So, \( \frac{6}{25} > \frac{5}{25} \). The product is greater than the proper fraction.
Comparing the product with the improper fraction: \( \frac{6}{25} \) vs \( \frac{6}{5} \). Convert \( \frac{6}{5} \) to \( \frac{30}{25} \). So, \( \frac{6}{25} < \frac{30}{25} \). The product is smaller than the improper fraction.
In simple words: When you multiply a small fraction (less than 1) by a big fraction (more than 1), the answer will be bigger than the small fraction but smaller than the big fraction.

🎯 Exam Tip: To compare a product with the original fractions, it's helpful to convert all fractions to a common denominator or to decimal form. This makes it easier to see which value is larger or smaller.

 

Question. (Page 24) Solve
(i) \( 5 \div \frac{2}{3} \)
(ii) \( 7 \div \frac{3}{4} \)
(iii) \( 6 \div \frac{1}{5} \)
Answer:
When dividing by a fraction, the rule is to "invert and multiply". This means you flip the second fraction (find its reciprocal) and then multiply it by the first number or fraction.
(i) \( 5 \div \frac{2}{3} = 5 \times \frac{3}{2} = \frac{5 \times 3}{2} = \frac{15}{2} \)
(ii) \( 7 \div \frac{3}{4} = 7 \times \frac{4}{3} = \frac{7 \times 4}{3} = \frac{28}{3} \)
(iii) \( 6 \div \frac{1}{5} = 6 \times \frac{5}{1} = \frac{6 \times 5}{1} = \frac{30}{1} = 30 \)
In simple words: When you divide by a fraction, you flip the fraction over and then multiply. For example, dividing by \( \frac{2}{3} \) is the same as multiplying by \( \frac{3}{2} \).

🎯 Exam Tip: Remember to always flip only the second fraction (the divisor), not the first one. Ensure you multiply the numerators and denominators correctly after flipping.

 

Question. (Page 24) Fill in the blanks
(i) \( 2\frac{3}{5} + 2 = \frac{13}{5} + 2 = \text{.......} \)
(ii) \( \frac{8}{3} + 5 = \text{.......} = \text{.......} \)
(iii) \( 2\frac{2}{3} - 3 = \text{.......} = \text{.......} \)
Answer:
(i) To add a whole number to a mixed fraction (or an improper fraction), you can convert the whole number into a fraction with the same denominator, or simply add the whole number part and then the fractional part. In this case, convert the whole number 2 to a fraction with denominator 5, which is \( \frac{10}{5} \).
\( 2\frac{3}{5} + 2 = \frac{13}{5} + 2 = \frac{13}{5} + \frac{10}{5} = \frac{13+10}{5} = \frac{23}{5} \)
(ii) To add \( \frac{8}{3} + 5 \), convert the whole number 5 to a fraction with denominator 3, which is \( \frac{15}{3} \).
\( \frac{8}{3} + 5 = \frac{8}{3} + \frac{15}{3} = \frac{8+15}{3} = \frac{23}{3} \)
(iii) To subtract 3 from \( 2\frac{2}{3} \), convert the mixed fraction to an improper fraction first. \( 2\frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{6+2}{3} = \frac{8}{3} \).
Now, convert the whole number 3 to a fraction with denominator 3, which is \( \frac{9}{3} \).
\( 2\frac{2}{3} - 3 = \frac{8}{3} - \frac{9}{3} = \frac{8-9}{3} = -\frac{1}{3} \)
In simple words: (i) To add 2 to \( \frac{13}{5} \), you can think of 2 as \( \frac{10}{5} \), so you add \( \frac{13}{5} + \frac{10}{5} \) to get \( \frac{23}{5} \). (ii) \( \frac{8}{3} + 5 \) becomes \( \frac{8}{3} + \frac{15}{3} \), which gives \( \frac{23}{3} \). (iii) For \( 2\frac{2}{3} - 3 \), change \( 2\frac{2}{3} \) to \( \frac{8}{3} \). Then subtract \( \frac{9}{3} \) (which is 3) from it, resulting in \( -\frac{1}{3} \).

🎯 Exam Tip: When adding or subtracting fractions and whole numbers, always make sure they have a common denominator. Convert mixed numbers to improper fractions first to simplify calculations.

 

Question. (Page 25) Solve
(i) \( \frac{3}{5} + \frac{1}{2} \)
(ii) \( 2\frac{1}{2} + \frac{3}{5} \)
(iii) \( 5\frac{1}{6} + \frac{9}{2} \)
Answer:
To add fractions, they must have a common denominator. Find the Least Common Multiple (LCM) of the denominators to get the common denominator.
(i) Denominators are 5 and 2. LCM(5, 2) = 10.
\( \frac{3}{5} + \frac{1}{2} = \frac{3 \times 2}{5 \times 2} + \frac{1 \times 5}{2 \times 5} = \frac{6}{10} + \frac{5}{10} = \frac{6+5}{10} = \frac{11}{10} \)
(ii) First, convert the mixed fraction \( 2\frac{1}{2} \) to an improper fraction: \( 2\frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{5}{2} \).
Now, add \( \frac{5}{2} + \frac{3}{5} \). Denominators are 2 and 5. LCM(2, 5) = 10.
\( \frac{5}{2} + \frac{3}{5} = \frac{5 \times 5}{2 \times 5} + \frac{3 \times 2}{5 \times 2} = \frac{25}{10} + \frac{6}{10} = \frac{25+6}{10} = \frac{31}{10} \)
(iii) First, convert the mixed fraction \( 5\frac{1}{6} \) to an improper fraction: \( 5\frac{1}{6} = \frac{(5 \times 6) + 1}{6} = \frac{31}{6} \).
Now, add \( \frac{31}{6} + \frac{9}{2} \). Denominators are 6 and 2. LCM(6, 2) = 6.
\( \frac{31}{6} + \frac{9}{2} = \frac{31}{6} + \frac{9 \times 3}{2 \times 3} = \frac{31}{6} + \frac{27}{6} = \frac{31+27}{6} = \frac{58}{6} \)
This fraction can be simplified by dividing both the numerator and denominator by 2.
\( \frac{58}{6} = \frac{58 \div 2}{6 \div 2} = \frac{29}{3} \)
In simple words: To add fractions, make sure the bottom numbers are the same first. If they're not, find a common bottom number. If you have mixed numbers, turn them into simple fractions before adding.

🎯 Exam Tip: Always find the least common multiple (LCM) of the denominators to make calculations easier and to get the smallest possible common denominator.

 

Question. (Page 26) Which number is smaller?
(i) 35.37 and 35.07
(ii) 262.327 and 262.372
Answer:
To compare decimal numbers, start by comparing the digits from left to right, just like with whole numbers. The first position where the digits differ tells you which number is larger or smaller.
(i) Comparing 35.37 and 35.07:
The whole number part (35) is the same for both.
Look at the tenths place: 3 in 35.37 and 0 in 35.07.
Since 0 is smaller than 3, 35.07 is the smaller number.
(ii) Comparing 262.327 and 262.372:
The whole number part (262) is the same for both.
The tenths place (3) is the same for both.
Look at the hundredths place: 2 in 262.327 and 7 in 262.372.
Since 2 is smaller than 7, 262.327 is the smaller number.
In simple words: To find the smaller decimal, look at the numbers from left to right. The first time a digit is smaller in one number than the other, that whole number is smaller.

🎯 Exam Tip: Line up the decimal points and compare digits column by column, starting from the leftmost digit, to accurately identify the larger or smaller decimal number.

 

Question. (Page 29) Find the value of
(i) \( 2\frac{2}{5} \)
Answer:
(i) To find the value of the mixed fraction \( 2\frac{2}{5} \), you can convert it to an improper fraction or a decimal. Let's convert it to an improper fraction first.
To convert \( 2\frac{2}{5} \) to an improper fraction: Multiply the whole number (2) by the denominator (5) and add the numerator (2). Keep the same denominator (5).
\( 2\frac{2}{5} = \frac{(2 \times 5) + 2}{5} = \frac{10 + 2}{5} = \frac{12}{5} \)
To convert this to a decimal, divide the numerator by the denominator:
\( \frac{12}{5} = 12 \div 5 = 2.4 \).
In simple words: The mixed number \( 2\frac{2}{5} \) can be written as the improper fraction \( \frac{12}{5} \), which is the same as the decimal number 2.4.

🎯 Exam Tip: Mixed numbers are a combination of a whole number and a proper fraction. Always be ready to convert them to improper fractions or decimals based on the context of the problem.

 

Question. (Page 32) Divide the given decimal numbers by 10, 100 and 1000?
(i) 132.4
(ii) 1.03
(iii) 40.033
(iv) 4.321
Answer:
When dividing a decimal number by 10, 100, or 1000, you shift the decimal point to the left by the number of zeros in the divisor. For example, for 10 (one zero), move it one place left; for 100 (two zeros), move it two places left; for 1000 (three zeros), move it three places left.
(i) For 132.4:
\( 132.4 \div 10 = 13.24 \)
\( 132.4 \div 100 = 1.324 \)
\( 132.4 \div 1000 = 0.1324 \)
(ii) For 1.03:
\( 1.03 \div 10 = 0.103 \)
\( 1.03 \div 100 = 0.0103 \)
\( 1.03 \div 1000 = 0.00103 \)
(iii) For 40.033:
\( 40.033 \div 10 = 4.0033 \)
\( 40.033 \div 100 = 0.40033 \)
\( 40.033 \div 1000 = 0.040033 \)
(iv) For 4.321:
\( 4.321 \div 10 = 0.4321 \)
\( 4.321 \div 100 = 0.04321 \)
\( 4.321 \div 1000 = 0.004321 \)
In simple words: To divide a decimal number by 10, 100, or 1000, just move the decimal point to the left. Move it one spot for 10, two spots for 100, and three spots for 1000.

🎯 Exam Tip: If there are not enough digits to the left of the decimal point, add zeros as placeholders before moving the decimal point (e.g., \( 1.03 \div 1000 \rightarrow 0001.03 \div 1000 \rightarrow 0.00103 \)).

 

Question. (iv) Divide 4.321 by 10, 100 and 1000.
Answer:
To divide 4.321 by 10:
\( 4.321 \div 10 = 4.321 \times \frac{1}{10} = \frac{4321}{1000} \times \frac{1}{10} = \frac{4321}{10000} = 0.4321 \)

To divide 4.321 by 100:
\( 4.321 \div 100 = 4.321 \times \frac{1}{100} = \frac{4321}{1000} \times \frac{1}{100} = \frac{4321}{100000} = 0.04321 \)

To divide 4.321 by 1000:
\( 4.321 \div 1000 = 4.321 \times \frac{1}{1000} = \frac{4321}{1000} \times \frac{1}{1000} = \frac{4321}{1000000} = 0.004321 \)
In simple words: When you divide a decimal number by 10, 100, or 1000, the decimal point moves to the left. The number of places it moves is equal to how many zeros are in the divisor (1, 2, or 3 places respectively). Moving the decimal point to the left makes the number smaller, which is what happens when you divide.

🎯 Exam Tip: Remember that dividing by powers of 10 always shifts the decimal point to the left. The number of places moved is always equal to the number of zeros in the power of 10 (e.g., one zero for 10, two for 100).

 

Question. Solve the following division problems:
(i) 6 ÷ 1.2
(ii) 9 ÷ 4.5
(iii) 48 ÷ 0.8
Answer:
(i) To solve \( 6 \div 1.2 \):
\( 6 \div 1.2 = \frac{6}{1.2} \)
To remove the decimal, multiply the numerator and denominator by 10:
\( \frac{6 \times 10}{1.2 \times 10} = \frac{60}{12} = 5 \)

(ii) To solve \( 9 \div 4.5 \):
\( 9 \div 4.5 = \frac{9}{4.5} \)
To remove the decimal, multiply the numerator and denominator by 10:
\( \frac{9 \times 10}{4.5 \times 10} = \frac{90}{45} = 2 \)

(iii) To solve \( 48 \div 0.8 \):
\( 48 \div 0.8 = \frac{48}{0.8} \)
To remove the decimal, multiply the numerator and denominator by 10:
\( \frac{48 \times 10}{0.8 \times 10} = \frac{480}{8} = 60 \)
In simple words: When you need to divide by a decimal number, first make that decimal number a whole number. You can do this by multiplying both numbers in the division by 10 (or 100, etc.) until the bottom number has no decimal. After that, divide them as you would with whole numbers.

🎯 Exam Tip: Always make the divisor (the number you are dividing by) a whole number before you start dividing. This makes the calculation much simpler and reduces the chance of making mistakes.

 

Question. Solve the following problems:
(i) 7.75 ÷ 0.25
(ii) 5.6 × 1.4
(iii) 42.8 × 0.02
Answer:
(i) To solve \( 7.75 \div 0.25 \):
To make the divisor a whole number, multiply both numbers by 100:
\( 7.75 \div 0.25 = \frac{7.75 \times 100}{0.25 \times 100} = \frac{775}{25} = 31 \)

(ii) To solve \( 5.6 \times 1.4 \):
First, multiply the numbers as if there were no decimals: \( 56 \times 14 = 784 \).
Count the total decimal places in the original numbers: 5.6 has one, and 1.4 has one. So, the total is \( 1 + 1 = 2 \) decimal places.
Place the decimal point in the product so that it has two decimal places: \( 7.84 \).

(iii) To solve \( 42.8 \times 0.02 \):
First, multiply the numbers as if there were no decimals: \( 428 \times 2 = 856 \).
Count the total decimal places in the original numbers: 42.8 has one, and 0.02 has two. So, the total is \( 1 + 2 = 3 \) decimal places.
Place the decimal point in the product so that it has three decimal places: \( 0.856 \).
In simple words: For division with decimals, shift the decimal point in both numbers so the number you're dividing by becomes a whole number. For multiplication with decimals, ignore the decimal points while multiplying, then put the decimal point back in the answer by counting how many total decimal places were in the numbers you started with. These steps help keep the calculations clear and accurate.

🎯 Exam Tip: When multiplying decimals, the number of decimal places in your final answer must always be the sum of the decimal places in the numbers you multiplied. For division, ensure the divisor is a whole number first.