RS Aggarwal Solutions for Class 6 Chapter 7 Decimals

Exercise 7.1

Question 1. Write the following decimals in the place value table:
(i) 52.5
(ii) 12.57
(iii) 15.05
(iv) 74.059
(v) 0.503
Answer:

Question 2. The decimals shown in the above place value table can be written as follows:
(i) 307.12
(ii) 9543.025
(iii) 12.503
Answer:

Question 3. Write each of the following as decimals:
(i) One hundred seventy five and four hundredths.
(ii) Zero and twenty one hundredths
(iii) Nine and four thousandths
(iv) Zero and four hundred fifty nine thousandths
Answer:
(i) 175.04
(ii) 0.21
(iii) 0.459

Question 4. Convert each of the following to decimal form:
(i) \( 65 + \frac{2}{10} + \frac{7}{100} \)
(ii) \( 45 + \frac{9}{100} \)
(iii) \( 88 + \frac{5}{10} + \frac{2}{1000} \)
(iv) \( \frac{3}{10} + \frac{7}{1000} \)
Answer:
(i) There are 6 tens, 5 ones and 7 hundredths. Thus, the decimal number equals 65.27
(ii) There are 4 tens, 5 ones and 9 hundredths. Thus, the decimal number equals 45.09
(iii) There are 8 tens, 8 ones, 5 tenths and 2 thousandths. Thus, the decimal number equals 88.502
(iv) There are 3 tenths, 5 ones and 7 hundredths. Thus, the decimal number equals 0.307

Question 5. Write each of the following as decimals:
(i) Five and four tenths
(ii) Twelve four hundredths
(iii) Nine and seven hundred five thousandths
(iv) Zero point five two six
(v) Forty seven and six thousandths
(vi) Eight thousandths
(vii) Nineteen and nineteen hundredths
Answer:
(i) 5 + 4/10 = 5.4
(ii) 12 + 4/100 = 12.4
(iii) 9 + 705/1000 = 9.705
(iv) 0.526
(v) 47 + 6/1000 = 47.006
(vi) 8/1000 = 0.008
(vii) 19 + 13/100 = 19.19

Exercise 7.2

Question 1. Convert each of the following to decimal form:
(i) Three tenths
(ii) Two ones and five tenths
(iii) Thirty and one tenths
(iv) Twenty two and six tenths
(v) One hundred, two ones and three tenths
Answer:
(i) 3/10 = 0.3
(ii) 2 + 5/10 = 2.5
(iii) 30 + 1/10 = 30.1
(iv) 22 + 6/10 = 22.6
(v) 100 + 2 + 3/10 = 102.3

Question 2. Convert each of the following to decimal form:
(i) \( 30 + 6 + \frac{2}{10} \)
(ii) \( 700 + 5 + \frac{7}{10} \)
(iii) \( 100 + 60 + 5 + \frac{1}{10} \)
(iv) \( 200 + 70 + 9 + \frac{5}{10} \)
Answer:
(i) There are 3 tens, 6 ones and 2 tenths. Thus, the decimal equals 36.2
(ii) There are 7 hundreds, 5 ones and 7 tenths. Thus the decimal equals 705.7
(iii) There are 2 hundreds, 6 tens, 5 ones and 1 tenths. Thus the decimal equals 265.1
(iv) There are 2 hundreds, 7 tens, 9 ones and 5 tenths. Thus, the decimal equals 279.5

Question 3. Write each of the following as a decimal:
(i) \( \frac{22}{10} \)
(ii) \( \frac{3}{2} \)
(iii) \( \frac{2}{5} \)
Answer:
(i) Since the denominator is ten, the decimal is 2.2
(ii) Adjusting the denominator to 10, we have \( \frac{3}{2} \) \( \frac{3(5)}{2(5)} = \frac{15}{10} = 1.5 \)
(iii) Adjusting the denominator to 10, we have \( \frac{2}{5} \) \( \frac{2(2)}{5(2)} = \frac{4}{10} = 0.4 \)

Question 4. Express each of the following as a decimal:
(i) \( \frac{4}{0}25 \)
(ii) \( \frac{3}{9}210 \)
(iii) \( \frac{4}{3}5 \)
(iv) \( \frac{2}{5}12 \)
Answer:
(i) To express as a decimal, we must make the denominator 10 by multiplying it by a number. But, to keep the fraction's value the same, we should also multiply the numerator by the same number. Thus, we get

\( = 40 + \frac{2(25)}{2} = 40 + \frac{4}{10} = 40.4 \)

(ii) \( \frac{39}{2}10 = 39 + \frac{2}{10} \)

Here, the denominator is 10.

Thus, the decimal is 39.2

(iii) \( \frac{4}{3}5 = 4 + \frac{3}{5} \)

To express as a decimal, we must make the denominator by 10 by multiplying it by a number. but, to keep the fraction's value the same, we should also multiply the numerator by the same number. Thus we get,

\( = 4 + \frac{3(3)(25)}{2} \)

\( = 4 + \frac{6}{10} = 4.6 \)

(iv) \( \frac{25}{1}2 = 25 + \frac{1}{2} \)

To express as a decimal, we must make the denominator 10 by multiplying it by a number. But, to keep the fraction's value the same, we should also multiply the numerator by the same number. Thus, we get

\( = 25 + \frac{1(52)}{5} \)

\( = 25 + \frac{5}{10} = 25.5 \)

Question 5. Express each of the following as a fraction:
(i) 3.8
(ii) 21.2
(iii) 6.4
(iv) 1
Answer:
(i) 3.8
\( = 3 + 8 \) tenths
\( = 3 + \frac{8}{10} \)
\( = \frac{3(1010) + 8}{10} = \frac{3010 + 8}{10} = \frac{3810} = \frac{19}{5} \)

(ii) 21.2
\( = 21 + 2 \) tenths
\( = 21 + \frac{2}{10} = 21(1010) + \frac{2}{10} = \frac{21010 + 2}{10} = \frac{21210} = \frac{1065}{10} \)

(iii) 6.4
\( = 6 + 4 \) tenths
\( = 6 + \frac{4}{10} \)
\( = \frac{6(1010) + 4}{10} = \frac{6010 + 4}{10} = \frac{325}{10} \)

(iv) 1

Since the only number after the decimal is 0, the fraction is 1

Question 6. Represent the following number on the number line.
(i) 0.2
(ii) 1.9
(iii) 1.1
(iv) 2.5
Answer:
(i)


(ii)


(iii)


(iv)

Question 7. Identify which two whole numbers each decimal lies between:
(i) 0.8 is between the two whole numbers 0 and 1
(ii) 5.1 is between the two whole number 5 and 6.
(iii) 2.6 is between 2 and 3
(iv) 6.4 is between 6 and 7
Answer:
(i) Since 0.8 is 8 units from 0 and 2 units from 1, it is nearer to 1
(ii) Since 5.1 is 1 unit from 5 and 9 units from 6, it is nearer to 5
(iii) Since 2.6 is 6 units from 2 and 4 units from 3, it is nearer to 3
(iv) Since 6.4 is 4 units from 6 and 6 units from 7, it is nearer to 6

9.0 is itself a whole number, that is 9

4.9 is between 4 and 5

Since 4.9 is 9 units from 4 and 1 unit from 5, it is nearer to 5

Question 8. Write the decimal number represented by the points on the given number line A, B, C, D

Answer: A) 0.8, since A is at the eighth place between 0 and 1
B) 1.3, since B is at the third place between 1 and 2
C) 1.9, since C is at the ninth place between 1 and 2
D) 2.6, since D is at the sixth place between 2 and 3

Disclaimer: the image given in the book is not consistent; as the number of periods between 0 and 1 is ten but the number of periods between 1 and 2 are seven. So, ignoring the position of the given numbers 1, 2 and 3, it has been assumed that there are ten periods between every two consecutive numbers starting from the first point taken as zero.

Exercise 7.3

Question 1. Write each of the following as decimals:
(i) Five hundred twenty five and forty hundredths
(ii) Twelve and thirty five thousandths
(iii) Fifteen and seventeen thousandths
(iv) Eighty eight and forty eight hundredths
Answer:
(i) 525 + 40/100 = 525.40
(ii) 12 + 35/1000 = 12.035
(iii) 15 + 17/1000 = 15.017
(iv) 88 + 48/100 = 88.48

Question 2. Convert each of the following to decimal form:
(i) \( 137 + \frac{5}{100} \)
(ii) \( \frac{20 + 9 + 4}{100} \)
Answer:
(i) There is 1 hundred, 3 tens, 7 ones and 5 hundredths. Thus, the decimal is 137.05
(ii) There are 2 tens, 9 ones and 4 hundredths. Thus, the decimal is 29.04

Question 3. Write each of the following as a decimal:
(i) \( \frac{8}{100} \)
(ii) \( \frac{300}{1000} \)
(iii) \( \frac{18}{1000} \)
(iv) \( \frac{208}{100} \)
Answer:
(i) There are 8 hundredths. Thus, the decimal is 0.08
(ii) When reduced to lowest terms, the fraction equals 3/10. There are 3 tenths. Thus, the decimal is 0.3
(iii) There are eighteen thousandths. Thus the decimal is 0.018
(iv) \( \frac{208}{100} = \frac{200}{100} + \frac{8}{100} = 2 + \frac{8}{100} \)

There are 2 and 8 hundredths. Thus, the decimal is 2.08

888/1000

\( \frac{888}{1000} = \frac{800}{1000} + \frac{80}{1000} + \frac{8}{1000} = \frac{8}{10} + \frac{8}{100} + \frac{8}{1000} \)

There are 8 tenths, 8 hundredths and 8 thousandths. Thus, the decimal is 0.888

Question 4. Express each of the following as a decimal:
(i) \( 12\frac{1}{4} \)
(ii) \( 7\frac{1}{8} \)
(iii) \( 5\frac{1}{20} \)
Answer:
(i) \( 12\frac{1}{4} = 12 + \frac{1}{4} = 12 + \frac{25}{4(25)} = 12 + \frac{25}{100} = 12.25 \)

(ii) \( 7 + \frac{1(1258)}{(125)} = 7 + \frac{125}{1000} = 7.125 \)

(iii) \( 5 + \frac{1(520)}{(5)} = 5 + \frac{5}{10} = 5.05 \)

Question 5. Express each of the following as a fraction:
(i) 0.04
(ii) \( 7\frac{1}{8} \)
(iii) \( 5\frac{1}{20} \)
(iv) 1.20
(v) 17.38
Answer:
(i)

Question 6. (i) 2 tens, 9 ones, 4 tenths and 1 hundredths.
Answer: We have 2 tens, 9 ones, 4 tenths, and 1 hundredth. So the decimal representation is 29.41
In simple words: Add up the place values: 20 + 9 + 0.4 + 0.01, which gives you 29.41.

Exam Tip: Always write down each place value separately, then add them together. Make sure you position the decimal point correctly - tenths and hundredths go to the right of it.

Question 6. (ii) 3 tens, we have3 tens, 4 tenths, 8 hundredths and 3 thousandths.
Answer: We have 3 tens, 4 tenths, 8 hundredths, and 3 thousandths. So the decimal representation is 30.483
In simple words: Combine the values: 30 + 0.4 + 0.08 + 0.003, which equals 30.483.

Exam Tip: Count carefully how many places after the decimal point you need - each place value (tenths, hundredths, thousandths) has its own position.

Question 6. (iii) 1 hundred, 3 tens, 7 ones and 5 hundredths.
Answer: We have 1 hundred, 3 tens, 7 ones, and 5 hundredths. So the decimal representation is 137.05
In simple words: Put together: 100 + 30 + 7 + 0.05, which gives 137.05.

Exam Tip: Watch for missing place values - if a position is skipped (like tenths), put a 0 in that spot.

Question 6. (iv) 7 tenths, 6 hundredths and 4 thousandths.
Answer: We have 7 tenths, 6 hundredths, and 4 thousandths. So the decimal representation is 0.764
In simple words: Combine: 0.7 + 0.06 + 0.004, which equals 0.764.

Exam Tip: When there are no whole numbers, always put a 0 before the decimal point - never write just .764.

Question 6. (v) 2 tens, 3 ones, 2 tenths and 6 thousandths.
Answer: We have 2 tens, 3 ones, 2 tenths, and 6 thousandths. So the decimal representation is 23.206
In simple words: Collect: 20 + 3 + 0.2 + 0.006, which gives 23.206.

Exam Tip: Notice the hundredths place is 0 - fill it in even though it's not mentioned, because it sits between tenths and thousandths.

Question 6. (vi) 7 hundreds, 2 tens, 5 ones and 9 hundredths.
Answer: We have 7 hundreds, 2 tens, 5 ones, and 9 hundredths. So the decimal representation is 725.09
In simple words: Add: 700 + 20 + 5 + 0.09, which equals 725.09.

Exam Tip: Remember to put 0 in the tenths place when it's not given but hundredths are - order matters for decimals.

Exercise 7.4

Question 1. Express the following fractions as decimals:
(i) \( \frac{23}{10} \)
(ii) \( \frac{139}{100} \)
(iii) \( \frac{4375}{1000} \)
(iv) \( 12\frac{1}{2} \)
(v) \( 75\frac{1}{4} \)
(vi) \( 25\frac{1}{8} \)
(vii) \( 18\frac{3}{24} \)
(viii) \( \frac{3}{9}735 \)
(ix) \( \frac{1}{5}125 \)
(x) \( \frac{111}{250} \)
Answer:
(i) \( \frac{23}{10} = 2.3 \)
(ii) \( \frac{139}{100} = 1.39 \)
(iii) \( \frac{4375}{1000} = 4.375 \)
(iv) \( 12\frac{1}{2} = 12.5 \)
(v) \( 75\frac{1}{4} = 75.25 \)
(vi) \( 25\frac{1}{8} = 25.125 \)
(vii) \( 18\frac{3}{24} = 18.125 \)
(viii) \( \frac{3}{9}735 = 39.2 \)
(ix) \( \frac{1}{5}125 = 15.04 \)
(x) \( \frac{111}{250} = 0.444 \)
In simple words: Divide the numerator by the denominator. For fractions with denominators 10, 100, or 1000, count the zeros and move the decimal point that many places to the left in the numerator.

Exam Tip: Always check your decimal placement by counting zeros in the denominator - one zero means one decimal place, two zeros means two decimal places, and so on.

Question 2. Express the following decimals as fractions in the lowest form:
(i) 0.5
(ii) 2.5
(iii) 0.60
(iv) 0.18
(v) 5.25
(vi) 7.125
(vii) 15.004
(viii) 20.375
(ix) 600.75
Answer:
(i) \( 0.5 = \frac{5}{10} = \frac{1}{2} \)
(ii) \( 2.5 = \frac{25}{10} = \frac{5}{2} \)
(iii) \( 0.60 = \frac{60}{100} = \frac{3}{5} \)
(iv) \( 0.18 = \frac{18}{100} = \frac{9}{50} \)
(v) \( 5.25 = \frac{525}{100} = \frac{21}{4} \)
(vi) \( 7.125 = \frac{7125}{1000} = \frac{201}{8} \)
(vii) \( 15.004 = \frac{15004}{1000} = \frac{3751}{250} \)
(viii) \( 20.375 = \frac{20375}{1000} = \frac{163}{8} \)
(ix) \( 600.75 = \frac{60075}{100} = \frac{2403}{14} \)
In simple words: Write the decimal as a fraction with the correct denominator (10, 100, 1000, etc.), then reduce it by finding the highest common factor of the top and bottom numbers.

Exam Tip: Always simplify to the lowest terms - check if both numerator and denominator can be divided by the same number until they cannot be reduced further.

Exercise 7.5

Question 1. Fill in the blanks by using > or < to complete the following.
(i) 25.35 > 8.47
(ii) 20.695 < 20.93
(iii) 0.39 < 0.72
(iv) 0.109 < 0.83
(v) 0.236 > 0.201
(vi) 0.93 < 0.99
Answer:
(i) Looking at the whole part: 25 is bigger than 8, so the statement is correct as shown.
(ii) The whole parts match at 20. Comparing tenths: 6 is less than 9, so the statement is correct as shown.
(iii) Both whole parts are 0. Comparing tenths: 3 is less than 7, so the statement is correct as shown.
(iv) Both whole parts are 0. Comparing tenths: 1 is less than 8, so the statement is correct as shown.
(v) Both whole parts are 0. Both tenths digits match at 2. Comparing hundredths: 3 is greater than 0, so the statement is correct as shown.
(vi) Both whole parts are 0. Both tenths digits match at 9. Comparing hundredths: 3 is less than 9, so the statement is correct as shown.
In simple words: Start by comparing the whole number parts. If they're equal, look at the tenths place. If those are equal, check the hundredths, and keep moving right until you find a difference.

Exam Tip: Always compare place by place from left to right - don't try to compare all the digits at once, which can lead to errors.

Question 2. Which is greater? Give reason for your answer.
(i) 1.008 < 1.800
(ii) 3.3 = 3.300
(iii) 5.64 > 5.603
(iv) 1.431 < 1.439
(v) 0.5 > 0.05
Answer:
(i) The whole parts are equal at 1. Looking at tenths: 0 is less than 8, so 1.800 is the larger number. The reason is that 1.800 has 8 tenths while 1.008 has 0 tenths.
(ii) Both numbers are the same. The trailing zeros after 3.3 do not change the value, so 3.3 = 3.300.
(iii) The whole parts are the same at 5. Tenths digits match at 6. Comparing hundredths: 4 is greater than 0, so 5.64 is larger. The reason is the difference in the hundredths place (4 versus 0).
(iv) The whole parts and tenths digits match at 1 and 4. Comparing hundredths: 3 is less than 3... wait, they're equal. Looking at thousandths: 1 is less than 9, so 1.439 is larger. The reason is the thousands place differs (1 versus 9).
(v) Both whole parts are 0. Comparing tenths: 5 is greater than 0, so 0.5 is the bigger number. The reason is that 0.5 has 5 tenths while 0.05 has only 0 tenths (the 5 is in the hundredths place).
In simple words: Always align the numbers by their decimal points and check digit by digit from left to right. The first position where digits differ tells you which number is bigger.

Exam Tip: Be careful with zeros - trailing zeros don't change value, but zeros in the middle of the number (like between the decimal point and a non-zero digit) do matter.

Exercise 7.6

Question 1. Express as rupees (Rs) using decimals
(i) 15 paisa
(ii) 5 paisa
(iii) 350 paisa
(iv) 2 rupees 60 paisa
Answer:
(i) We know that 100 paisa = Rs. 1. So 1 paisa = Rs. 1/100. Therefore, 15 paisa = 15/100 = Rs. 0.15
(ii) We know that 100 paisa = Rs. 1. So 1 paisa = Rs. 1/100. Therefore, 5 paisa = 5/100 = Rs. 0.05
(iii) We know that 100 paisa = Rs. 1. So 1 paisa = Rs. 1/100. Therefore, 350 paisa = 350/100 = Rs. 3.50
(iv) We know that 100 paisa = Rs. 1. So 1 paisa = Rs. 1/100. Therefore, 2 rupees 60 paisa = 2 + 60/100 = Rs. 2.60
In simple words: Since 100 paisa equals 1 rupee, divide the number of paisa by 100 to get the rupee value in decimals. If you have rupees and paisa together, add them after converting paisa to rupees.

Exam Tip: Remember the conversion: 100 paisa = 1 rupee. Always place the decimal point correctly - two digits after it for paisa conversions.

Question 2. Express as metres (m) using decimals
(i) 15 cm
(ii) 8 cm
(iii) 135 cm
(iv) 3 m 65 cm
Answer:
(i) We know that 100 cm = 1 m. So 1 cm = 1/100 m. Therefore, 15 cm = \( 15 \left(\frac{1}{100} m\right) = \frac{15}{100} m = 0.15 m \)
(ii) We know that 100 cm = 1 m. So 1 cm = 1/100 m. Therefore, 8 cm = \( 8 \left(\frac{1}{100} m\right) = \frac{8}{100} m = 0.08 m \)
(iii) We know that 100 cm = 1 m. So 1 cm = 1/100 m. Therefore, 135 cm = \( 135 \left(\frac{1}{100} m\right) = \frac{135}{100} m = 1.35 m \)
(iv) We know that 100 cm = 1 m. So 1 cm = 1/100 m. Therefore, 3 m 65 cm = \( 3 + \frac{65}{100} m = 3.65 m \)
In simple words: Since 100 centimetres make 1 metre, split any centimetre amount by 100 to convert it. If metres and centimetres are given together, add the metre values after changing centimetres.

Exam Tip: The conversion is straightforward - always divide centimetres by 100. Make sure the decimal point goes in the right position (two places from the right).

Question 3. Express as centimetre (cm) using decimals
(i) 5 mm
(ii) 60 mm
(iii) 175 mm
(iv) 4 cm 5 mm
Answer:
(i) We know that 10 mm = 1 cm. So 1 mm = 1/10 cm. Therefore, 5 mm = 5/10 cm = 0.5 cm
(ii) We know that 10 mm = 1 cm. So 1 mm = 1/10 cm. Therefore, 60 mm = 60/10 cm = 6 cm
(iii) We know that 10 mm = 1 cm. So 1 mm = 1/10 cm. Therefore, 175 mm = 175/10 cm = 17.5 cm
(iv) We know that 10 mm = 1 cm. So 1 mm = 1/10 cm. Therefore, 4 cm 5 mm = 4 + 5/10 = 4.5 cm
In simple words: Since 10 millimetres equal 1 centimetre, divide the millimetre amount by 10 to find the centimetre value. When both are given, add the centimetre values once you've converted.

Exam Tip: The ratio is 10 millimetres per 1 centimetre - divide by 10 and the decimal goes one place from the right.

Question 4. Express as kilogram (km) using decimals
(i) 5 m
(ii) 55 m
(iii) 555 m
(iv) 5555 m
(v) 15 km 35 m
Answer:
(i) We know that 1000 m = 1 km. So 1 m = 1/1000 km. Therefore, 5 m = 5/1000 km = 0.005 km
(ii) We know that 1000 m = 1 km. So 1 m = 1/1000 km. Therefore, 55 m = 55/1000 km = 0.055 km
(iii) We know that 1000 m = 1 km. So 1 m = 1/1000 km. Therefore, 555 m = 555/1000 km = 0.555 km
(iv) We know that 1000 m = 1 km. So 1 m = 1/1000 km. Therefore, 5555 m = 5555/1000 km = 5.555 km
(v) For mixed measurements: 15 km 35 m = 15 + 35/1000 km = 15.035 km
In simple words: Because 1000 metres equal 1 kilometre, divide the metre total by 1000. When both kilometres and metres are present, put the metres portion as a decimal after finding its equivalent fraction.

Exam Tip: The key relationship is 1000 metres = 1 kilometre. Dividing by 1000 shifts the decimal point three places to the left.

Question 5. Express each of the following without using decimals
(i) 8g
(ii) 150 g
(iii) 2750 g
(iv) 5 kg 750 g
(v) 36 kg 50 g
Answer:
(i) Since 1000 g equals 1 kg, we know that 1 g is 1/1000 of a kg, which is 0.001 kg. Therefore, 8 g becomes 8/1000 kg, giving us 0.008 kg.
(ii) Given that 1000 g equals 1 kg, 1 g is 1/1000 kg or 0.001 kg. So, 150 g is 150/1000 kg, which equals 0.150 kg.
(iii) Since 1000 g equals 1 kg, 1 g is 1/1000 kg or 0.001 kg. Thus, 2750 g becomes 2750/1000 kg, which equals 2.750 kg.
(iv) Because 1000 g equals 1 kg, 1 g is 1/1000 kg or 0.001 kg. So, 5 kg 750 g is 5 plus 750/1000 kg, which equals 5.750 kg.
(v) Given that 1000 g equals 1 kg, 1 g is 1/1000 kg or 0.001 kg. Thus, 36 kg 50 g is 36 plus 50/1000 kg, which equals 36.050 kg.

Exam Tip: Always identify the conversion factor first (like 1000 g = 1 kg), then divide the gram amount by 1000 to get kilograms. Add any whole kilograms at the end.

Question 6. Express each of the following without using decimals
(i) Rs.5.25
(ii) 8.354 kg
(iii) 3.05 km
(iv) 7.54 m
(v) 15.005 kg
(vi) 12.05 m
Answer:
(i) Since 100 paisa equals 1 rupee, 1 paisa is 1/100 of a rupee. Therefore, Rs 5.25 becomes 5 plus 0.25, which is 5 plus 25/100, or Rs 5 and 25 paisa.
(ii) Because 100 g equals 1 kg, 1 g is 1/1000 kg. So, 8.354 kg becomes 8 plus 0.354, which is 8 plus 354/1000, or 8 kg 354 g.
(iii) We know that 10 mm equals 1 cm. Thus, 1 mm is 1/10 cm. So, 3.5 cm becomes 3 plus 0.5, which is 3 plus 5/10, or 3 cm 5 mm.
(iv) Since 1000 m equals 1 km, 1 m is 1/1000 of a km. Thus, 3.05 km becomes 3 plus 0.05, which is 3 plus 5/100, or 3 plus 50/1000 km, equalling 3 km 50 m.
(v) Given that 100 cm equals 1 m, 1 cm is 1/100 of a m. Therefore, 7.54 m becomes 7 plus 0.54, which is 7 plus 54/100, or 7 m 54 cm.
(vi) Because 1 kg equals 1000 g, 15.005 kg becomes 15 plus 0.005, which is 15 plus 5/1000, or 15 kg 5 g. Also, 1 m equals 100 cm, so 12.05 m becomes 12 plus 0.05, which is 12 plus 5/100, or 12 m 5 cm.

Exam Tip: Match the decimal places to the correct subunit conversion (e.g., 0.05 with denominator 100 for cm, 0.005 with denominator 1000 for grams), then write the answer as whole units and subunits.

Exercise 7.7

Question 1. Choose the decimal(s) from the brackets which are not equivalent to the given decimals:
(i) 0.8 (0.80, 0.85, 0.800, 0.08)
(ii) 25.1 (25.01, 25.10, 25.100, 25.001)
(iii) 45.05 (45.050, 15.005, 45.500, 45.0500)
Answer:
(i) The decimals 0.85 and 0.08 are not equivalent to 0.8. In 0.85, there is a 5 in the hundredths position, whereas 0.8 has nothing in the hundredths spot. In 0.08, the digit 0 sits in the tenths position, while in 0.8, the digit 8 sits in the tenths position.
(ii) The decimals 25.01 and 25.001 are not equivalent to 25.1. In 25.01, the digit 0 is in the tenths place, while in 25.1, the digit 1 is in the tenths place.
(iii) The decimals 45.005 and 45.500 are not equivalent to 45.05. In 45.005, the digit 0 is in the hundredths place, while in 45.05, the digit 5 is in the hundredths place. In 45.500, the digit 5 is in the tenths place, whereas in 45.05, the digit 5 is in the hundredths place.

Exam Tip: Focus on place values - tenths, hundredths, and thousandths. Adding trailing zeros does not change value, but changing digit position does.

Question 2. Which of the following are like decimals?
Answer:
(i) 0.34, 0.07, 5.35, 24.70 - These form like decimals because each has the same count of digits following the decimal point (two digits).
(ii) 45.05, 4.505, 20.55, 20.5 - These are unlike decimals because they contain differing counts of digits after the decimal point.
(iii) 8.80, 17.08, 8.94, 0.27 - These are like decimals since each has exactly two digits appearing after the decimal point.
(iv) 4.50, 16.80, 0.700, 7.08 - These form unlike decimals because the number of digits following the decimal point varies among them.

Exam Tip: Like decimals must have identical counts of decimal places; trailing zeros can be added or removed to make them alike.

Question 3. Which of the following statements are correct?
Answer:
(i) This is correct - the two decimals share an identical count of digits after the decimal point, differing only by 2.
(ii) Correct - these three decimals show varying counts of digits past the decimal point.
(iii) Wrong - these two decimals have different counts of digits following the decimal point.
(iv) Wrong - these three decimals demonstrate varying counts of digits beyond the decimal point.
(v) Correct - all three decimals possess the same number of digits after the decimal point.

Exam Tip: A statement about like decimals is correct only if all numbers involved have the exact same decimal place count; check each one carefully before deciding.

Question 4. Convert each of the following sets of unlike decimals to like decimal:
Answer:
(i) Between the two decimals 7.85 and 7.8, the former contains more decimal places (specifically two), so we adjust 7.8 to match by adding a trailing zero, making it 7.80. The resulting like decimals are 7.80 and 7.85.
(ii) Among the pair 2.02 and 3.2, the first has more decimal places (two), so we change 3.2 by appending a trailing zero to get 3.20. The resulting like decimals are 2.02 and 3.20.
(iii) For the group of three - 0.6, 5.8, and 12.765 - the last one contains the greatest number of decimal places (three), so we modify the other two: 0.6 becomes 0.600 and 5.8 becomes 5.800. The adjusted like decimals are 0.600, 5.800, and 12.765.
(iv) In the set 5.296, 5.2, and 5.29, the first has the highest decimal place count (three), so we revise: 5.2 becomes 5.200 and 5.29 becomes 5.290. The adjusted like decimals are 5.296, 5.200, and 5.290.
(v) Among 4.3294, 13.29, and 132.9, the initial number shows the greatest decimal place count (four), so we modify all: 13.29 becomes 13.2900 and 132.9 becomes 132.9400. The adjusted like decimals are 4.3294, 13.2900, and 132.9400.

Exam Tip: Identify which decimal has the most places, then append zeros to all the others until they match that count. This preserves the original values while making them "like."

Exercise 7.8

Question 1. Find the sum of each of the following:
Answer:
(i) 102.360 + 7.054 + 0.800 = 110.214
(ii) 0.060 + 4.108 + 91.500 = 95.668
(iii) 312.800 + 290.020 + 128.457 = 731.277
(iv) 113.285 + 6.700 + 9.340 + 30.080 = 370.421
(v) 18.0030 + 41.7000 + 10.9500 + 5.0570 = 75.7100

Exam Tip: Line up the decimal points before adding. Use place value columns to keep digits organized and avoid careless errors in the tenths, hundredths, and thousandths places.

Question 2. Add the following:
Answer:
(i) 41.80 + 39.24 + 5.01 + 62.60 = 148.65
(ii) 4.702 + 4.200 + 6.020 + 1.270 = 16.192
(iii) 18.030 + 146.300 + 0.829 + 5.324 = 170.483

Exam Tip: Make sure all decimals are aligned vertically by their decimal points before performing addition.

Question 3. Find the sum of each of the following:
Answer:
(i) 0.007 + 8.500 + 30.080 = 38.587
(ii) 280.69 + 25.20 + 38.00 = 343.89
(iii) 25.650 + 9.005 + 3.700 + 38.355 = 38.355
(iv) 27.076 + 0.550 + 0.004 = 27.630

Exam Tip: Double-check your arithmetic by adding from right to left, starting with the smallest place value. Verify that carrying is done correctly across decimal places.

Question 4. Radhika's mother gave her Rs.10.50 and her father gave her Rs.15.80. Find the total amount given to by her parents?
Answer: Radhika's mother handed her Rs 10.50. Her father handed her Rs 15.80. To find the total money Radhika got, we add these two amounts: (10.50 + 15.80) equals Rs 26.30.

Exam Tip: Always identify what you are adding in word problems. Line up decimal points, then add the amounts in the normal way.

Question 5. Rahul bought 4 kg 60 g of grapes and 5 kg 300 g of mangoes. Find the weight of the fruits he bought in all?
Answer: The weight of the grapes is 4 kg 90 g, which converts to 4.090 kg. The weight of the grapes is 2 kg 60 g, converting to 2.060 kg. The weight of the mangoes is 5 kg 300 g, which converts to 5.300 kg. Therefore, the total weight of all fruits purchased by Rahul equals (4.090 + 2.060 + 5.300) kg, giving us 11.450 kg. The total weight of the fruits is 11.450 kg.

Exam Tip: Convert mixed units (kg and g) to decimal form first, then add. Always check that your final answer makes sense in the original units.

Question 6. Nasreen bought 3m 20 cm cloth for her shirt and 2 m 5 cm cloth for her skirt. Find the total cloth bought by her?
Answer: The cloth needed for the shirt is 3 m 20 cm, which equals 3.20 m. The cloth needed for the skirt is 2 m 50 m, which equals 2.05 m. The total cloth purchased by Nasreen is (3.20 + 2.05) m, which equals 5.25 m. The total cloth bought by Nasreen is 5.25 m.

Exam Tip: Convert compound measurements (m and cm) to decimals before adding. Be careful with conversions - 5 cm is 0.05 m, not 0.5 m.

Question 7. Sunita travels 15 km 268 m by bus, 7 km 7 m by car and 500 m by foot in order to reach her school. How far is her school from her residence?
Answer: The distance covered by bus is 15 km 268 m, which is 15.268 km. The distance covered by car is 7 km 7m, which is 7.007 km. The distance travelled on foot is 500 m, which is 0.500 km. The total distance travelled by Sunita is (15.268 + 7.07 + 0.500) km, which equals 22.775 km. Therefore, the total distance covered by Sunita is 22.775 km.

Exam Tip: Always convert all measurements to the same unit before adding. Double-check the conversion (e.g., 7 m is 0.007 km, not 0.07 km).

Exercise 7.9

Question 1. Subtract:
Answer:
1. 46.23 - 37.5 = 8.73
2. 128.40 - 53.05 = 75.35
3. 45.03 - 27.80 = 17.23
4. 23.930 - 5.946 = 17.984

Exam Tip: Align decimal points vertically before subtracting. Add trailing zeros if needed to ensure both numbers have the same number of decimal places.

Question 2. Subtract:
Answer:
1. 9.756 - 6.280 = 3.476
2. 21.05 - 15.27 = 5.78
3. 18.50 - 6.79 = 11.71
4. 48.10 - 0.37 = 47.73
5. 108.032 - 86.800 = 21.232

Exam Tip: Work from right to left, starting with the smallest place value. Borrow from the next column when needed, keeping the decimal point in its original position.

Question 3. The sum of two numbers is 100. If one of them is 78.01. Find the other one?
Answer: When you add two numbers together and get 100, and one of those numbers is 78.01, you need to find what the second number is. Let's say the other number is x. So, 78.01 + x = 100. To find x, subtract 78.01 from both sides: x = 100 - 78.01, which gives x = 21.99. Therefore, the other number is 21.99.

Exam Tip: Always set up an equation when two numbers or values are related — this makes the problem systematic and reduces errors in calculation.

Question 4. Waheeda's school is at a distance 5 km 350 m from her house. She travels 1 km 70 m on foot and the rest she travels by bus. How much distance does she travel by bus?
Answer: The total distance from Waheeda's house to school is 5 km 350 m, which equals 5.350 km. She walks 1 km 70 m on foot, which is 1.070 km. The remaining distance is covered by bus. Distance by bus = 5.350 - 1.070 = 4.280 km. So, Waheeda travels 4.280 km by bus.

Exam Tip: Convert all measurements to the same unit (kilometres or metres) before performing any calculation to avoid mistakes.

Question 5. Raju bought a book of Rs.35.65, he gave Rs.50.35 to the shopkeeper. How much money did he pay back to the shopkeeper?
Answer: The book costs Rs. 35.65. Raju gave Rs. 50.35 to the shopkeeper. To find the change, subtract the cost from the amount given: 50.35 - 35.65 = Rs. 14.70. Therefore, the shopkeeper returned Rs. 14.70 to Raju.

Exam Tip: Always check that the amount given is greater than the cost; the difference is the change received.

Question 6. Raju bought a water melon weighing 5 kg 200g. Out of this she gave 2 kg 750 g to her neighbor. What is the weight of the watermelon left with ruby?
Answer: The watermelon initially weighed 5 kg 200 g, which equals 5.200 kg. She gave 2 kg 750 g to her neighbour, which equals 2.750 kg. The remaining weight is 5.200 - 2.750 = 2.450 kg. Thus, the weight of the watermelon left with Ruby is 2.450 kg.

Exam Tip: Convert all weights to the same decimal form (kilogrammes) for straightforward subtraction without confusion.

Question 7. Victor drove 89.05 km on Saturday and 73.9 km on Sunday. How many kilometers did he drive more on Saturday?
Answer: Victor travelled 89.05 km on Saturday and 73.9 km on Sunday. To find how much more he drove on Saturday, subtract Sunday's distance from Saturday's distance: 89.05 - 73.9 = 15.15 km. Therefore, Victor drove 15.15 km more on Saturday than on Sunday.

Exam Tip: When comparing two quantities, always subtract the smaller from the larger to get a positive difference.

Question 8. Raju bought a book of Rs.35.65. he gave Rs.50.35 to the shopkeeper. How much money did he pay back to the shopkeeper?
Answer: The cost of the book is Rs. 35.65. Raju handed Rs. 50.35 to the shopkeeper. The amount returned is found by subtracting: 50.35 - 35.65 = Rs. 14.70. Thus, the shopkeeper gave back Rs. 14.70 to Raju.

Exam Tip: Money problems require careful alignment of decimal points during subtraction to ensure accuracy.

Question 9. Gopal travelled 125.5 km by bus, 14.25 km by pony and the rest of the distance to kedarnath on foot. If he covered a total distance of 150 km, how much did he travel on foot?
Answer: Gopal's total journey was 150 km. He went 125.5 km by bus and 14.25 km by pony. The distance on foot is the remainder. Adding the bus and pony distances: 125.5 + 14.25 = 139.75 km. Distance on foot = 150 - 139.75 = 10.25 km. Therefore, Gopal travelled 10.25 km on foot.

Exam Tip: When a total is divided among different modes or parts, add all known parts first, then subtract from the total to find the unknown part.

Question 10. 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 cloth is left with her?
Answer: The original cloth measured 20 m 5 cm, which equals 20.05 m. She removed 4 m 50 cm for a curtain, equal to 4.50 m. The remaining cloth is 20.05 - 4.50 = 15.55 m. Therefore, 15.55 m of cloth is left with Tina.

Exam Tip: Always convert mixed measurements (metres and centimetres) to a single decimal unit before subtraction.

Question 11. Vineeta bought a book of Rs. 18.9, a pen of Rs. 8.50 and some papers for Rs. 5.05. She gave fifty rupee to the shopkeeper. How much balance did she get back?
Answer: Vineeta's purchases were: a book at Rs. 18.90, a pen at Rs. 8.50, and papers at Rs. 5.05. The total cost is 18.90 + 8.50 + 5.05 = Rs. 32.45. She paid Rs. 50. The change she received is 50 - 32.45 = Rs. 17.55. Therefore, the balance returned was Rs. 17.55.

Exam Tip: For multi-item purchases, sum all costs first, then find the change by subtracting the total from the amount paid.

Question 12. Tanuj walked 8.62 km on Monday, 7.05 km on Tuesday, and some distance on Wednesday. If he walked 21.01 km in three days, how much distance did he walk on Wednesday?
Answer: Over three days, Tanuj walked a total of 21.01 km. On Monday he walked 8.62 km and on Tuesday 7.05 km. Adding these two days: 8.62 + 7.05 = 15.67 km. The distance on Wednesday is 21.01 - 15.67 = 5.34 km. Therefore, Tanuj walked 5.34 km on Wednesday.

Exam Tip: When finding an unknown value within a total, add all known values and subtract from the total to get the answer.

Exercise 7.10

Question 1. Find the value.
Answer:
1. 3/10 equals 0.3
2. 7/100 equals 0.07
3. 4/1000 equals 0.004
4. The value of 37/10000 is 0.0037
5. The place value of 5 in 0.04532 is 5/1000
6. The value of 231/1000 is 0.231
7. The value of 3/5 × 100 is 3.005 (or 60/100 = 0.60)
8. The value of 3/25 is 0.12
9. The value of \( 2\frac{1}{25} \) is 2.04
10. \( \frac{4}{7} \) × 8 equals 4.875
11. \( 2 + \frac{3}{10} + \frac{5}{100} \) equals 2.35
12. \( \frac{3}{100} + \frac{5}{1000} \) equals 0.0305
13. 1 cm equals 0.01 m
14. 1 m equals 0.001 km
15. 2 kg 5 gm equals 2.005 kg
16. 15 litres and 15 ml equals 15.015 litres
17. Like decimals are 5.5, 6.6, 7.7, 8.8
18. The value of 0.5 + 0.005 + 5.05 is 5.555
19. 0.35 - 0.035 equals 0.0315
20. 2.5 + 3.05 - 4.005 equals 1.545
21. Which is greater among 2.3, 2.03, 2.33, 2.05? Answer: 2.33

Exam Tip: Conversion of fractions to decimals is fundamental - identify the denominator (10, 100, 1000, etc.) and place the decimal point accordingly from the right.