ML Aggarwal Class 6 Maths Solutions Chapter 07 Decimals

 

Exercise 7.1

 

Question 1. Write each of the following decimal numbers in words:
(i) 30.5
(ii) 0.03
(iii) 108.56
(iv) 47.20
(v) 5.008
(vi) 26.039
Answer: (i) 30.5 is expressed as thirty point five. (ii) 0.03 is expressed as zero point zero three. (iii) 108.56 is expressed as one hundred eight point five six. (iv) 47.20 is expressed as forty seven point two zero. (v) 5.008 is expressed as five point zero zero eight. (vi) 26.039 is expressed as twenty six point zero three nine.
In simple words: When saying a decimal aloud, speak the whole number part first, then say "point," and then speak each digit after the decimal separately.

Exam Tip: Always pronounce each digit individually after the decimal point - do not try to read the fractional part as a single whole number.

 

Question 2. Write each of the following decimal numbers in the place value table:
(i) 4.2
(ii) 0.3
(iii) 205.9
(iv) 0.29
(v) 2.08
(vi) 7200.812
(vii) 38.007
Answer: Place each digit in its correct position according to the powers of 10, with the decimal point separating the whole number part from the fractional part. For example, in 205.9, the digit 2 goes in the hundreds place, 0 in the tens place, 5 in the ones place, and 9 in the tenths place. Similarly, 7200.812 has 7 in the thousands, 2 in the hundreds, 0 in both the tens and ones places, 8 in the tenths, 1 in the hundredths, and 2 in the thousandths place.

In simple words: Each position to the left of the decimal point gets bigger (ones, tens, hundreds, thousands), and each position to the right gets smaller (tenths, hundredths, thousandths). Place each digit in its correct box.

Exam Tip: Always align the decimal point vertically when filling a place value table - this helps ensure each digit lands in the right column.

 

Question 3. Write the following decimal numbers in the expanded form:
(i) 123.7
(ii) 43.06
(iii) 509.306
Answer:
(i) For 123.7: Break down as \( 1 \times 100 + 2 \times 10 + 3 \times 1 + 7 \times \frac{1}{10} = 100 + 20 + 3 + \frac{7}{10} \)
(ii) For 43.06: Break down as \( 4 \times 10 + 3 \times 1 + 0 \times \frac{1}{10} + 6 \times \frac{1}{100} = 40 + 3 + \frac{6}{100} \)
(iii) For 509.306: Break down as \( 5 \times 100 + 0 \times 10 + 9 \times 1 + 3 \times \frac{1}{10} + 0 \times \frac{1}{100} + 6 \times \frac{1}{1000} = 500 + 9 + \frac{3}{10} + \frac{6}{1000} \)
In simple words: Expanded form shows every digit multiplied by its place value. Add all these products together, and you get back the original number.

Exam Tip: Write each digit and its place value separately - this shows you understand what each position means. Do not skip the zeros in place value breakdown.

 

Question 4. Write each of the following as a decimal number:
(i) \( 200 + 60 + 5 + \frac{3}{10} \)
(ii) \( 50 + \frac{1}{10} + \frac{6}{100} \)
(iii) \( 70 + 6 + \frac{7}{10} + \frac{9}{1000} \)
(iv) \( 600 + 7 + \frac{3}{100} + \frac{6}{1000} \)
Answer:
(i) \( 200 + 60 + 5 + \frac{3}{10} = 265.3 \)
(ii) \( 50 + \frac{1}{10} + \frac{6}{100} = 50.16 \)
(iii) \( 70 + 6 + \frac{7}{10} + \frac{9}{1000} = 76.709 \)
(iv) \( 600 + 7 + \frac{3}{100} + \frac{6}{1000} = 607.036 \)
In simple words: Add up all the whole number parts first, then place each fraction digit in the correct spot after the decimal point.

Exam Tip: Line up the decimal points and place digits carefully - tenths go one place after the decimal, hundredths go two places, and thousandths go three places.

 

Question 5. Write each of the following as decimals:
(i) Two ones and five tenths
(ii) Two tens and nine tenths
(iii) Six hundred point eight
(iv) Two hundred five and five hundredths
(v) Seven and fifteen thousandths
Answer:
(i) Two ones and five tenths = \( 2 + \frac{5}{10} = 2.5 \)
(ii) Two tens and nine tenths = \( 20 + \frac{9}{10} = 20.9 \)
(iii) Six hundred point eight = \( 600 + \frac{8}{10} = 600.8 \)
(iv) Two hundred five and five hundredths = \( 205 + \frac{5}{100} = 205.05 \)
(v) Seven and fifteen thousandths = \( 7 + \frac{15}{1000} = 7.015 \)
In simple words: The words before "and" or "point" tell you the whole number part. The words after tell you what fractions go in the decimal places.

Exam Tip: "Tenths" means one place after the decimal; "hundredths" means two places; "thousandths" means three places. Use zeros as placeholders when needed.

 

Question 6. Write the number given in the following place value table in decimal form:

Answer: (i) 7102.306 (ii) 211.902 (iii) 3053.015 (iv) 70.03 (v) 5.40 (vi) 719.028
In simple words: Read across each row from left to right. Write the digits in order, placing the decimal point before the tenths column.

Exam Tip: Always place the decimal point in the right location - it should come before the tenths place. Check your digit arrangement by verifying each digit matches its place value column.

 

Question 7. Show the following decimal numbers on the number line:
(i) 0.4
(ii) 1.9
(iii) 1.1
(iv) 2.5
Answer: Between consecutive whole numbers on the number line, the space is divided into 10 equal parts. Each part shows an increase of 0.1. Position 0.4 is four-tenths of the way from 0 to 1. Position 1.1 is one-tenth of the way from 1 to 2. Position 1.9 is nine-tenths of the way from 1 to 2. Position 2.5 is halfway between 2 and 3.
In simple words: Find the whole number below the decimal (like 1 for 1.1). Count that many places to the right. Then count the tenths digit (like 1) more marks. Mark that spot.

Exam Tip: Count carefully from the correct whole number mark, then move right the right number of tenths. Always divide each unit into 10 equal segments for accuracy.

 

Question 8. Write the decimal numbers represented by the points A, B, C and D on the given number line:
Answer: By reading the position of each point on the number line where the unit length is divided into 10 equal parts, the decimal values are: A = 0.8, B = 1.3, C = 2.2, D = 2.9
In simple words: Look at where each marked point sits between the whole numbers. Count how many tenths from the left whole number the point is, then write it as a decimal.

Exam Tip: Read the position carefully by counting the number of small divisions from the nearest whole number on the left.

 

Question 9. Between which two numbers in tenths place on the number line does each of the given number lie?
(i) 0.06
(ii) 0.45
(iii) 0.66
(iv) 0.92
Answer: (i) The number 0.06 lies between 0.0 and 0.1. (ii) The number 0.45 lies between 0.4 and 0.5. (iii) The number 0.66 lies between 0.6 and 0.7. (iv) The number 0.92 lies between 0.9 and 1.0
In simple words: Look at the first decimal place (the tenths). The number lies between the tenths mark below it and the tenths mark above it.

Exam Tip: When two decimal numbers have the same tenths digit, the one with the smaller hundredths digit is smaller overall.

 

Exercise 7.2

 

Question 1. Write the following decimal fractions as decimal numbers:
(i) \( \frac{531}{10} \)
(ii) \( \frac{422}{100} \)
(iii) \( \frac{58301}{1000} \)
(iv) \( \frac{7}{10} \)
(v) \( \frac{3}{100} \)
(vi) \( \frac{37}{1000} \)
Answer: (i) \( \frac{531}{10} = 53.1 \) (ii) \( \frac{422}{100} = 4.22 \) (iii) \( \frac{58301}{1000} = 58.301 \) (iv) \( \frac{7}{10} = 0.7 \) (v) \( \frac{3}{100} = 0.03 \) (vi) \( \frac{37}{1000} = 0.037 \)
In simple words: The denominator tells you how many decimal places to have. For 10 (tenths), move the decimal one place left. For 100 (hundredths), move two places. For 1000 (thousandths), move three places.

Exam Tip: Count the number of zeros in the denominator - this equals the number of decimal places. Add leading zeros if the numerator is smaller than the denominator.

 

Question 2. Write the following decimal numbers as decimal fractions:
(i) 54.01
(ii) 318.105
(iii) 0.37
(iv) 0.047
(v) 0.03
(vi) 34.5
Answer: (i) 54.01 has 2 decimal places, so the denominator is 100. Hence, \( 54.01 = \frac{5401}{100} \) (ii) 318.105 has 3 decimal places, so the denominator is 1000. Hence, \( 318.105 = \frac{318105}{1000} \) (iii) 0.37 has 2 decimal places, so the denominator is 100. Hence, \( 0.37 = \frac{37}{100} \) (iv) 0.047 has 3 decimal places, so the denominator is 1000. Hence, \( 0.047 = \frac{47}{1000} \) (v) 0.03 has 2 decimal places, so the denominator is 100. Hence, \( 0.03 = \frac{3}{100} \) (vi) 34.5 has 1 decimal place, so the denominator is 10. Hence, \( 34.5 = \frac{345}{10} \)
In simple words: Count how many digits come after the decimal point. That number tells you how many zeros go in your denominator (10, 100, or 1000). Remove the decimal and that becomes your numerator.

Exam Tip: The denominator is always a power of 10 (10, 100, 1000...) with a number of zeros equal to the number of decimal places.

 

Question 3. Write the following decimal numbers as fractions in lowest terms:
(i) 0.8
(ii) 0.04
(iii) 0.125
(iv) 0.225
(v) 0.066
(vi) 0.092
Answer: (i) \( 0.8 = \frac{8}{10} = \frac{4}{5} \) (ii) \( 0.04 = \frac{4}{100} = \frac{1}{25} \) (iii) \( 0.125 = \frac{125}{1000} = \frac{1}{8} \) (iv) \( 0.225 = \frac{225}{1000} = \frac{9}{40} \) (v) \( 0.066 = \frac{66}{1000} = \frac{33}{500} \) (vi) \( 0.092 = \frac{92}{1000} = \frac{23}{250} \)
In simple words: First convert to a decimal fraction using the power of 10. Then divide both the top and bottom by the largest number that goes into both evenly - keep doing this until you cannot simplify anymore.

Exam Tip: Always reduce the fraction to simplest form using the greatest common divisor (GCD). Check your answer by dividing the numerator by the denominator - it should equal the original decimal.

 

Question 4. Convert the following decimal numbers into mixed fractions:
Answer: To convert a decimal to a mixed fraction, first separate the whole number part from the decimal part. Convert the decimal part to a fraction using the place value method, then simplify. Combine the whole number and the simplified fraction to create the mixed fraction. For example, 4.6 becomes \( 4 + \frac{6}{10} = 4 + \frac{3}{5} = 4\frac{3}{5} \)
In simple words: Keep the whole number in front. Change only the decimal part to a simplified fraction. Put them together as a mixed number.

Exam Tip: Always simplify the fractional part by finding the GCD of the numerator and denominator. This shows you understand how to reduce fractions properly.

 

Question 4. Write the repeating decimal for each of the following and use a bar to show the repetend.
(i) \( \frac { 1 }{ 9 } \)
(ii) \( \frac { -4 }{ 3 } \)
(iii) \( \frac { 1 }{ 6 } \)
Answer:
(i) \( \frac { 1 }{ 9 } = 0.\overline{1} \)
(ii) \( \frac { -4 }{ 3 } = -1.\overline{3} \)
(iii) \( \frac { 1 }{ 6 } = 0.1\overline{6} \) — In this case, only the digit 6 repeats, not the 1 that comes before it.
In simple words: Divide the numerator by the denominator. When a digit (or group of digits) appears over and over without stopping, place a bar over it. The bar shows that this part repeats endlessly.

Exam Tip: Always place the bar only over the digits that actually repeat. Non-repeating digits at the start must not be covered by the bar.

 

Question 5. Convert the following fractions into decimal numbers:
(i) \( \frac{4}{5} \)
(ii) \( \frac{6}{25} \)
(iii) \( \frac{112}{125} \)
(iv) \( \frac{3}{4} \)
(v) \( \frac{3}{8} \)
(vi) \( 7\frac{3}{40} \)
Answer:
(i) To change \( \frac{4}{5} \) to a decimal, multiply both the numerator and denominator by 2:
\( \frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10} = 0.8 \)

The decimal form is 0.8.

(ii) To change \( \frac{6}{25} \) to a decimal, multiply both the numerator and denominator by 4:
\( \frac{6}{25} = \frac{6 \times 4}{25 \times 4} = \frac{24}{100} = 0.24 \)

The decimal form is 0.24.

(iii) To change \( \frac{112}{125} \) to a decimal, multiply both the numerator and denominator by 8:
\( \frac{112}{125} = \frac{112 \times 8}{125 \times 8} = \frac{896}{1000} = 0.896 \)

The decimal form is 0.896.

(iv) To change \( \frac{3}{4} \) to a decimal, multiply both the numerator and denominator by 25:
\( \frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100} = 0.75 \)

The decimal form is 0.75.

(v) To change \( \frac{3}{8} \) to a decimal, multiply both the numerator and denominator by 125:
\( \frac{3}{8} = \frac{3 \times 125}{8 \times 125} = \frac{375}{1000} = 0.375 \)

The decimal form is 0.375.

(vi) To change \( 7\frac{3}{40} \) to a decimal, first convert the mixed number:
\( 7\frac{3}{40} = 7 + \frac{3}{40} \)

Now multiply the fraction by \( \frac{25}{25} \):
\( \frac{3}{40} = \frac{3 \times 25}{40 \times 25} = \frac{75}{1000} = 0.075 \)

Therefore: \( 7\frac{3}{40} = 7 + 0.075 = 7.075 \)

The decimal form is 7.075.
In simple words: To turn a fraction into a decimal, make the denominator 10, 100, 1000, or another power of 10 by multiplying both the top and bottom by the same number. Then read off the decimal directly.

Exam Tip: When the denominator is 5, 25, or 125 (factors of 10, 100, or 1000), this method works perfectly. Always check that your final denominator is a power of 10.

 

Question 6. Convert the following unlike decimal numbers to like decimal numbers:
(i) 17.5, 3.912
(ii) 5.04, 13.1902
(iii) 2.451, 3.7, 28.34
(iv) 3.1, 2.678, 27.0103
Answer:
(i) The maximum number of decimal places among these numbers is 3. We will change each number to have exactly 3 decimal places by adding zeros where needed.

17.5 = 17.500
3.912 = 3.912

Since 17.500 and 3.912 now have the same number of decimal places, they are like decimals.

(ii) The maximum number of decimal places among these numbers is 4. We will change each number to have exactly 4 decimal places by adding zeros where needed.

5.04 = 5.0400
13.1902 = 13.1902

Since 5.0400 and 13.1902 now have the same number of decimal places, they are like decimals.

(iii) The maximum number of decimal places among these numbers is 3. We will change each number to have exactly 3 decimal places by adding zeros where needed.

2.451 = 2.451
3.7 = 3.700
28.34 = 28.340

Since 2.451, 3.700, and 28.340 now have the same number of decimal places, they are like decimals.

(iv) The maximum number of decimal places among these numbers is 4. We will change each number to have exactly 4 decimal places by adding zeros where needed.

3.1 = 3.1000
2.678 = 2.6780
27.0103 = 27.0103

Since 3.1000, 2.6780, and 27.0103 now have the same number of decimal places, they are like decimals.
In simple words: Like decimals are numbers that have the same number of digits after the decimal point. Just add zeros to the right to make them match.

Exam Tip: Count the decimal places in each number, find the maximum, and add zeros to all the others to reach that maximum. This makes comparing and adding/subtracting easier.

 

Question 7. In each of the following pairs of decimal numbers, state which number is greater:
(i) 0.3, 0.4
(ii) 1, 0.99
(iii) 1.09, 1.093
(iv) 0.5, 0.05
Answer:
(i) The two decimals are 0.3 and 0.4. The whole number parts are the same. Looking at the tenths digits, 3 is less than 4.

Therefore, 0.4 is greater.

(ii) The two decimals are 1 and 0.99. Comparing the whole number parts, 1 is greater than 0.

Therefore, 1 is greater.

(iii) The two decimals are 1.09 and 1.093. Write them as like decimals: 1.090 and 1.093. Looking at the decimal parts, 090 is less than 093.

Therefore, 1.093 is greater.

(iv) The two decimals are 0.5 and 0.05. Write them as like decimals: 0.50 and 0.05. Looking at the decimal parts, 50 is greater than 05.

Therefore, 0.5 is greater.
In simple words: Compare the whole number parts first. If they are the same, look at the tenths place, then the hundredths, and so on, from left to right.

Exam Tip: Convert numbers to like decimals before comparing - this makes it much simpler to see which is bigger.

 

Question 8. In each of the following pairs of decimal numbers, state which number is smaller:
(i) 45.78, 345.8
(ii) 37.701, 37.71
(iii) 5.907, 5.903
Answer:
(i) The two decimals are 45.78 and 345.8. Looking at the whole number parts, 45 is less than 345.

Therefore, 45.78 is smaller.

(ii) The two decimals are 37.701 and 37.71. Write them as like decimals: 37.701 and 37.710. Looking at the decimal parts, 701 is less than 710.

Therefore, 37.701 is smaller.

(iii) The two decimals are 5.907 and 5.903. The whole number parts are equal. Looking at the decimal parts, 907 is greater than 903.

Therefore, 5.903 is smaller.
In simple words: When comparing two numbers, first check the whole number parts. If those are the same, look at each decimal place from left to right until you find a difference.

Exam Tip: Always write both numbers as like decimals before comparing the decimal parts - this prevents careless mistakes with place values.

 

Question 9. Arrange the following decimal numbers in ascending order:
(i) 27.35, 27.305, 2.7, 2.543
(ii) 4.53, 4.07, 29.1, 0.9, 0.709
Answer:
(i) The maximum number of decimal places is 3. We convert each number to have 3 decimal places:

27.350, 27.305, 2.700, 2.543

Comparing these, we get: 2.543 < 2.700 < 27.305 < 27.350

The numbers in ascending order are:
2.543 < 2.7 < 27.305 < 27.35

(ii) The maximum number of decimal places is 3. We convert each number to have 3 decimal places:

4.530, 4.070, 29.100, 0.900, 0.709

Comparing these, we get: 0.709 < 0.900 < 4.070 < 4.530 < 29.100

The numbers in ascending order are:
0.709 < 0.9 < 4.07 < 4.53 < 29.1
In simple words: First, add zeros to make all numbers have the same number of decimal places. Then compare the whole number parts, then the decimal parts from left to right.

Exam Tip: Always convert to like decimals before arranging - this reduces errors. Compare whole numbers first, since they determine the order in most cases.

 

Question 10. Arrange the following decimal numbers in descending order:
(i) 3.303, 33.03, 3.3, 30.33
(ii) 72.5, 2.75, 27.505, 0.275, 2.507
Answer:
(i) The maximum number of decimal places is 3. We convert each number to have 3 decimal places:

3.303, 33.030, 3.300, 30.330

Comparing these, we get: 33.030 > 30.330 > 3.303 > 3.300

The numbers in descending order are:
33.03 > 30.33 > 3.303 > 3.3

(ii) The maximum number of decimal places is 3. We convert each number to have 3 decimal places:

72.500, 2.750, 27.505, 0.275, 2.507

Comparing these, we get: 72.500 > 27.505 > 2.750 > 2.507 > 0.275

The numbers in descending order are:
72.5 > 27.505 > 2.75 > 2.507 > 0.275
In simple words: Make all numbers have the same number of decimal places first. Then arrange them from largest to smallest by comparing the whole numbers first, then the decimal parts.

Exam Tip: In descending order, the number with the largest whole number part comes first. If whole numbers are equal, check the decimal parts from left to right.

 

Exercise 7.3

 

Question 1. Add:
(i) 17.5, 8.8
(ii) 9.999, 0.03
(iii) 5.87, 1.03, 0.1
(iv) 23.71, 9.9, 4.023
(v) 4.5, 16.024, 7.99
(vi) 8.79, 23.001, 5.41, 0.875
Answer:
(i) Arrange the decimals vertically and add:
\[\begin{align} &17.500\\ &+\phantom{0}8.800\\ \hline &26.300 \end{align}\]

The sum is 26.3.

(ii) Convert to like decimals with 3 decimal places: 9.999 and 0.030. Arrange vertically and add:
\[\begin{align} &9.999\\ &+0.030\\ \hline &10.029 \end{align}\]

The sum is 10.029.

(iii) Convert to like decimals with 2 decimal places: 5.87, 1.03, 0.10. Arrange vertically and add:
\[\begin{align} &5.87\\ &1.03\\ &+0.10\\ \hline &7.00 \end{align}\]

The sum is 7.

(iv) Convert to like decimals with 3 decimal places: 23.710, 9.900, 4.023. Arrange vertically and add:
\[\begin{align} &23.710\\ &9.900\\ &+4.023\\ \hline &37.633 \end{align}\]

The sum is 37.633.

(v) Convert to like decimals with 3 decimal places: 4.500, 16.024, 7.990. Arrange vertically and add:
\[\begin{align} &4.500\\ &16.024\\ &+7.990\\ \hline &28.514 \end{align}\]

The sum is 28.514.

(vi) Convert to like decimals with 3 decimal places: 8.790, 23.001, 5.410, 0.875. Arrange vertically and add:
\[\begin{align} &8.790\\ &23.001\\ &5.410\\ &+0.875\\ \hline &38.076 \end{align}\]

The sum is 38.076.
In simple words: Make all numbers have the same number of decimal places by adding zeros. Then line them up so the decimal points match, and add like you normally do with whole numbers.

Exam Tip: Always align decimal points vertically before adding. Start by converting all numbers to like decimals - this prevents alignment errors and careless mistakes.

 

Question 2. Calculate:
(i) 5.82 - 2.65
(ii) 19.01 - 12.234
(iii) 15.4 + 3.015 - 14.237
(iv) 7.4 - 2.19 - 0.456 - 3.5
(v) 19.27 - 3.6 - 8.812 + 0.84
(vi) 6.4 - 2.351 - 1.45 - 0.999
Answer:
(i) Arrange vertically and subtract:
\[\begin{align} &5.82\\ &-2.65\\ \hline &3.17 \end{align}\]

Therefore, 5.82 - 2.65 = 3.17.

(ii) Convert to like decimals with 3 decimal places: 19.010 and 12.234. Arrange vertically and subtract:
\[\begin{align} &19.010\\ &-12.234\\ \hline &6.776 \end{align}\]

Therefore, 19.01 - 12.234 = 6.776.

(iii) Convert to like decimals with 3 decimal places: 15.400, 3.015, 14.237.

Given expression = 15.400 + 3.015 - 14.237

First, add 15.400 + 3.015:
\[\begin{align} &15.400\\ &+3.015\\ \hline &18.415 \end{align}\]

Then subtract 14.237:
\[\begin{align} &18.415\\ &-14.237\\ \hline &4.178 \end{align}\]

Therefore, 15.4 + 3.015 - 14.237 = 4.178.

(iv) Convert to like decimals with 3 decimal places: 7.400, 2.190, 0.456, 3.500.

Given expression = 7.400 - 2.190 - 0.456 - 3.500
= 7.400 - (2.190 + 0.456 + 3.500)

First, add 2.190 + 0.456 + 3.500:
\[\begin{align} &2.190\\ &0.456\\ &+3.500\\ \hline &6.146 \end{align}\]

Then subtract from 7.400:
\[\begin{align} &7.400\\ &-6.146\\ \hline &1.254 \end{align}\]

Therefore, 7.4 - 2.19 - 0.456 - 3.5 = 1.254.

(v) Convert to like decimals with 3 decimal places: 19.270, 3.600, 8.812, 0.840.

Given expression = 19.270 - 3.600 - 8.812 + 0.840

First, add the amounts being subtracted: 3.600 + 8.812 = 12.412

Then, add the amounts being added: 19.270 + 0.840 = 20.110

Finally: 20.110 - 12.412 = 7.698

Therefore, 19.27 - 3.6 - 8.812 + 0.84 = 7.698.

(vi) Convert to like decimals with 3 decimal places: 6.400, 2.351, 1.450, 0.999.

Given expression = 6.400 - 2.351 - 1.450 - 0.999
= 6.400 - (2.351 + 1.450 + 0.999)

First, add 2.351 + 1.450 + 0.999:
\[\begin{align} &2.351\\ &1.450\\ &+0.999\\ \hline &4.800 \end{align}\]

Then subtract from 6.400:
\[\begin{align} &6.400\\ &-4.800\\ \hline &1.600 \end{align}\]

Therefore, 6.4 - 2.351 - 1.45 - 0.999 = 1.600 or 1.6.
In simple words: Convert all numbers to like decimals first. Line up the decimal points. Add or subtract each column, starting from the right, just like you do with whole numbers.

Exam Tip: In expressions with mixed operations, group positive and negative terms separately, then combine. This reduces the chance of sign errors and makes the working clearer.

 

Question 3. (v) Compute 19.27 - 3.6 - 8.812 + 0.84.
Answer: First, express each number with three decimal places: 19.270 - 3.600 - 8.812 + 0.840. Group and add the positive terms: 19.270 + 0.840 = 20.110. Add the negative terms: 3.600 + 8.812 = 12.412. Now subtract: 20.110 - 12.412 = 7.698.
In simple words: Convert all numbers to the same number of decimal places, then group the numbers being added together and the numbers being subtracted together. Add each group, then do the final subtraction.

Exam Tip: Always align decimal points vertically and write all numbers with the same number of decimal places before performing operations.

 

Question 3. (vi) Compute 6.4 - 2.351 - 1.45 - 0.999.
Answer: Write all numbers with three decimal places: 6.400 - 2.351 - 1.450 - 0.999. Rewrite as 6.400 - (2.351 + 1.450 + 0.999). First add: 2.351 + 1.450 + 0.999 = 4.800. Then subtract: 6.400 - 4.800 = 1.600.
In simple words: Make all numbers have the same decimal places. When subtracting many numbers, add them together first, then take that sum away from the first number.

Exam Tip: Group subtracted values and add them before subtracting from the main number - this reduces errors.

 

Question 4. What number added to 0.756 gives 1?
Answer: Let the unknown number be x. We set up the equation: 0.756 + x = 1. Solving for x: x = 1 - 0.756. Rewrite both numbers with three decimal places: x = 1.000 - 0.756. Subtract in column form: 1.000 - 0.756 = 0.244. Therefore, the required number is 0.244.
In simple words: If you add 0.756 to 0.244, you get exactly 1. So 0.244 is the missing number.

Exam Tip: Set up an equation and isolate the variable. Always align the decimal points when subtracting.

 

Question 5. By how much should 17.45 be decreased to get 7.9702?
Answer: Let the decrease be x. Then: 17.45 - x = 7.9702. Solving for x: x = 17.45 - 7.9702. Write both with four decimal places: x = 17.4500 - 7.9702. Perform subtraction in column form: 17.4500 - 7.9702 = 9.4798. Therefore, 17.45 must be decreased by 9.4798.
In simple words: If you start at 17.45 and take away 9.4798, you end up at 7.9702.

Exam Tip: Express both numbers with the same number of decimal places (the larger count needed) before subtracting to avoid mistakes.

 

Exercise 7.4

 

Question 1. Evaluate the following:
(i) 3.7 × 4.5
(ii) 12.08 × 9.3
(iii) 238.06 × 7.5
(iv) 0.79 × 32.4
(v) 3.6 × 1.4 × 0.7
(vi) 9.01 × 2.5 × 1.6
Answer:
(i) Set aside the decimal points and multiply: 37 × 45 = 1665. The numbers together have 1 + 1 = 2 decimal places. Place the decimal point so the result has 2 places: 3.7 × 4.5 = 16.65.
(ii) Ignore decimals and multiply: 1208 × 93 = 112344. Together they have 2 + 1 = 3 decimal places. Place the decimal: 12.08 × 9.3 = 112.344.
(iii) Ignore decimals: 23806 × 75 = 1785450. Total decimal places: 2 + 1 = 3. Result: 238.06 × 7.5 = 1785.450 = 1785.45.
(iv) Ignore decimals: 79 × 324 = 25596. Total decimal places: 2 + 1 = 3. Result: 0.79 × 32.4 = 25.596.
(v) Ignore decimals: 36 × 14 × 7 = 504 × 7 = 3528. Total decimal places: 1 + 1 + 1 = 3. Result: 3.6 × 1.4 × 0.7 = 3.528.
(vi) Ignore decimals: 901 × 25 × 16 = 22525 × 16 = 360400. Total decimal places: 2 + 1 + 1 = 4. Result: 9.01 × 2.5 × 1.6 = 36.0400 = 36.04.
In simple words: Multiply the numbers as if there were no decimal points. Then count how many decimal places all the original numbers had together, and place that many decimal places in your answer.

Exam Tip: Count total decimal places carefully - a common error is miscounting how many decimal places to move in the final answer.

 

Question 2. Calculate the following:
(i) 70.756 ÷ 4
(ii) 2.46 ÷ 6
(iii) 3.016 ÷ 8
(iv) 8.64 ÷ 3.6
(v) 72.8 ÷ 0.04
(vi) 0.144 ÷ 0.02
Answer:
(i) Using long division: 70.756 ÷ 4 = 17.689.
(ii) Using long division: 2.46 ÷ 6 = 0.41.
(iii) Using long division: 3.016 ÷ 8 = 0.377.
(iv) To divide by a decimal, multiply both dividend and divisor by 10: 8.64 ÷ 3.6 = (8.64 × 10) ÷ (3.6 × 10) = 86.4 ÷ 36. Using long division: 86.4 ÷ 36 = 2.4.
(v) To divide by 0.04, multiply both by 100: 72.8 ÷ 0.04 = (72.8 × 100) ÷ (0.04 × 100) = 7280 ÷ 4 = 1820.
(vi) To divide by 0.02, multiply both by 100: 0.144 ÷ 0.02 = (0.144 × 100) ÷ (0.02 × 100) = 14.4 ÷ 2 = 7.2.
In simple words: For division by a whole number, use long division. For division by a decimal, multiply both the number being divided and the divisor by 10, 100, or 1000 to make the divisor whole, then divide.

Exam Tip: When the divisor is a decimal, eliminate the decimal point by multiplying both numbers by the same power of 10 before performing division.

 

Question 3. Multiply each of the following numbers by 10, 100 and 1000 (orally):
(i) 4.7
(ii) 3.45
(iii) 0.234
Answer: When a decimal is multiplied by 10, 100, or 1000, the decimal point moves to the right by one, two, or three places respectively.
(i) For 4.7: 4.7 × 10 = 47; 4.7 × 100 = 470; 4.7 × 1000 = 4700.
(ii) For 3.45: 3.45 × 10 = 34.5; 3.45 × 100 = 345; 3.45 × 1000 = 3450.
(iii) For 0.234: 0.234 × 10 = 2.34; 0.234 × 100 = 23.4; 0.234 × 1000 = 234.
In simple words: Multiplying by 10 shifts the decimal point one place right. Multiplying by 100 shifts it two places right. Multiplying by 1000 shifts it three places right.

Exam Tip: Remember that multiplying by powers of 10 moves the decimal point to the right, making the number larger.

 

Question 4. Divide each of the following numbers by 10, 100 and 1000 (orally):
(i) 4.7
(ii) 3.45
(iii) 23.01
Answer: When a decimal is divided by 10, 100, or 1000, the decimal point moves to the left by one, two, or three places respectively.
(i) For 4.7: 4.7 ÷ 10 = 0.47; 4.7 ÷ 100 = 0.047; 4.7 ÷ 1000 = 0.0047.
(ii) For 3.45: 3.45 ÷ 10 = 0.345; 3.45 ÷ 100 = 0.0345; 3.45 ÷ 1000 = 0.00345.
(iii) For 23.01: 23.01 ÷ 10 = 2.301; 23.01 ÷ 100 = 0.2301; 23.01 ÷ 1000 = 0.02301.
In simple words: Dividing by 10 shifts the decimal point one place left. Dividing by 100 shifts it two places left. Dividing by 1000 shifts it three places left.

Exam Tip: Division by powers of 10 moves the decimal point left, making the number smaller. Count the zeros in 10, 100, or 1000 to know how many places to move.

 

Question 5. Find the value of the following:
(i) (3.5)²
(ii) (0.4)³
Answer:
(i) We compute: (3.5)² = 3.5 × 3.5. Ignoring decimals: 35 × 35 = 1225. The two factors together have 1 + 1 = 2 decimal places. Therefore (3.5)² = 12.25.
(ii) We compute: (0.4)³ = 0.4 × 0.4 × 0.4. Ignoring decimals: 4 × 4 × 4 = 64. The three factors together have 1 + 1 + 1 = 3 decimal places. Therefore (0.4)³ = 0.064.
In simple words: Multiply the numbers without looking at decimal points. Count the total decimal places in all the factors you multiplied together, then place that many decimal places in your answer.

Exam Tip: For powers of decimals, multiply as normal then count decimal places: if you multiply n factors each with d decimal places, the result has n × d decimal places total.

 

Exercise 7.5

 

Question 1. Express as rupees using decimals:
(i) 75 paise
(ii) 1025 paise
(iii) 63 rupees 9 paise
Answer: We know that 1 paisa = \( \frac{1}{100} \) rupee = Rs. 0.01.
(i) 75 paise = \( \frac{75}{100} \) = Rs. 0.75.
(ii) 1025 paise = \( \frac{1025}{100} \) = Rs. 10.25.
(iii) 63 rupees 9 paise = Rs. 63 + \( \frac{9}{100} \) = Rs. 63 + Rs. 0.09 = Rs. 63.09.
In simple words: To turn paise into rupees, divide by 100. If you have rupees and paise, write the paise as a decimal and add it to the rupees.

Exam Tip: Remember that 100 paise = 1 rupee, so divide paise by 100 to convert to rupees.

 

Question 2. Express as cm using decimals:
(i) 8 mm
(ii) 263 mm
(iii) 13 cm 3 mm
Answer: We know that 1 mm = \( \frac{1}{10} \) cm = 0.1 cm.
(i) 8 mm = \( \frac{8}{10} \) = 0.8 cm.
(ii) 263 mm = \( \frac{263}{10} \) = 26.3 cm.
(iii) 13 cm 3 mm = 13 cm + \( \frac{3}{10} \) = 13 cm + 0.3 cm = 13.3 cm.
In simple words: Since 10 mm = 1 cm, divide millimetres by 10 to get centimetres. If you have both cm and mm, write the mm part as a decimal and add.

Exam Tip: 10 mm = 1 cm, so always divide mm by 10 when converting to cm.

 

Question 3. Express as metres using decimals:
(i) 6 cm
(ii) 528 cm
(iii) 7 m 55 cm
Answer: We know that 1 cm = \( \frac{1}{100} \) m = 0.01 m.
(i) 6 cm = \( \frac{6}{100} \) = 0.06 m.
(ii) 528 cm = \( \frac{528}{100} \) = 5.28 m.
(iii) 7 m 55 cm = 7 m + \( \frac{55}{100} \) = 7 m + 0.55 m = 7.55 m.
In simple words: Since 100 cm = 1 m, divide centimetres by 100 to get metres. If you have both m and cm, write the cm as a decimal and add.

Exam Tip: 100 cm = 1 m, so divide cm by 100 to convert to metres.

 

Question 4. Express as km using decimals:
(i) 5 m
(ii) 888 m
(iii) 15 km 88 m
Answer: We know that 1 m = \( \frac{1}{1000} \) km = 0.001 km.
(i) 5 m = \( \frac{5}{1000} \) = 0.005 km.
(ii) 888 m = \( \frac{888}{1000} \) = 0.888 km.
(iii) 15 km 88 m = 15 km + \( \frac{88}{1000} \) = 15 km + 0.088 km = 15.088 km.
In simple words: Since 1000 m = 1 km, divide metres by 1000 to get kilometres. If you have both km and m, write the m as a decimal and add.

Exam Tip: 1000 m = 1 km, so divide metres by 1000 when converting to kilometres.

 

Question 5. Express as kg using decimals:
(i) 37 g
Answer: We know that 1 g = \( \frac{1}{1000} \) kg = 0.001 kg. Therefore, 37 g = \( \frac{37}{1000} \) = 0.037 kg.
In simple words: Since 1000 g = 1 kg, divide grams by 1000 to turn them into kilograms.

Exam Tip: 1000 g = 1 kg, so divide grams by 1000 to convert to kilograms.

 

Question 5. Express 37 g, 100 g, and 5 kg 8 g in kilograms using decimals.
(i) 37 g
(ii) 100 g
(iii) 5 kg 8 g
Answer: To convert grams to kilograms, we use the fact that 1 g = 1/1000 kg = 0.001 kg.
(i) 37 g = 37/1000 kg = 0.037 kg
(ii) 100 g = 100/1000 kg = 0.1 kg
(iii) 5 kg 8 g = 5 kg + 8/1000 kg = 5 kg + 0.008 kg = 5.008 kg
In simple words: To change grams into kilograms, divide by 1000. This moves the decimal point three places to the left.

Exam Tip: Always remember that 1000 g = 1 kg. When combining whole kilograms with grams, add the decimal forms carefully — 5 kg 8 g is not 5.8 kg, it is 5.008 kg.

 

Question 6. Express as kL using decimals:
(i) 6 L
(ii) 555 L
(iii) 3 kL 95 L
Answer: Since 1 L = 1/1000 kL = 0.001 kL, we divide each quantity by 1000.
(i) 6 L = 6/1000 kL = 0.006 kL
(ii) 555 L = 555/1000 kL = 0.555 kL
(iii) 3 kL 95 L = 3 kL + 95/1000 kL = 3 kL + 0.095 kL = 3.095 kL
In simple words: Litres become kilolitres by dividing by 1000. For mixed amounts like 3 kL 95 L, write the kilolitres first, then add the decimal form of the litres.

Exam Tip: Similar to grams and kilograms, 1000 L = 1 kL. Take care not to write 3 kL 95 L as 3.95 kL — the correct form is 3.095 kL.

 

Question 7. Anita bought 2 m 70 cm cloth for her shirt and 2 m 85 cm cloth for her trouser. Find the total length of the cloth bought by her.
Answer: First, convert each length to metres in decimal form. Length of cloth for shirt = 2 m 70 cm = 2 m + 70/100 m = 2 m + 0.70 m = 2.70 m. Length of cloth for trouser = 2 m 85 cm = 2 m + 85/100 m = 2 m + 0.85 m = 2.85 m. Now add both amounts: 2.70 + 2.85 = 5.55 m. So the total length of cloth Anita purchased is 5.55 m or 5 m 55 cm.
In simple words: Change each measurement to metres with decimals, then add them together like normal numbers.

Exam Tip: Always convert mixed measurements (metres and centimetres) to decimal form before adding. Line up the decimal points in column addition.

 

Question 8. Sunita travelled 15 km 268 m by bus, 7 km 7 m by car and 500 m on foot in order to reach her school. How far is her school from her residence?
Answer: Convert each distance to kilometres in decimal form. Distance travelled by bus = 15 km 268 m = 15 km + 268/1000 km = 15 km + 0.268 km = 15.268 km. Distance travelled by car = 7 km 7 m = 7 km + 7/1000 km = 7 km + 0.007 km = 7.007 km. Distance travelled on foot = 500 m = 500/1000 km = 0.500 km. Total distance = 15.268 + 7.007 + 0.500 = 22.775 km. Therefore, Sunita's school is 22.775 km or 22 km 775 m from her residence.
In simple words: Convert all distances to kilometres with decimals, then add them in a column to find the total.

Exam Tip: When a distance is given in two units (km and m), convert carefully by dividing metres by 1000. Ensure decimal points are aligned when adding multiple numbers.

 

Question 9. Rahul bought 4 kg 90 g apples, 2 kg 60 g grapes and 5 kg 300 g mangoes. Find the total weight of all the fruits he bought.
Answer: Convert each weight to kilograms in decimal form. Weight of apples = 4 kg 90 g = 4 kg + 90/1000 kg = 4 kg + 0.090 kg = 4.090 kg. Weight of grapes = 2 kg 60 g = 2 kg + 60/1000 kg = 2 kg + 0.060 kg = 2.060 kg. Weight of mangoes = 5 kg 300 g = 5 kg + 300/1000 kg = 5 kg + 0.300 kg = 5.300 kg. Total weight = 4.090 + 2.060 + 5.300 = 11.450 kg. So the total weight of all the fruits Rahul bought is 11.450 kg or 11 kg 450 g.
In simple words: Write each fruit's weight as a decimal in kilograms, then add all three decimals together.

Exam Tip: Convert grams to kilograms by dividing by 1000 and writing as a decimal. Always align decimal points when adding in columns.

 

Question 10. Rani has Rs.18.50. She bought one ice cream for Rs.11.75. How much money does she have now?
Answer: Money Rani had = Rs.18.50. Cost of ice cream = Rs.11.75. Money left = 18.50 - 11.75. Subtracting in column form: 18.50 - 11.75 = 6.75. Therefore, Rani now has Rs.6.75.
In simple words: Subtract the amount spent from the amount she had to find what is left.

Exam Tip: Line up decimal points before subtracting money amounts. Borrow from the ones place if needed, just like with whole numbers.

 

Question 11. Tina had 20 m 5 cm long cloth. She cuts 4 m 50 cm length of cloth from this for making a curtain. How much length of cloth is left with her?
Answer: Total length of cloth = 20 m 5 cm = 20 m + 5/100 m = 20 m + 0.05 m = 20.05 m. Length of cloth cut for curtain = 4 m 50 cm = 4 m + 50/100 m = 4 m + 0.50 m = 4.50 m. Length of cloth left = 20.05 - 4.50 = 15.55 m. Therefore, the length of cloth left with Tina is 15.55 m or 15 m 55 cm.
In simple words: Change both measurements to metres with decimals, then subtract one from the other.

Exam Tip: Always line up decimal points in column subtraction. The number of decimal places should be the same for both numbers being subtracted.

 

Question 12. Ruby bought a watermelon weighing 5 kg 300 g. Out of which she gave 2 kg 680 g to her neighbour. What is the weight of the watermelon left with Ruby?
Answer: Weight of watermelon = 5 kg 300 g = 5 kg + 300/1000 kg = 5 kg + 0.300 kg = 5.300 kg. Weight given to neighbour = 2 kg 680 g = 2 kg + 680/1000 kg = 2 kg + 0.680 kg = 2.680 kg. Weight left = 5.300 - 2.680 = 2.620 kg. Therefore, the weight of the watermelon left with Ruby is 2.620 kg or 2 kg 620 g.
In simple words: Convert both weights to decimals, then subtract the amount given away from the original amount.

Exam Tip: Ensure both numbers have the same number of decimal places before subtracting. Borrow from the whole number part if the decimal part in the first number is smaller.

 

Question 13. The cost of 1 metre of cloth is Rs.35.80. What will be cost of 9.8 metres of cloth?
Answer: Cost of 1 metre of cloth = Rs.35.80. Cost of 9.8 metres = Rs.35.80 × 9.8. To multiply, ignore the decimal points and calculate 3580 × 98 = 350840. Count the total decimal places in both numbers: 35.80 has 2 decimal places and 9.8 has 1 decimal place, giving 2 + 1 = 3 decimal places. Place the decimal point in the result: 35.80 × 9.8 = 350.840 = 350.84. So the cost of 9.8 metres of cloth is Rs.350.84.
In simple words: Multiply as if there are no decimals, then count how many decimal places are in both numbers and put the decimal point that many places from the right.

Exam Tip: The total number of decimal places in the answer equals the sum of decimal places in the two numbers being multiplied. Check your answer by estimation — 36 × 10 is roughly 360.

 

Question 14. Farida bought some bags of cement, each weighing 49.8 kg. If the total weight of all the bags is 1792.8 kg, how many bags did she buy?
Answer: Weight of each bag = 49.8 kg. Total weight of all bags = 1792.8 kg. Number of bags = 1792.8 ÷ 49.8. To remove decimals, multiply both numerator and denominator by 10: Number of bags = (1792.8 × 10) ÷ (49.8 × 10) = 17928 ÷ 498. Performing the division: 17928 ÷ 498 = 36. Therefore, Farida bought 36 bags of cement.
In simple words: Divide total weight by weight per bag. Multiply both numbers by 10 to remove decimals, then divide as usual.

Exam Tip: To divide decimals, multiply both the dividend and divisor by the same power of 10 until both are whole numbers. Then divide normally.

 

Objective Type Questions - Mental Maths

 

Question 1. Fill in the following blanks:
(i) The decimal point in a decimal number is placed between ones digit and .... digit.
(ii) The place value of the digit 3 in the decimal number 15.437 is .....
(iii) The decimal number 27.025 has ..... decimal places.
(iv) The decimal number 5.06 is read as five point .....
(v) If an object is divided into 1000 equal parts, then its 27 parts are represented by .....
(vi) Two decimal numbers having different number of ..... are called unlike decimal numbers.
(vii) 4 tens, 3 ones, 2 tenths, 0 hundredths and 5 thousandths in decimal form is written as .....
(viii) The smallest decimal number upto three decimal places is ......
(ix) The largest four digit decimal number less than 1 using the digits 1, 5, 3 and 8 once is ...
Answer:
(i) The decimal point in a decimal number is placed between ones digit and tenths digit.
(ii) The place value of the digit 3 in the decimal number 15.437 is \( \frac{3}{100} \).
(iii) The decimal number 27.025 has 3 decimal places.
(iv) The decimal number 5.06 is read as five point zero six.
(v) If an object is divided into 1000 equal parts, then its 27 parts are represented by 0.027.
(vi) Two decimal numbers having different number of decimal places are called unlike decimal numbers.
(vii) 4 tens, 3 ones, 2 tenths, 0 hundredths and 5 thousandths in decimal form is written as 43.205.
(viii) The smallest decimal number upto three decimal places is 0.001.
(ix) The largest four digit decimal number less than 1 using the digits 1, 5, 3 and 8 once is 0.8531.
In simple words: Decimals use place values based on powers of 10. The tenths place is right after the decimal point, followed by hundredths, then thousandths.

Exam Tip: Know the names of each decimal place: tenths (first), hundredths (second), thousandths (third). Also remember that unlike decimals have different numbers of digits after the decimal point.

 

Question 2. State whether the following statements are true (T) or false (F):
(i) Every decimal number can be represented by a point on a number line.
(ii) Fractions with denominator 10, 100, 1000, ..... are called decimal fractions.
(iii) A decimal number having 3 decimal places can be written as a fraction with denominator 1000.
(iv) The value of a decimal number remains the same if any number of extra zeros are written at the end of a decimal number.
(v) If a decimal number is multiplied by 10, then the decimal point moves by one place to the left.
Answer:
(i) True. Every decimal number corresponds to a unique point on a number line.
(ii) True. By definition, fractions with denominator 10, 100, 1000, ... are called decimal fractions.
(iii) True. For a decimal number with 3 decimal places, the denominator of its decimal fraction is 1000.
(iv) True. For example, 3.7 = 3.70 = 3.700.
(v) False. When a decimal number is multiplied by 10, the decimal point moves by one place to the right, not to the left.
In simple words: Decimal fractions are fractions where the denominator is a power of 10. Trailing zeros don't change the value, but multiplying by 10 always shifts the decimal right.

Exam Tip: Statement (v) is a common source of confusion. Multiplication by 10 moves the decimal to the RIGHT; division by 10 moves it to the LEFT.

 

Multiple Choice Questions

 

Question 3. Five and seven hundredths is equal to
(a) 5.7
(b) 5.07
(c) 5.70
(d) 0.57
Answer: (b) 5.07
Five and seven hundredths means 5 + 7/100 = 5 + 0.07 = 5.07.
In simple words: "Seven hundredths" is the fraction 7/100, which is 0.07. Add this to 5 to get 5.07.

Exam Tip: Be careful about the place value. "Seven hundredths" goes in the hundredths place (second spot after the decimal), not the tenths place.

 

Question 4. Sixty three thousandths is equal to
(a) 0.63
(b) 0.603
(c) 0.063
(d) 0.630
Answer: (c) 0.063
Sixty three thousandths = 63/1000 = 0.063.
In simple words: Thousandths is the third place after the decimal. The number 63 goes in that position, with zeros filling the first two places: 0.063.

Exam Tip: Distinguish between hundredths (0.xy) and thousandths (0.0xy). The word "thousandths" signals three decimal places.

 

Question 5. \( 3\frac{7}{100} \) is equal to
(a) 3.07
(b) 3.7
(c) 3.70
(d) 3.007
Answer: (a) 3.07
\( 3\frac{7}{100} = 3 + \frac{7}{100} = 3 + 0.07 = 3.07 \)
In simple words: A mixed fraction with 7/100 means 7 in the hundredths place, which is the second spot after the decimal.

Exam Tip: The denominator tells you which decimal place to use: 10 is tenths, 100 is hundredths, 1000 is thousandths.

 

Question 6. \( 5\frac{3}{1000} \) is equal to
(a) 5.03
(b) 5.3
(c) 5.003
(d) 5.0003
Answer: (c) 5.003
\( 5\frac{3}{1000} = 5 + \frac{3}{1000} = 5 + 0.003 = 5.003 \)
In simple words: With a denominator of 1000, the 3 goes in the thousandths place, which is three positions after the decimal point.

Exam Tip: The number of zeros in the denominator (10 has one, 100 has two, 1000 has three) tells you how many decimal places to use.

 

Question 7. The place value of the digit 7 in the decimal number 5.0378 is
(a) 7
(b) \( \frac{7}{10} \)
(c) \( \frac{7}{100} \)
(d) \( \frac{7}{1000} \)
Answer: (d) \( \frac{7}{1000} \)
In the number 5.0378, the digit 7 is in the thousandths place. So the place value of 7 is \( \frac{7}{1000} \).
In simple words: Count positions after the decimal: tenths (first), hundredths (second), thousandths (third). The 7 is at position three, so its value is 7/1000.

Exam Tip: Write out the decimal as an expanded form to identify place values clearly: 5.0378 = 5 + 0 + 0.03 + 0.007 + 0.0008.

 

Question 8. The place value of the digit 0 in the decimal number 13.405 is
(a) 0
(b) \( \frac{1}{10} \)
(c) \( \frac{1}{100} \)
(d) none of the options
Answer: (a) 0
In the number 13.405, the digit 0 is in the hundredths place. The place value of 0 at any position is 0.
In simple words: No matter where a 0 digit appears, its place value is always zero.

Exam Tip: The place value of the digit 0 is always 0, regardless of its position in the number.

 

Question 9. The value of \( 5 + \frac{7}{10} + \frac{3}{1000} \) is
(a) 5.73
(b) 5.703
(c) 5.073
(d) 0.753
Answer: (b) 5.703
\( 5 + \frac{7}{10} + \frac{3}{1000} = 5 + 0.7 + 0.003 = 5.703 \)
In simple words: Write each fraction as a decimal using its place value, then add them: 5 + 0.7 (tenths) + 0.003 (thousandths) = 5.703.

Exam Tip: When adding fractions with different denominators (10, 100, 1000), convert each to decimal form first, then add in the standard way.

 

Question 10. The value of \( \frac{3}{25} \) is
(a) 1.2
(b) 0.012
(c) 0.12
(d) none of these
Answer: (c) 0.12
In simple words: To convert a fraction to a decimal, multiply the top and bottom by a number that makes the bottom 10, 100, 1000, etc. Here, \( \frac{3}{25} = \frac{3 \times 4}{25 \times 4} = \frac{12}{100} = 0.12 \).

Exam Tip: Always look for a multiplier that turns the denominator into a power of 10 - this is the fastest way to convert any fraction to a decimal.

 

Question 11. The value of \( 5\frac{1}{25} \) is
(a) 5.4
(b) 5.25
(c) 5.04
(d) 5.004
Answer: (c) 5.04
In simple words: Separate the whole number (5) from the fractional part. Convert \( \frac{1}{25} \) to 0.04 by multiplying by \( \frac{4}{4} \), then add them: 5 + 0.04 = 5.04.

Exam Tip: For mixed numbers, convert the fraction part first, then add it to the whole number - never try to convert the entire mixed number in one step.

 

Question 12. The decimal number not equivalent to 5.7 is
(a) 5.70
(b) 5.07
(c) 5.700
(d) 5.7000
Answer: (b) 5.07
In simple words: Adding zeros to the right of a decimal does not change its value - 5.7, 5.70, 5.700, and 5.7000 are all the same. But 5.07 is different because the zero is before the 7, making it much smaller.

Exam Tip: Trailing zeros (zeros at the end) don't matter, but zeros in the middle or between the decimal point and other digits change the value completely.

 

Question 13. When 0.04 is written as a fraction in its simplest form, then the sum of numerator and denominator is
(a) 7
(b) 21
(c) 26
(d) 104
Answer: (c) 26
In simple words: Write 0.04 as \( \frac{4}{100} \). Simplify by dividing both top and bottom by their greatest common factor, 4, to get \( \frac{1}{25} \). Add the numerator and denominator: 1 + 25 = 26.

Exam Tip: Always reduce fractions to simplest form before adding the numerator and denominator - using the unreduced form will give a wrong answer.

 

Question 14. 13.572 correct to the tenth place is
(a) 10
(b) 13.57
(c) 14.5
(d) 13.6
Answer: (d) 13.6
In simple words: Look at the hundredths digit (7) to decide whether to round the tenths digit (5) up or down. Since 7 is at least 5, round up: 13.572 becomes 13.6.

Exam Tip: Always check the digit to the right of the place you are rounding - if it is 5 or more, round up; if it is less than 5, round down.

 

Question 15. 1 g is equal to
(a) 0.1 kg
(b) 0.01 kg
(c) 0.001 kg
(d) 0.0001 kg
Answer: (c) 0.001 kg
In simple words: There are 1000 grams in 1 kilogram. So 1 gram = \( \frac{1}{1000} \) kg = 0.001 kg.

Exam Tip: Memorise metric conversions: 1 kg = 1000 g. This lets you quickly convert any gram value to kilograms by moving the decimal point three places to the left.

 

Question 16. 2 km 7 m is equal to
(a) 2.7 km
(b) 2.07 km
(c) 2.007 km
(d) 2.0007 km
Answer: (c) 2.007 km
In simple words: There are 1000 metres in 1 kilometre. So 7 m = \( \frac{7}{1000} \) km = 0.007 km. Add this to 2 km: 2 + 0.007 = 2.007 km.

Exam Tip: When combining different metric units, always convert the smaller unit first, then add - never try to mix units directly in the answer.

 

Question 17. Among 2.34, 2.43, 2.344 and 2.4, the greatest number is
(a) 2.34
(b) 2.43
(c) 2.344
(d) 2.4
Answer: (b) 2.43
In simple words: Make all numbers have the same number of decimal places: 2.340, 2.430, 2.344, 2.400. Now compare: 2.430 is the largest because 43 is bigger than 34, 34, and 40 in the hundredths places.

Exam Tip: Always write decimals with the same number of decimal places before comparing - this makes it easy to see which is bigger at each position.

 

Question 18. 5.2 - 3.6 is equal to
(a) 0.16
(b) 2.6
(c) 0.26
(d) 1.6
Answer: (d) 1.6
In simple words: Line up the decimal points and subtract: 5.2 minus 3.6 gives 1.6. You can check: 3.6 + 1.6 = 5.2.

Exam Tip: Always align decimal points vertically before subtracting - misaligning them is the most common source of errors in decimal subtraction.

 

Question 19. A decimal number lying between 2.2 and 2.22 is
(a) 2.12
(b) 2.23
(c) 2.219
(d) 2.3
Answer: (c) 2.219
In simple words: Write both 2.2 and 2.22 with three decimal places: 2.200 and 2.220. Now check which option falls between them. 2.219 is between 2.200 and 2.220, so it is the answer.

Exam Tip: When finding a number between two decimals, use three decimal places to check all options clearly - this prevents mistakes.

 

Question 20. 0.023 lies between
(a) 0.2 and 0.3
(b) 0.02 and 0.03
(c) 0.029 and 0.03
(d) 0.026 and 0.024
Answer: (b) 0.02 and 0.03
In simple words: Look at 0.023 with three decimal places: it is 0.023. In comparison, 0.02 becomes 0.020 and 0.03 becomes 0.030. Clearly, 0.020 < 0.023 < 0.030, so 0.023 is between 0.02 and 0.03.

Exam Tip: Always write numbers with equal decimal places when checking if one lies between two others - this prevents errors from misreading place values.

 

Question 21. 0.7499 lies between
(a) 0.7 and 0.74
(b) 0.759 and 0.799
(c) 0.749 and 0.75
(d) 0.74992 and 0.75
Answer: (c) 0.749 and 0.75
In simple words: Write all numbers with four decimal places: 0.7490, 0.7499, 0.7500. Since 0.7490 < 0.7499 < 0.7500, the number 0.7499 lies between 0.749 and 0.75.

Exam Tip: Use the same number of decimal places throughout to compare - this is especially important when the numbers are very close to each other.

 

Question 22. Which of the following decimal numbers is the greatest?
(a) 0.182
(b) 0.038
(c) 0.219
(d) 0.291
Answer: (d) 0.291
In simple words: All numbers have three decimal places already. Look at the tenths place first: 0.291 has 2 in the tenths place, while 0.182 and 0.219 also have 1 or 2. But 0.291 has 9 in the hundredths place, which is larger than any of the others.

Exam Tip: Compare decimal numbers from left to right - start at the tenths place, then the hundredths, then the thousandths, until you find a place where one number is larger.

 

Question 23. Which of the following decimal numbers is the smallest?
(a) 0.108
(b) 1.08
(c) 0.801
(d) 0.81
Answer: (a) 0.108
In simple words: Write all as three-decimal numbers: 0.108, 1.080, 0.801, 0.810. The number 0.108 is smallest because it has 0 in the tenths place, while all others have at least 1 in the tenths place.

Exam Tip: When comparing decimals, the tenths place is the most important - a number with 0 tenths will always be smaller than one with 1 or more tenths.

 

Question 24. 0.003 × 0.2 is equal to
(a) 0.6
(b) 0.06
(c) 0.006
(d) 0.0006
Answer: (d) 0.0006
In simple words: Ignore the decimal points and multiply: 3 × 2 = 6. Now count the total decimal places in both numbers: 0.003 has 3 places and 0.2 has 1 place, giving 4 places total. Place the decimal 4 places from the right: 0.0006.

Exam Tip: To multiply decimals, ignore the points, multiply as whole numbers, then count decimal places in both factors to place the decimal in the answer.

 

Question 25. 0.45 ÷ 0.9 is equal to
(a) 50
(b) 5
(c) 0.5
(d) 0.05
Answer: (c) 0.5
In simple words: Multiply both the dividend and divisor by 10 to remove decimals: \( \frac{0.45 \times 10}{0.9 \times 10} = \frac{4.5}{9} \). Divide: 4.5 ÷ 9 = 0.5.

Exam Tip: To divide decimals, multiply both numbers by the same power of 10 to remove decimals from the divisor first - this makes the division much simpler.

 

Question 26. Statement I: In the number 34.17, the whole number part is 34 and the decimal part is 0.17.
Statement II: \( 34.17 = 3 \times 10 + 4 \times 1 + 1 \times \frac{1}{10} + 7 \times \frac{1}{100} \)
(a) Statement I is true but statement II is false.
(b) Statement I is false but statement II is true.
(c) Both Statement I and statement II are true.
(d) Both Statement I and statement II are false.
Answer: (c) Both Statement I and statement II are true.
In simple words: Statement I is correct - 34 is everything to the left of the decimal point, and 0.17 is everything to the right. Statement II shows the expanded form correctly: 3 tens, 4 ones, 1 tenth, and 7 hundredths add up to 34.17. Both are true.

Exam Tip: Understand both expanded form and standard form - questions often ask you to move between them, so knowing both formats helps you spot errors quickly.

 

Question 27. Statement I: \( \frac{1}{20} \) is a terminating fraction.
Statement II: The denominator of \( \frac{1}{20} \) is the 20th multiple of its numerator.
(a) Statement I is true but statement II is false.
(b) Statement I is false but statement II is true.
(c) Both Statement I and statement II are true.
(d) Both Statement I and statement II are false.
Answer: (c) Both Statement I and statement II are true.
In simple words: Statement I: Convert \( \frac{1}{20} \) by multiplying by \( \frac{5}{5} \) to get \( \frac{5}{100} = 0.05 \), which terminates. Statement II: The numerator is 1 and the denominator is 20. Since 20 = 20 × 1, the denominator is the 20th multiple of the numerator. Both statements are true.

Exam Tip: A terminating decimal is one that ends - fractions with denominators made only of 2s and 5s (like 20 = 4 × 5) always terminate.

 

Question 28. Statement I: According to a report from 2021, about 24.62% of our country's total land area is covered by forests. When we round this percentage to the nearest whole number, it represents \( \frac{1}{4} \) th of the total land area.
Statement II: 24.62 rounded off to the nearest whole number is 25. Also, 25% is \( \frac{1}{4} \).
(a) Statement I is true but statement II is false.
(b) Statement I is false but statement II is true.
(c) Both Statement I and statement II are true.
(d) Both Statement I and statement II are false.
Answer: (c) Both Statement I and statement II are true.
In simple words: Statement II: Look at 24.62 - the tenths digit is 6, which is at least 5, so round up to 25. Also, 25% = \( \frac{25}{100} = \frac{1}{4} \). So Statement II is true. Statement I: Since 24.62% rounds to 25%, and 25% = \( \frac{1}{4} \), Statement I is also true.

Exam Tip: Memorise key percentages: 25% = \( \frac{1}{4} \), 50% = \( \frac{1}{2} \), 75% = \( \frac{3}{4} \) - these appear often in exams and save time.

 

Question 29. Statement I: 42.500 and 42.123 are two unlike decimal numbers.
Statement II: 42.500 = 42.5
(a) Statement I is true but statement II is false.
(b) Statement I is false but statement II is true.
(c) Both Statement I and statement II are true.
(d) Both Statement I and statement II are false.
Answer: (b) Statement I is false but statement II is true.
In simple words: Statement I: Both 42.500 and 42.123 have three decimal places, so they are like decimals, not unlike. So Statement I is false. Statement II: Extra zeros at the end do not change the value of a decimal, so 42.500 = 42.5 is true.

Exam Tip: Like decimals have the same number of decimal places; unlike decimals have different numbers. Trailing zeros can always be added or removed without changing value.

 

Question 30. Statement I: The cost of 68.45 m of cloth at the rate of Rs.400 per metre is Rs.273.80 × 100
Statement II: 40.0 kL = 40,000.00 L
(a) Statement I is true but statement II is false.
(b) Statement I is false but statement II is true.
(c) Both Statement I and statement II are true.
(d) Both Statement I and statement II are false.
Answer: (c) Both Statement I and statement II are true.
In simple words: Statement I: Cost = 68.45 × 400. Working this out: 68.45 × 400 = 27,380 = Rs.273.80 × 100. Statement I is true. Statement II: 1 kL = 1000 L, so 40.0 kL = 40 × 1000 = 40,000 L = 40,000.00 L. Statement II is true.

Exam Tip: Metric conversions: 1 kL = 1000 L. Always multiply or divide by the right power of 10 when converting units.

 

Check Your Progress - Question 1. Convert the following decimal numbers into fractions (in lowest terms):
(i) 6.015
(ii) 0.876
(iii) 23.375
Answer:
(i) 6.015 has three decimal places, so it equals \( \frac{6015}{1000} \). Find the GCD of 6015 and 1000, which is 5. Divide: \( \frac{6015 \div 5}{1000 \div 5} = \frac{1203}{200} \).

(ii) 0.876 equals \( \frac{876}{1000} \). The GCD of 876 and 1000 is 4. Divide: \( \frac{876 \div 4}{1000 \div 4} = \frac{219}{250} \).

(iii) 23.375 equals \( \frac{23375}{1000} \). The GCD of 23375 and 1000 is 125. Divide: \( \frac{23375 \div 125}{1000 \div 125} = \frac{187}{8} \).
In simple words: Write each decimal as a fraction with 10, 100, or 1000 in the denominator based on how many decimal places it has. Then simplify by dividing the top and bottom by their greatest common factor.

Exam Tip: Always check if the fraction can be reduced further by finding the GCD - examiners expect the simplest form, and missing this step costs marks.

 

Question 1. Write the following fractions as decimal numbers:
(i) \( \frac{5}{8} \)
(ii) \( 2\frac{31}{125} \)
(iii) \( 13\frac{7}{40} \)
Answer: To convert fractions to decimals, multiply the numerator and denominator by a power of 10 to make the denominator equal to 10, 100, 1000, or another similar number.
(i) \( \frac{5}{8} = \frac{5 \times 125}{8 \times 125} = \frac{625}{1000} = 0.625 \)
(ii) \( 2\frac{31}{125} = 2 + \frac{31}{125} = 2 + \frac{31 \times 8}{125 \times 8} = 2 + \frac{248}{1000} = 2 + 0.248 = 2.248 \)
(iii) \( 13\frac{7}{40} = 13 + \frac{7}{40} = 13 + \frac{7 \times 25}{40 \times 25} = 13 + \frac{175}{1000} = 13 + 0.175 = 13.175 \)
In simple words: Change the bottom number to 10, 100, or 1000 by multiplying. Multiply the top by the same amount. Then write it as a decimal.

Exam Tip: Always identify which power of 10 the denominator can become - this tells you what to multiply by. Check your final decimal by reversing the process.

 

Question 2. Arrange the following decimal numbers in ascending order:
(i) 123.8, 74.205, 74.209, 7.4209
(ii) 85.01, 85.1, 85.001, 85.103
Answer:
(i) First, make all decimals the same length by adding zeros. Since the longest has 4 decimal places, we write:
123.8000, 74.2050, 74.2090, 7.4209
Now compare: 7.4209 is the smallest (starts with 7), then 74.2050 and 74.2090 (both start with 74), then 123.8000 (largest). Between 74.2050 and 74.2090, we compare the third decimal: 0 is less than 9.
In ascending order: \( 7.4209 < 74.205 < 74.209 < 123.8 \)
(ii) Make all decimals equal length. The longest has 3 decimal places:
85.010, 85.100, 85.001, 85.103
Comparing: First digits are all 85, so look at the tenths place. 0 is smallest, then 1. Check the remaining digits.
In ascending order: \( 85.001 < 85.01 < 85.1 < 85.103 \)
In simple words: Add zeros so all numbers have the same number of digits after the decimal point. Then line them up and compare left to right.

Exam Tip: Always count the maximum decimal places first and add trailing zeros to match - this prevents comparison errors. Compare one place at a time from left to right.

 

Question 3. Arrange the following decimal numbers in descending order:
(i) 6.45, 4.65, 6.405, 64.5
(ii) 73.5, 35.7, 7.35, 7.035
Answer:
(i) Write all with equal decimal places. The longest has 3 decimal places:
6.450, 4.650, 6.405, 64.500
The largest digit in the ones place is 6 (from 64.5), so 64.500 comes first. Next are the 6.xxx numbers: 6.450 and 6.405. Since 4 is greater than 4 in the tenths place of both, check the hundredths: 5 is greater than 0. The smallest is 4.650.
In descending order: \( 64.5 > 6.45 > 6.405 > 4.65 \)
(ii) Write with equal decimal places. The longest has 3 decimal places:
73.500, 35.700, 7.350, 7.035
Largest ones digit is 7 (from 73.5), so 73.500 first. Next is 35.700. Then 7.350 and 7.035; the tenths digits are 3 and 0, so 7.350 is larger.
In descending order: \( 73.5 > 35.7 > 7.35 > 7.035 \)
In simple words: Add zeros to make the same decimal length. Put the biggest number first and work down to the smallest.

Exam Tip: For descending order, start by comparing the whole number part first - it's the quickest way to spot the largest and smallest values.

 

Question 4. If the school bags of Garima and Nakul weigh 5.2 kg and 4.832 kg respectively, find (i) the total weight (ii) the difference in weight of the bags
Answer: Convert both to like decimals with three decimal places: 5.200 kg and 4.832 kg.
(i) Total weight = 5.200 + 4.832
\[ \begin{array}{r} 5.200 \\ + 4.832 \\ \hline 10.032 \end{array} \]
The combined weight is 10.032 kg.
(ii) Difference in weight = 5.200 - 4.832
\[ \begin{array}{r} 5.200 \\ - 4.832 \\ \hline 0.368 \end{array} \]
The difference is 0.368 kg.
In simple words: Add or subtract by lining up the decimal points. The bag weights become the same length by adding zeros, then you can add or subtract easily.

Exam Tip: Always align decimal points vertically before adding or subtracting - misalignment is a common error that changes the answer entirely.

 

Question 5. Evaluate the following:
(i) 31.42 - 17.853 - 6.43
(ii) 13.01 - 5.428 - 3.703 + 2.99
Answer:
(i) Write each number with three decimal places: 31.420, 17.853, 6.430
Rewrite as: 31.420 - (17.853 + 6.430)
First, add: 17.853 + 6.430 = 24.283
Then, subtract: 31.420 - 24.283 = 7.137
(ii) Write each number with three decimal places: 13.010, 5.428, 3.703, 2.990
Rewrite as: (13.010 + 2.990) - (5.428 + 3.703)
Add the positive terms: 13.010 + 2.990 = 16.000
Add the negative terms: 5.428 + 3.703 = 9.131
Subtract: 16.000 - 9.131 = 6.869
In simple words: Make all numbers have the same decimal places. Group additions together and subtractions together. Calculate each group, then find the final answer.

Exam Tip: Grouping additions and subtractions (rather than calculating left to right) reduces errors and makes working clearer to mark.

 

Question 6. By how much does the sum of 15.453 and 31.647 exceed the sum of 18.47 and 19.506?
Answer: Calculate the first sum: 15.453 + 31.647
\[ \begin{array}{r} 15.453 \\ + 31.647 \\ \hline 47.100 \end{array} \]
Calculate the second sum. Convert to like decimals: 18.470 and 19.506
\[ \begin{array}{r} 18.470 \\ + 19.506 \\ \hline 37.976 \end{array} \]
Find the difference: 47.100 - 37.976
\[ \begin{array}{r} 47.100 \\ - 37.976 \\ \hline 9.124 \end{array} \]
The first sum exceeds the second sum by 9.124.
In simple words: Add the first two numbers to get one total. Add the next two numbers to get another total. Then subtract the smaller total from the bigger total.

Exam Tip: Break the problem into steps: find both sums first, then find their difference. Show each calculation separately for clarity.

 

Question 7. Convert 2435 m to km and express the result as a mixed fraction.
Answer: Recall that 1 m = \( \frac{1}{1000} \) km.
So, 2435 m = \( \frac{2435}{1000} \) km = 2.435 km
To express as a fraction in lowest terms, find the GCD of 2435 and 1000. Both are divisible by 5:
\( \frac{2435}{1000} = \frac{487}{200} \)
Convert to a mixed fraction: \( \frac{487}{200} = 2\frac{87}{200} \)
Therefore, 2435 m = 2.435 km = \( 2\frac{87}{200} \) km
In simple words: One metre is one-thousandth of a kilometre. So divide 2435 by 1000 to get 2.435. Then reduce the fraction and change it to a mixed number.

Exam Tip: When converting units, always recall the conversion factor. Here, 1000 m = 1 km, so divide by 1000. Always simplify fractions to lowest terms before presenting.

 

Question 8. Namita travels 20 km 50 m every day. Out of this she travels 10 km 200 m by bus and the rest by auto. How much distance does she travel by auto?
Answer: Total distance = 20 km 50 m
Convert to decimal: 20 km + \( \frac{50}{1000} \) km = 20 km + 0.050 km = 20.050 km
Distance by bus = 10 km 200 m
Convert to decimal: 10 km + \( \frac{200}{1000} \) km = 10 km + 0.200 km = 10.200 km
Distance by auto = Total - By bus = 20.050 - 10.200
\[ \begin{array}{r} 20.050 \\ - 10.200 \\ \hline 9.850 \end{array} \]
Namita travels 9.850 km or 9 km 850 m by auto.
In simple words: Change metres to kilometres by dividing by 1000 and adding to the km part as a decimal. Then subtract the bus distance from the total distance.

Exam Tip: For mixed unit problems (km and m), convert everything to the same unit first. Remember: 1000 m = 1 km, so move the decimal point 3 places.

 

Question 9. Asha purchased 5 kg 400 g rice, 2 kg 20 g sugar and 10 kg 850 g flour. Find the total weight of his purchases.
Answer: Convert each to kilograms:
Weight of rice = 5 kg 400 g = 5 kg + \( \frac{400}{1000} \) kg = 5 kg + 0.400 kg = 5.400 kg
Weight of sugar = 2 kg 20 g = 2 kg + \( \frac{20}{1000} \) kg = 2 kg + 0.020 kg = 2.020 kg
Weight of flour = 10 kg 850 g = 10 kg + \( \frac{850}{1000} \) kg = 10 kg + 0.850 kg = 10.850 kg
Total weight = 5.400 + 2.020 + 10.850
\[ \begin{array}{r} 5.400 \\ 2.020 \\ + 10.850 \\ \hline 18.270 \end{array} \]
The total weight is 18.270 kg or 18 kg 270 g.
In simple words: Change grams to kilograms by dividing by 1000. Write each item as a decimal in kilograms. Then add all three together.

Exam Tip: Always convert all measurements to the same unit (in this case kg) before adding. Line up decimal points carefully to avoid arithmetic mistakes.

 

Question 10. 1 kg of pure milk contains 0.263 kg of fat. How much fat is there in 15.5 kg of milk?
Answer: Fat in 1 kg = 0.263 kg
Fat in 15.5 kg = 0.263 × 15.5
Ignore decimal points and multiply: 263 × 155 = 40765
Count the total decimal places: 0.263 has 3 decimal places, 15.5 has 1 decimal place. Total = 3 + 1 = 4 decimal places.
Therefore: 0.263 × 15.5 = 4.0765
There is 4.0765 kg of fat in 15.5 kg of milk.
In simple words: Multiply the two numbers ignoring the decimals first. Then count how many decimal places there are in both numbers added together. Place the decimal that many places from the right in your answer.

Exam Tip: For decimal multiplication, the key rule is: count total decimal places in both factors, then place the decimal that many digits from the right in the product.

 

Question 11. The product of two numbers is 15.275. If one number is 4.7, find the other.
Answer: If the product of two numbers = 15.275 and one number = 4.7, then the other = \( \frac{15.275}{4.7} \)
To simplify, multiply both numerator and denominator by 10:
\( \frac{15.275 \times 10}{4.7 \times 10} = \frac{152.75}{47} \)
Perform the division:
\[ \begin{array}{c|l} & 3.25 \\ \hline 47 & 152.75 \\ & -141 \\ \hline & 117 \\ & -94 \\ \hline & 235 \\ & -235 \\ \hline & 0 \end{array} \]
The other number is 3.25.
In simple words: To find the missing number, divide the product by the number you know. Use long division to get your answer.

Exam Tip: Always verify by multiplying back: 4.7 × 3.25 should equal 15.275. This check confirms your answer is correct.

 

Question 12. On her birthday, Ayushi is taking her 5 friends to a movie and treats them with cold drink. The cost of a ticket is Rs.150 and a cold drink costs Rs.28.50. How much Ayushi has to spend?
Answer: Total number of persons = Ayushi + 5 friends = 6
Cost of one ticket = Rs.150
Cost of 6 tickets = 150 × 6 = Rs.900
Cost of one cold drink = Rs.28.50
Cost of 6 cold drinks = 28.50 × 6
Multiply: 28.50 × 6 = 171.00 = Rs.171
Total amount to spend = 900 + 171
\[ \begin{array}{r} 900.00 \\ + 171.00 \\ \hline 1071.00 \end{array} \]
Ayushi has to spend Rs.1,071.
In simple words: Count how many people need tickets and cold drinks. Multiply each price by the number of people. Then add the two totals.

Exam Tip: When multiplying money, treat it like any decimal number. After finding both costs, add them carefully by aligning decimal points.

 

Question 13. Write digits in the boxes of the number: 3⬜6⬜.8⬜⬜ to obtain (i) greatest possible number (ii) smallest possible number. Repetition of digits in a number is not allowed.
Answer: The number has the form 3⬜6⬜.8⬜⬜, where 3, 6, and 8 are already used. We must fill four boxes with four different digits from {0, 1, 2, 4, 5, 7, 9}.
(i) For the greatest number: Use the four largest available digits in decreasing order from left to right. The largest available are 9, 7, 5, 4. Place them as: 3967.854
Greatest possible number = 3967.854
(ii) For the smallest number: Use the four smallest available digits in increasing order from left to right. The smallest available are 0, 1, 2, 4. Place them as: 3061.824
Smallest possible number = 3061.824
In simple words: For the biggest number, pick the largest unused digits and put them in the blanks from left to right. For the smallest, pick the smallest unused digits and fill the blanks from left to right.

Exam Tip: Remember that the leftmost blank affects the number more than blanks further right. Always fill from left to right with largest digits for maximum, smallest for minimum.

 

Question 14. Arrange the following numbers in descending order: \( 5\frac{3}{4} \), 5.721, \( 5\frac{7}{8} \), \( 5\frac{17}{25} \), 5.693
Answer: Convert each mixed fraction to a decimal:
\( 5\frac{3}{4} = 5 + \frac{3}{4} = 5 + \frac{3 \times 25}{4 \times 25} = 5 + \frac{75}{100} = 5 + 0.75 = 5.75 \)
\( 5\frac{7}{8} = 5 + \frac{7}{8} = 5 + \frac{7 \times 125}{8 \times 125} = 5 + \frac{875}{1000} = 5 + 0.875 = 5.875 \)
\( 5\frac{17}{25} = 5 + \frac{17}{25} = 5 + \frac{17 \times 4}{25 \times 4} = 5 + \frac{68}{100} = 5 + 0.68 = 5.68 \)
All numbers in decimal form: 5.75, 5.721, 5.875, 5.68, 5.693
Convert to like decimals with three decimal places: 5.750, 5.721, 5.875, 5.680, 5.693
Compare: 5.875 has the largest hundredths digit (8), so it comes first. Next is 5.750. Then 5.721. Then 5.693. Finally 5.680 (smallest).
In descending order: \( 5\frac{7}{8} > 5\frac{3}{4} > 5.721 > 5.693 > 5\frac{17}{25} \)
In simple words: Change all the mixed numbers to decimals. Add zeros to make them all the same length. Then arrange them from biggest to smallest.

Exam Tip: When a question mixes fractions and decimals, convert all to the same form (decimal is usually easiest). This prevents comparison mistakes.

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