RS Aggarwal Class 7 Mathematics Solutions Chapter 3 Decimals

Decimals

Exercise 3A

Question 1. Express each of the following decimals as a fraction in its lowest terms:
(i) 0.8
(ii) 0.75
(iii) 0.06
(iv) 0.285
Answer:
(i) \( 0.8 = \frac{8}{10} = \frac{8 \div 2}{10 \div 2} = \frac{4}{5} \)
(ii) \( 0.75 = \frac{75}{100} = \frac{75 \div 25}{100 \div 25} = \frac{3}{4} \)
(iii) \( 0.06 = \frac{6}{100} = \frac{6 \div 2}{100 \div 2} = \frac{3}{50} \)
(iv) \( 0.285 = \frac{285}{1000} = \frac{285 \div 5}{1000 \div 5} = \frac{57}{200} \)

Exam Tip: To write a decimal as a fraction, place the digits after the point over 10, 100 or 1000 depending on how many decimal places there are, then cancel using the HCF to reach the lowest terms.

Question 2. Express each of the following as a mixed fraction:
(i) 5.6
(ii) 12.25
(iii) 6.004
(iv) 4.625
Answer:
(i) \( 5.6 = \frac{56}{10} = \frac{56 \div 2}{10 \div 2} = \frac{28}{5} = 5\frac{3}{5} \)
(ii) \( 12.25 = \frac{1225}{100} = \frac{1225 \div 25}{100 \div 25} = \frac{49}{4} = 12\frac{1}{4} \)
(iii) \( 6.004 = \frac{6004}{1000} = \frac{6004 \div 4}{1000 \div 4} = \frac{1501}{250} = 6\frac{1}{250} \)
(iv) \( 4.625 = \frac{4625}{1000} = \frac{4625 \div 125}{1000 \div 125} = \frac{37}{8} = 4\frac{5}{8} \)

Exam Tip: Once the fraction is fully reduced, divide the numerator by the denominator to split it into a whole number part and a proper fraction part.

Question 3. Express each of the following as a decimal:
(i) \( \frac{47}{10} \)
(ii) \( \frac{156}{100} \)
Answer:
(i) On dividing, we get \( \frac{47}{10} = 4.7 \)
(ii) On dividing, we get \( \frac{156}{100} = 1.56 \)

Exam Tip: Dividing by 10, 100 or 1000 simply moves the decimal point left by 1, 2 or 3 places - there is no need for long division in such cases.

Question 4. Convert the following into like decimals:
(i) 6.5, 16.03, 0.274 and 119.4
(ii) 3.5, 0.67, 15.6 and 4
Answer:
(i) Making the number of decimal places equal in each number gives us 6.500, 16.030, 0.274 and 119.400.
(ii) Making the number of decimal places equal in each number gives us 3.50, 0.67, 15.60 and 4.00.

Exam Tip: Like decimals have the same number of digits after the point. Simply add trailing zeros to the shorter decimals - this never changes their value.

Question 5. Which is greater in each of the following?
(i) 78.23, 69.85
(ii) 3.406, 3.46
(iii) 5.68, 5.86
(iv) 14.05, 14.005
(v) 1.85, 1.805
(vi) 0.98, 1.07
Answer:
(i) Looking at the whole number parts, 78 > 69. So, 78.23 > 69.85.
(ii) Turning the decimals into like decimals gives 3.406 and 3.460. The whole numbers are equal (3 = 3), the tenths digits are equal (4 = 4), and the hundredths digit 6 > 0. So, 3.406 < 3.46.
(iii) The whole number parts are equal (5 = 5). Looking at the tenths digit, 6 < 8. So, 5.68 < 5.86.
(iv) Turning the decimals into like decimals gives 14.050 and 14.005. The whole numbers match (14 = 14) and the tenths digits match (0 = 0), but the hundredths digit 5 > 0. So, 14.05 > 14.005.
(v) Turning the decimals into like decimals gives 1.850 and 1.805. The whole numbers match (1 = 1) and the tenths digits match (8 = 8), but the hundredths digit 5 > 0. So, 1.85 > 1.805.
(vi) The whole number parts show 0 < 1. So, 0.98 < 1.07.

Exam Tip: When comparing decimals, always compare digit by digit from left to right, starting with the whole number part, and convert to like decimals first if the number of decimal places differs.

Question 6. Arrange the following decimals in ascending order:
(i) 4.6, 7.4, 4.58, 7.32, 4.06
(ii) 0.5, 5.5, 0.05, 5.05, 5.55
(iii) 6.84, 6.48, 6.8, 6.4, 6.08
(iv) 2.2, 2.202, 2.02, 22.2, 2.002
Answer:
(i) Turning the decimals into like decimals, we get 4.60, 7.40, 4.58, 7.32 and 4.06. It is clear that 4.06 < 4.58 < 4.60 < 7.32 < 7.40. So, the decimals arranged from smallest to largest are 4.06, 4.58, 4.6, 7.32 and 7.4.
(ii) Turning the decimals into like decimals, we get 0.50, 5.50, 0.05, 5.05 and 5.55. It is clear that 0.05 < 0.50 < 5.05 < 5.50 < 5.55. So, the decimals arranged from smallest to largest are 0.05, 0.5, 5.05, 5.5 and 5.55.
(iii) Turning the decimals into like decimals, we get 6.84, 6.48, 6.80, 6.40 and 6.08. It is clear that 6.08 < 6.40 < 6.48 < 6.80 < 6.84. So, the decimals arranged from smallest to largest are 6.08, 6.4, 6.48, 6.8 and 6.84.
(iv) Turning the decimals into like decimals, we get 2.200, 2.202, 2.020, 22.200 and 2.002. It is clear that 2.002 < 2.020 < 2.200 < 2.202 < 22.200. So, the decimals arranged from smallest to largest are 2.002, 2.02, 2.2, 2.202 and 22.2.

Exam Tip: Convert every number to the same number of decimal places first - this makes it much easier to spot the correct ascending order without mistakes.

Question 7. Arrange the following decimals in descending order:
(i) 7.4, 8.34, 74.4, 7.44, 0.74
(ii) 2.6, 2.26, 2.06, 2.007, 2.3
Answer:
(i) Turning the decimals into like decimals, we get 7.40, 8.34, 74.40, 7.44 and 0.74. It is clear that 74.40 > 8.34 > 7.44 > 7.40 > 0.74. So, the decimals arranged from largest to smallest are 74.4, 8.34, 7.44, 7.4 and 0.74.
(ii) Turning the decimals into like decimals, we get 2.600, 2.260, 2.060, 2.007 and 2.300. It is clear that 2.600 > 2.300 > 2.260 > 2.060 > 2.007. So, the decimals arranged from largest to smallest are 2.6, 2.3, 2.26, 2.06 and 2.007.

Exam Tip: For descending order, the same like-decimal method applies - just read the comparison the other way round, from largest down to smallest.

Question 8. Express 45 mm in cm, m and km.
Answer:
\( 45 \text{ mm} = \frac{45}{10} \text{ cm} = 4.5 \text{ cm} \)
\( 4.5 \text{ cm} = \frac{4.5}{100} \text{ m} = 0.045 \text{ m} \)
\( 0.045 \text{ m} = \frac{0.045}{1000} \text{ km} = 0.000045 \text{ km} \)
So, 45 mm = 4.5 cm = 0.045 m = 0.000045 km.

Exam Tip: Learn the standard unit chain: 10 mm = 1 cm, 100 cm = 1 m, 1000 m = 1 km. Converting to a bigger unit always means dividing.

Question 9. Express each of the following as rupees using decimals:
(i) 8 paise
(ii) 9 rupees 75 paise
(iii) 8 rupees 5 paise
Answer:
(i) \( 8 \text{ paise} = \text{Rs. } \frac{8}{100} = \text{Rs. } 0.08 \)
(ii) \( 9 \text{ rupees } 75 \text{ paise} = \text{Rs. } \left(9 + \frac{75}{100}\right) = \text{Rs. } (9 + 0.75) = \text{Rs. } 9.75 \)
(iii) \( 8 \text{ rupees } 5 \text{ paise} = \text{Rs. } \left(8 + \frac{5}{100}\right) = \text{Rs. } (8 + 0.05) = \text{Rs. } 8.05 \)

Exam Tip: Since 100 paise make Rs. 1, treat the paise value as a fraction over 100 and add it to the rupee part to get the decimal form.

Question 10. Express each of the following as kilometres, using decimals:
(i) 65 m
(ii) 284 m
(iii) 3 km 5 m
Answer:
(i) \( 65 \text{ m} = \frac{65}{1000} \text{ km} = 0.065 \text{ km} \)
(ii) \( 284 \text{ m} = \frac{284}{1000} \text{ km} = 0.284 \text{ km} \)
(iii) \( 3 \text{ km } 5 \text{ m} = \left(3 + \frac{5}{1000}\right) \text{ km} = (3 + 0.005) \text{ km} = 3.005 \text{ km} \)

Exam Tip: Since 1000 m make 1 km, divide the metre value by 1000 to switch to kilometres, and add it to any whole km already given.

Exercise 3B

Question 1. Add: 16.00, 8.70, 0.94, 6.80 and 7.77
Answer:
We first change the given decimals into like decimals, getting 16.00, 8.70, 0.94, 6.80, 7.77.
Writing these numbers in column form and adding them gives 40.21.
So, the total of the given decimals is 40.21.

Exam Tip: Line up the decimal points exactly one below the other before adding - this is the most common place marks are lost in this type of sum.

Question 2. Add: 18.600, 206.370, 8.008, 26.400 and 6.900
Answer:
We first change the given decimals into like decimals, getting 18.600, 206.370, 8.008, 26.400, 6.900.
Writing these numbers in column form and adding them gives 266.278.
So, the total of the given decimals is 266.278.

Exam Tip: Line up the decimal points exactly one below the other before adding - this is the most common place marks are lost in this type of sum.

Question 3. Add: 63.50, 9.70, 0.80, 26.66 and 12.17
Answer:
We first change the given decimals into like decimals, getting 63.50, 9.70, 0.80, 26.66, 12.17.
Writing these numbers in column form and adding them gives 112.83.
So, the total of the given decimals is 112.83.

Exam Tip: Line up the decimal points exactly one below the other before adding - this is the most common place marks are lost in this type of sum.

Question 4. Add: 17.400, 86.390, 9.435, 8.800 and 0.060
Answer:
We first change the given decimals into like decimals, getting 17.400, 86.390, 9.435, 8.800, 0.060.
Writing these numbers in column form and adding them gives 122.085.
So, the total of the given decimals is 122.085.

Exam Tip: Line up the decimal points exactly one below the other before adding - this is the most common place marks are lost in this type of sum.

Question 5. Add: 26.900, 19.740, 231.769 and 0.048
Answer:
We first change the given decimals into like decimals, getting 26.900, 19.740, 231.769, 0.048.
Writing these numbers in column form and adding them gives 278.457.
So, the total of the given decimals is 278.457.

Exam Tip: Line up the decimal points exactly one below the other before adding - this is the most common place marks are lost in this type of sum.

Question 6. Add: 23.800, 8.940, 0.078 and 214.600
Answer:
We first change the given decimals into like decimals, getting 23.800, 8.940, 0.078, 214.600.
Writing these numbers in column form and adding them gives 247.418.
So, the total of the given decimals is 247.418.

Exam Tip: Line up the decimal points exactly one below the other before adding - this is the most common place marks are lost in this type of sum.

Question 7. Add: 6.606, 66.600, 666.000, 0.066 and 0.660
Answer:
We first change the given decimals into like decimals, getting 6.606, 66.600, 666.000, 0.066, 0.660.
Writing these numbers in column form and adding them gives 739.932.
So, the total of the given decimals is 739.932.

Exam Tip: Line up the decimal points exactly one below the other before adding - this is the most common place marks are lost in this type of sum.

Question 8. Add: 9.090, 0.909, 99.900, 9.990 and 0.099
Answer:
We first change the given decimals into like decimals, getting 9.090, 0.909, 99.900, 9.990, 0.099.
Writing these numbers in column form and adding them gives 119.988.
So, the total of the given decimals is 119.988.

Exam Tip: Line up the decimal points exactly one below the other before adding - this is the most common place marks are lost in this type of sum.

Question 9. Subtract 14.79 from 72.43.
Answer:
The larger number 72.43 is placed on top in column form and 14.79 is subtracted from it.
\( 72.43 - 14.79 = 57.64 \)

Exam Tip: Always place the bigger decimal on top and line up the decimal points before subtracting, adding zeros to make the number of decimal places equal.

Question 10. Subtract 36.74 from 52.60.
Answer:
The larger number 52.60 is placed on top in column form and 36.74 is subtracted from it.
\( 52.60 - 36.74 = 15.86 \)

Exam Tip: Always place the bigger decimal on top and line up the decimal points before subtracting, adding zeros to make the number of decimal places equal.

Question 11. Subtract 13.876 from 22.000.
Answer:
The larger number 22.000 is placed on top in column form and 13.876 is subtracted from it.
\( 22.000 - 13.876 = 8.124 \)

Exam Tip: Always place the bigger decimal on top and line up the decimal points before subtracting, adding zeros to make the number of decimal places equal.

Question 12. Subtract 15.079 from 24.160.
Answer:
The larger number 24.160 is placed on top in column form and 15.079 is subtracted from it.
\( 24.160 - 15.079 = 9.081 \)

Exam Tip: Always place the bigger decimal on top and line up the decimal points before subtracting, adding zeros to make the number of decimal places equal.

Question 13. Subtract 0.680 from 1.007.
Answer:
The larger number 1.007 is placed on top in column form and 0.680 is subtracted from it.
\( 1.007 - 0.680 = 0.327 \)

Exam Tip: Always place the bigger decimal on top and line up the decimal points before subtracting, adding zeros to make the number of decimal places equal.

Question 14. Subtract 0.4678 from 5.0500.
Answer:
The larger number 5.0500 is placed on top in column form and 0.4678 is subtracted from it.
\( 5.0500 - 0.4678 = 4.5822 \)

Exam Tip: Always place the bigger decimal on top and line up the decimal points before subtracting, adding zeros to make the number of decimal places equal.

Question 15. Subtract 2.5307 from 8.0000.
Answer:
The larger number 8.0000 is placed on top in column form and 2.5307 is subtracted from it.
\( 8.0000 - 2.5307 = 5.4693 \)

Exam Tip: Always place the bigger decimal on top and line up the decimal points before subtracting, adding zeros to make the number of decimal places equal.

Question 16. Subtract 6.732 from 9.001.
Answer:
The larger number 9.001 is placed on top in column form and 6.732 is subtracted from it.
\( 9.001 - 6.732 = 2.269 \)

Exam Tip: Always place the bigger decimal on top and line up the decimal points before subtracting, adding zeros to make the number of decimal places equal.

Question 17. Subtract 5.746 from 9.100.
Answer:
The larger number 9.100 is placed on top in column form and 5.746 is subtracted from it.
\( 9.100 - 5.746 = 3.354 \)

Exam Tip: Always place the bigger decimal on top and line up the decimal points before subtracting, adding zeros to make the number of decimal places equal.

Question 18. What number should be added to 63.58 to get 92?
Answer:
The number needed = \( 92 - 63.58 = 28.42 \)
So, 28.42 needs to be added to 63.58 to reach 92.

Exam Tip: Turn a "what should be added" question into a straightforward subtraction: bigger number minus the given number.

Question 19. What number should be subtracted from 8.1 to get 0.813?
Answer:
The number needed = \( 8.100 - 0.813 = 7.287 \)
So, 7.287 needs to be taken away from 8.1 to leave 0.813.

Exam Tip: Convert both numbers into like decimals first, then subtract the smaller result from 8.1 to find the missing value.

Question 20. By how much should 32.67 be increased to get 60.1?
Answer:
The number needed = \( 60.10 - 32.67 = 27.43 \)
So, 32.67 needs to go up by 27.43 to reach 60.1.

Exam Tip: An "increase by" question is solved the same way as an addition-missing-number question: subtract the smaller value from the target.

Question 21. By how much should 74.3 be decreased to get 26.87?
Answer:
The number needed = \( 74.30 - 26.87 = 47.43 \)
So, 74.3 needs to go down by 47.43 to reach 26.87.

Exam Tip: A "decreased by" question always uses subtraction: take the smaller target value away from the starting number.

Question 22. Rohit bought some articles from a shop costing Rs. 23.75, Rs. 2.85 and Rs. 15.90, and gave the shopkeeper a Rs. 50 note. How much money did he get back?
Answer:
Total amount spent by Rohit on the articles = \( \text{Rs. } (23.75 + 2.85 + 15.90) = \text{Rs. } 42.50 \)
Money handed to the shopkeeper = Rs. 50
So, the change returned by the shopkeeper = \( \text{Rs. } (50 - 42.50) = \text{Rs. } 7.50 \)
So, Rohit got back Rs. 7.50.

Exam Tip: For money word problems, always add up all the costs first, then subtract the total from the amount tendered to find the change.

Exercise 3C

Question 1. Multiply:
(i) \( 73.92 \times 10 \)
(ii) \( 7.54 \times 10 \)
(iii) \( 84.003 \times 10 \)
(iv) \( 0.83 \times 10 \)
(v) \( 0.7 \times 10 \)
(vi) \( 0.032 \times 10 \)
Answer:
(i) \( 73.92 \times 10 = 739.2 \) [decimal point moves right by 1 place]
(ii) \( 7.54 \times 10 = 75.4 \) [decimal point moves right by 1 place]
(iii) \( 84.003 \times 10 = 840.03 \) [decimal point moves right by 1 place]
(iv) \( 0.83 \times 10 = 8.3 \) [decimal point moves right by 1 place]
(v) \( 0.7 \times 10 = 7 \) [decimal point moves right by 1 place]
(vi) \( 0.032 \times 10 = 0.32 \) [decimal point moves right by 1 place]

Exam Tip: Multiplying by 10, 100 or 1000 just shifts the decimal point right by 1, 2 or 3 places - no actual multiplication working is needed.

Question 2. Multiply:
(i) \( 2.397 \times 100 \)
(ii) \( 6.83 \times 100 \)
(iii) \( 2.9 \times 100 \)
(iv) \( 0.08 \times 100 \)
(v) \( 0.6 \times 100 \)
(vi) \( 0.003 \times 100 \)
Answer:
(i) \( 2.397 \times 100 = 239.7 \) [decimal point moves right by 2 places]
(ii) \( 6.83 \times 100 = 683 \) [decimal point moves right by 2 places]
(iii) \( 2.9 \times 100 = 290 \) [decimal point moves right by 2 places]
(iv) \( 0.08 \times 100 = 8 \) [decimal point moves right by 2 places]
(v) \( 0.6 \times 100 = 60 \) [decimal point moves right by 2 places]
(vi) \( 0.003 \times 100 = 0.3 \) [decimal point moves right by 2 places]

Exam Tip: Multiplying by 10, 100 or 1000 just shifts the decimal point right by 1, 2 or 3 places - no actual multiplication working is needed.

Question 3. Multiply:
(i) \( 6.7314 \times 1000 \)
(ii) \( 0.182 \times 1000 \)
(iii) \( 0.076 \times 1000 \)
(iv) \( 6.25 \times 1000 \)
(v) \( 4.8 \times 1000 \)
(vi) \( 0.06 \times 1000 \)
Answer:
(i) \( 6.7314 \times 1000 = 6731.4 \) [decimal point moves right by 3 places]
(ii) \( 0.182 \times 1000 = 182 \) [decimal point moves right by 3 places]
(iii) \( 0.076 \times 1000 = 76 \) [decimal point moves right by 3 places]
(iv) \( 6.25 \times 1000 = 6250 \) [decimal point moves right by 3 places]
(v) \( 4.8 \times 1000 = 4800 \) [decimal point moves right by 3 places]
(vi) \( 0.06 \times 1000 = 60 \) [decimal point moves right by 3 places]

Exam Tip: Multiplying by 10, 100 or 1000 just shifts the decimal point right by 1, 2 or 3 places - no actual multiplication working is needed.

Question 5. Multiply:
(i) \( 5.4 \times 16 \)
(ii) \( 3.65 \times 19 \)
(iii) \( 0.854 \times 12 \)
(iv) \( 36.78 \times 48 \)
(v) \( 4.125 \times 86 \)
(vi) \( 104.06 \times 75 \)
(vii) \( 6.032 \times 124 \)
(viii) \( 0.0146 \times 69 \)
(ix) \( 0.00125 \times 327 \)
(x) \( 54.5 \times 1.76 \)
(xi) \( 0.045 \times 2.4 \)
(xii) \( 1.245 \times 6.4 \)
Answer:
(i) \( 54 \times 16 = 864 \), so \( 5.4 \times 16 = 86.4 \) [1 decimal place]
(ii) \( 365 \times 19 = 6935 \), so \( 3.65 \times 19 = 69.35 \) [2 decimal places]
(iii) \( 854 \times 12 = 10248 \), so \( 0.854 \times 12 = 10.248 \) [3 decimal places]
(iv) \( 3673 \times 48 = 176304 \), so \( 36.78 \times 48 = 1763.04 \) [2 decimal places]
(v) \( 4125 \times 86 = 354750 \), so \( 4.125 \times 86 = 354.750 = 354.75 \) [3 decimal places]
(vi) \( 10406 \times 75 = 780450 \), so \( 104.06 \times 75 = 7804.50 = 7804.5 \) [2 decimal places]
(vii) \( 6032 \times 124 = 747968 \), so \( 6.032 \times 124 = 747.968 \) [3 decimal places]
(viii) \( 146 \times 69 = 10074 \), so \( 0.0146 \times 69 = 1.0074 \) [4 decimal places]
(ix) \( 125 \times 327 = 40875 \), so \( 0.00125 \times 327 = 0.40875 \) [5 decimal places]
(x) \( 545 \times 176 = 95920 \), so \( 54.5 \times 1.76 = 95.920 = 95.92 \) [3 decimal places]
(xi) \( 45 \times 24 = 1080 \), so \( 0.045 \times 2.4 = 0.1080 = 0.108 \) [4 decimal places]
(xii) \( 1245 \times 64 = 79680 \), so \( 1.245 \times 6.4 = 7.9680 = 7.968 \) [4 decimal places]

Exam Tip: First multiply the digits as whole numbers, ignoring the decimal points completely. Then count the total decimal places in both factors together and place the point that many digits from the right in the answer.

Question 6. Find the product:
(i) \( 13 \times 1.3 \times 0.13 \)
(ii) \( 2.4 \times 1.5 \times 2.5 \)
(iii) \( 0.8 \times 3.5 \times 0.05 \)
(iv) \( 0.2 \times 0.02 \times 0.002 \)
(v) \( 11.1 \times 1.1 \times 0.11 \)
(vi) \( 2.1 \times 0.21 \times 0.021 \)
Answer:
(i) \( 13 \times 13 \times 13 = 2197 \). The total decimal places across the numbers = 3, so \( 13 \times 1.3 \times 0.13 = 2.197 \)
(ii) \( 24 \times 15 \times 25 = 9000 \). The total decimal places across the numbers = 3, so \( 2.4 \times 1.5 \times 2.5 = 9.000 = 9 \)
(iii) \( 8 \times 35 \times 5 = 1400 \). The total decimal places across the numbers = 4, so \( 0.8 \times 3.5 \times 0.05 = 0.1400 = 0.14 \)
(iv) \( 2 \times 2 \times 2 = 8 \). The total decimal places across the numbers = 6, so \( 0.2 \times 0.02 \times 0.002 = 0.000008 \)
(v) \( 111 \times 11 \times 11 = 13431 \). The total decimal places across the numbers = 4, so \( 11.1 \times 1.1 \times 0.11 = 1.3431 \)
(vi) \( 21 \times 21 \times 21 = 9261 \). The total decimal places across the numbers = 6, so \( 2.1 \times 0.21 \times 0.021 = 0.009261 \)

Exam Tip: When three decimals are multiplied together, add up all the decimal places from all three numbers at once before placing the point in the final product.

Question 7. Evaluate:
(i) \( (1.2)^2 \)
(ii) \( (0.7)^2 \)
(iii) \( (0.04)^2 \)
(iv) \( (0.11)^2 \)
Answer:
(i) \( (1.2)^2 = 1.2 \times 1.2 \). Since \( 12 \times 12 = 144 \) and the total decimal places = 2, \( (1.2)^2 = 1.44 \)
(ii) \( (0.7)^2 = 0.7 \times 0.7 \). Since \( 7 \times 7 = 49 \) and the total decimal places = 2, \( (0.7)^2 = 0.49 \)
(iii) \( (0.04)^2 = 0.04 \times 0.04 \). Since \( 4 \times 4 = 16 \) and the total decimal places = 4, \( (0.04)^2 = 0.0016 \)
(iv) \( (0.11)^2 = 0.11 \times 0.11 \). Since \( 11 \times 11 = 121 \) and the total decimal places = 4, \( (0.11)^2 = 0.0121 \)

Exam Tip: Squaring a decimal simply means multiplying it by itself - use the same digit-then-place-the-point method as any other decimal multiplication.

Question 8. Evaluate:
(i) \( (0.3)^3 \)
(ii) \( (0.05)^3 \)
(iii) \( (1.5)^3 \)
Answer:
(i) \( (0.3)^3 = 0.3 \times 0.3 \times 0.3 \). Since \( 3 \times 3 \times 3 = 27 \) and the total decimal places = 3, \( (0.3)^3 = 0.027 \)
(ii) \( (0.05)^3 = 0.05 \times 0.05 \times 0.05 \). Since \( 5 \times 5 \times 5 = 125 \) and the total decimal places = 6, \( (0.05)^3 = 0.000125 \)
(iii) \( (1.5)^3 = 1.5 \times 1.5 \times 1.5 \). Since \( 15 \times 15 \times 15 = 3375 \) and the total decimal places = 3, \( (1.5)^3 = 3.375 \)

Exam Tip: For a cube, multiply the number by itself three times and add up the decimal places from all three copies before placing the point.

Question 9. A bus covers a distance of 62.5 km in 1 hour. What distance will it cover in 18 hours?
Answer:
Distance covered by the bus in 1 hour = 62.5 km
So, distance covered in 18 hours = \( (62.5 \times 18) \text{ km} = 1125 \text{ km} \)
So, the bus can travel 1125 km in 18 hours.

Exam Tip: In "distance per hour" problems, simply multiply the given speed by the number of hours to get the total distance.

Question 10. The weight of 1 tin of oil is 16.8 kg. Find the weight of 45 such tins.
Answer:
Weight of 1 tin of oil = 16.8 kg
So, weight of 45 such tins = \( (16.8 \times 45) \text{ kg} = 756 \text{ kg} \)
So, the weight of 45 tins of oil is 756 kg.

Exam Tip: Multiply the weight of a single unit by the total number of units to find the combined weight - keep the decimal placement rule in mind.

Question 11. The weight of a bag of wheat is 97.8 kg. Find the weight of 500 such bags.
Answer:
Weight of 1 bag of wheat = 97.8 kg
So, weight of 500 such bags = \( (97.8 \times 500) \text{ kg} = 48900 \text{ kg} \)
So, the weight of 500 bags of wheat is 48900 kg.

Exam Tip: Multiplying by a round number like 500 can often be done by multiplying by 5 and shifting the decimal point two places to the right.

Question 12. The weight of a bag of sugar is 48.45 kg. Find the weight of 16 such bags.
Answer:
Weight of 1 bag of sugar = 48.450 kg
So, weight of 16 bags of sugar = \( (48.450 \times 16) \text{ kg} = 775.2 \text{ kg} \)
So, the weight of 16 bags of sugar is 775.2 kg.

Exam Tip: Multiply as whole numbers first, then place the decimal point using the total number of decimal places in the original figures.

Question 13. The capacity of a sauce bottle is 0.845 kg. Find the capacity of 72 such bottles.
Answer:
Capacity of 1 sauce bottle = 0.845 kg
So, capacity of 72 such bottles = \( (0.845 \times 72) \text{ kg} = 60.84 \text{ kg} \)
So, the capacity of 72 bottles of sauce will be 60.84 kg.

Exam Tip: Even with a small decimal capacity, the multiplication method stays the same: multiply as whole numbers, then insert the decimal point correctly.

Question 14. The weight of a bottle of jam is 925 g. Find the weight of 25 such bottles, in kg.
Answer:
Weight of 1 bottle of jam = \( 925 \text{ g} = 0.925 \text{ kg} \)
So, weight of 25 such bottles = \( (0.925 \times 25) \text{ kg} = 23.125 \text{ kg} \)
So, the weight of 25 bottles of jam will be 23.125 kg.

Exam Tip: When units are mixed (grams given, kg wanted), convert first to a single unit before doing the multiplication.

Question 15. The capacity of a drum of oil is 16.85 litres. Find the capacity of 48 such drums.
Answer:
Capacity of 1 drum of oil = 16.850 litres
So, capacity of 48 such drums = \( (16.850 \times 48) \text{ litres} = 808.800 \text{ litres} \)
So, the capacity of 48 drums of oil is 808.8 litres.

Exam Tip: Add trailing zeros to make the decimal easier to line up during multiplication, then drop unnecessary trailing zeros from the final answer.

Question 16. The cost of 1 kg of rice is Rs. 56.80. Find the cost of 16.25 kg of rice.
Answer:
Cost of 1 kg of rice = Rs. 56.80
So, cost of 16.25 kg of rice = \( \text{Rs. } (56.80 \times 16.25) = \text{Rs. } 923 \)
So, the cost of 16.25 kg of rice is Rs. 923.

Exam Tip: For rate-based money problems, multiply the price per unit by the quantity, counting decimal places from both numbers together.

Question 17. The cost of 1 metre of cloth is Rs. 108.50. Find the cost of 18.5 m of cloth.
Answer:
Cost of 1 m of cloth = Rs. 108.50
So, cost of 18.5 m of cloth = \( \text{Rs. } (108.50 \times 18.5) = \text{Rs. } 2007.25 \)
So, the cost of 18.5 m of cloth is Rs. 2007.25.

Exam Tip: Watch the decimal places carefully when both numbers being multiplied are themselves decimals - count places from both before placing the point.

Question 18. A car covers a distance of 8.6 km with 1 litre of petrol. What distance will it cover with 36.5 litres of petrol?
Answer:
Distance covered by the car with 1 litre of petrol = 8.6 km
So, distance covered with 36.5 litres of petrol = \( (8.6 \times 36.5) \text{ km} = 313.900 \text{ km} \)
So, the distance covered by the car with 36.5 litres of petrol is 313.9 km.

Exam Tip: Read "per litre" or "per unit" carefully - it tells you which number to multiply by the given quantity.

Question 19. A taxi driver charges Rs. 9.80 per km. What will he charge for a journey of 106.5 km?
Answer:
Charge for 1 km = Rs. 9.80
So, charge for 106.5 km = \( \text{Rs. } (9.80 \times 106.5) = \text{Rs. } 1043.70 \)
So, the taxi driver will charge Rs. 1043.70 for a journey of 106.5 km.

Exam Tip: A per-km fare question is another rate-times-quantity multiplication - keep the money value to two decimal places at the end.

Exercise 3D

Question 1. Divide:
(i) \( 131.6 \div 10 \)
(ii) \( 32.56 \div 10 \)
(iii) \( 4.38 \div 10 \)
(iv) \( 0.34 \div 10 \)
(v) \( 0.08 \div 10 \)
(vi) \( 0.062 \div 10 \)
Answer:
(i) \( 131.6 \div 10 = \frac{131.6}{10} = 13.16 \) [decimal point moves left by 1 place]
(ii) \( 32.56 \div 10 = \frac{32.56}{10} = 3.256 \) [decimal point moves left by 1 place]
(iii) \( 4.38 \div 10 = \frac{4.38}{10} = 0.438 \) [decimal point moves left by 1 place]
(iv) \( 0.34 \div 10 = \frac{0.34}{10} = 0.034 \) [decimal point moves left by 1 place]
(v) \( 0.08 \div 10 = \frac{0.08}{10} = 0.008 \) [decimal point moves left by 1 place]
(vi) \( 0.062 \div 10 = \frac{0.062}{10} = 0.0062 \) [decimal point moves left by 1 place]

Exam Tip: Dividing by 10, 100 or 1000 just shifts the decimal point left by 1, 2 or 3 places - there is no long division required.

Question 2. Divide:
(i) \( 137.2 \div 100 \)
(ii) \( 23.4 \div 100 \)
(iii) \( 4.7 \div 100 \)
(iv) \( 0.3 \div 100 \)
(v) \( 0.58 \div 100 \)
(vi) \( 0.02 \div 100 \)
Answer:
(i) \( 137.2 \div 100 = \frac{137.2}{100} = 1.372 \) [decimal point moves left by 2 places]
(ii) \( 23.4 \div 100 = \frac{23.4}{100} = 0.234 \) [decimal point moves left by 2 places]
(iii) \( 4.7 \div 100 = \frac{4.7}{100} = 0.047 \) [decimal point moves left by 2 places]
(iv) \( 0.3 \div 100 = \frac{0.3}{100} = 0.003 \) [decimal point moves left by 2 places]
(v) \( 0.58 \div 100 = \frac{0.58}{100} = 0.0058 \) [decimal point moves left by 2 places]
(vi) \( 0.02 \div 100 = \frac{0.02}{100} = 0.0002 \) [decimal point moves left by 2 places]

Exam Tip: Dividing by 10, 100 or 1000 just shifts the decimal point left by 1, 2 or 3 places - there is no long division required.

Question 3. Divide:
(i) \( 1286.5 \div 1000 \)
(ii) \( 354.16 \div 1000 \)
(iii) \( 38.9 \div 1000 \)
(iv) \( 4.6 \div 1000 \)
(v) \( 0.8 \div 1000 \)
(vi) \( 2 \div 1000 \)
Answer:
(i) \( 1286.5 \div 1000 = \frac{1286.5}{1000} = 1.2865 \) [decimal point moves left by 3 places]
(ii) \( 354.16 \div 1000 = \frac{354.16}{1000} = 0.35416 \) [decimal point moves left by 3 places]
(iii) \( 38.9 \div 1000 = \frac{38.9}{1000} = 0.0389 \) [decimal point moves left by 3 places]
(iv) \( 4.6 \div 1000 = \frac{4.6}{1000} = 0.0046 \) [decimal point moves left by 3 places]
(v) \( 0.8 \div 1000 = \frac{0.8}{1000} = 0.0008 \) [decimal point moves left by 3 places]
(vi) \( 2 \div 1000 = \frac{2}{1000} = 0.002 \) [decimal point moves left by 3 places]

Exam Tip: Dividing by 10, 100 or 1000 just shifts the decimal point left by 1, 2 or 3 places - there is no long division required.

Question 4. Convert each of the following fractions into a decimal:
(i) \( \frac{12}{8} \)
(ii) \( \frac{63}{15} \)
(iii) \( \frac{47}{20} \)
(iv) \( \frac{101}{25} \)
(v) \( \frac{31}{40} \)
(vi) \( \frac{11}{16} \)
Answer:
(i) \( \frac{12}{8} = \frac{3}{2} \). Dividing gives \( 12 \div 8 = 1.5 \)
(ii) \( \frac{63}{15} = \frac{21}{5} \). Dividing gives \( 63 \div 15 = 4.2 \)
(iii) \( \frac{47}{20} \). Dividing gives \( 47 \div 20 = 2.35 \)
(iv) \( \frac{101}{25} \). Dividing gives \( 101 \div 25 = 4.04 \)
(v) \( \frac{31}{40} \). Two zeros are annexed for the division, giving \( 31 \div 40 = 0.775 \)
(vi) \( \frac{11}{16} \). Four zeros are annexed for the division, giving \( 11 \div 16 = 0.6875 \)

Exam Tip: Reduce the fraction first if possible, then carry out ordinary long division, adding zeros after the decimal point of the numerator as needed until the division ends.

Question 5. Divide:
(i) 43.2 by 6
(ii) 60.48 by 12
(iii) 117.6 by 21
(iv) 217.44 by 18
(v) 2.575 by 25
(vi) 6.08 by 8
(vii) 0.765 by 9
(viii) 0.768 by 16
(ix) 0.175 by 25
(x) 0.3322 by 11
(xi) 2.13 by 15
(xii) 6.54 by 12
(xiii) 5.52 by 16
(xiv) 1.001 by 14
(xv) 0.477 by 18
Answer:
(i) \( 43.2 \div 6 = 7.2 \)
(ii) \( 60.48 \div 12 = 5.04 \)
(iii) \( 117.6 \div 21 = 5.6 \)
(iv) \( 217.44 \div 18 = 12.08 \)
(v) \( 2.575 \div 25 = 0.103 \)
(vi) \( 6.08 \div 8 = 0.76 \)
(vii) \( 0.765 \div 9 = 0.085 \)
(viii) \( 0.768 \div 16 = 0.048 \)
(ix) \( 0.175 \div 25 = \frac{0.175 \times 1000}{25 \times 1000} = \frac{175}{25000} = \frac{7}{1000} = 0.007 \)
(x) \( 0.3322 \div 11 = 0.0302 \)
(xi) \( 2.13 \div 15 = 0.142 \)
(xii) \( 6.54 \div 12 = 0.545 \)
(xiii) \( 5.52 \div 16 = 0.345 \)
(xiv) \( 1.001 \div 14 = 0.0715 \)
(xv) \( 0.477 \div 18 = 0.0265 \)

Exam Tip: When the dividend runs out of digits before the division is complete, keep annexing zeros after the decimal point and continue dividing until the remainder becomes zero.

Question 6. Divide:
(i) 16.46 by 20
(ii) 403.8 by 30
(iii) 19.2 by 80
(iv) 156.8 by 200
(v) 12.8 by 500
(vi) 18.08 by 400
Answer:
(i) \( 16.46 \div 20 = \frac{16.46}{20} = \frac{16.46 \times 100}{2000} = \frac{1646}{1000} = \frac{823}{1000} = 0.823 \)
(ii) \( 403.8 \div 30 = \frac{4038}{300} = \frac{1346}{100} = 13.46 \)
(iii) \( 19.2 \div 80 = \frac{192}{800} = \frac{24}{100} = 0.24 \)
(iv) \( 156.8 \div 200 = \frac{1568}{2000} = \frac{784}{1000} = 0.784 \)
(v) \( 12.8 \div 500 = \frac{128}{5000} = \frac{256}{10000} = 0.0256 \)
(vi) \( 18.08 \div 400 = \frac{1808}{40000} = \frac{452}{10000} = 0.0452 \)

Exam Tip: When dividing by a multiple of 10 such as 20, 30, 80 or 500, first multiply both the dividend and divisor by 10 to clear the decimal, then simplify the resulting fraction.

Question 7. Divide:
(i) 3.28 by 0.8
(ii) 0.288 by 0.9
(iii) 25.395 by 1.5
(iv) 2.0484 by 0.18
(v) 0.228 by 0.38
(vi) 0.8085 by 0.35
(vii) 21.976 by 1.64
(viii) 11.04 by 1.6
(ix) 6.612 by 11.6
(x) 0.076 by 0.19
(xi) 48 by 0.074
(xii) 16.578 by 5.4
(xiii) 28 by 0.56
(xiv) 3 by 80
Answer:
(i) \( 3.28 \div 0.8 = \frac{3.28 \times 10}{0.8 \times 10} = \frac{32.8}{8} = 4.1 \)
(ii) \( 0.288 \div 0.9 = \frac{2.88}{9} = 0.32 \)
(iii) \( 25.395 \div 1.5 = \frac{253.95}{15} = 16.93 \)
(iv) \( 2.0484 \div 0.18 = \frac{204.84}{18} = 11.38 \)
(v) \( 0.228 \div 0.38 = \frac{22.8}{38} = 0.6 \)
(vi) \( 0.8085 \div 0.35 = \frac{80.85}{35} = 2.31 \)
(vii) \( 21.976 \div 1.64 = \frac{2197.6}{164} = 13.4 \)
(viii) \( 11.04 \div 1.6 = \frac{110.4}{16} = 6.9 \)
(ix) \( 6.612 \div 11.6 = \frac{66.12}{116} = 0.57 \)
(x) \( 0.076 \div 0.19 = \frac{7.6}{19} = 0.4 \)
(xi) \( 48 \div 0.074 = \frac{48000}{74} = 2 \times 1000 = 2000 \)
(xii) \( 16.578 \div 5.4 = \frac{165.78}{54} = 3.07 \)
(xiii) \( 28 \div 0.56 = \frac{2800}{56} = \frac{100}{2} = 50 \)
(xiv) \( 3 \div 80 = \frac{3}{80} = 0.0375 \)

Exam Tip: When dividing by a decimal, multiply both numbers by 10, 100 or 1000 (whatever turns the divisor into a whole number) before carrying out the division.

Question 9. The cloth required to stitch one shirt is 1.8 m. How many shirts can be made from 45 m of cloth?
Answer:
Cloth required for 1 shirt = 1.8 m
So, number of shirts that can be made from 45 m of cloth = \( \frac{45}{1.8} = \frac{450}{18} = \frac{50}{2} = 25 \)
So, 25 shirts can be made from a 45 m piece of cloth.

Exam Tip: For "how many pieces from a total length" problems, divide the total length by the length needed for one piece.

Question 10. A car covers a distance of 22.8 km with 2.4 litres of petrol. What distance will it cover with 1 litre of petrol?
Answer:
Distance covered by the car with 2.4 litres of petrol = 22.8 km
So, distance covered with 1 litre of petrol = \( \frac{22.8}{2.4} = \frac{228}{24} = \frac{228 \div 12}{24 \div 12} = \frac{19}{2} = 9\frac{1}{2} \text{ km} \)
So, the distance the car covers with 1 litre of petrol is \( 9\frac{1}{2} \) km.

Exam Tip: To find the "per unit" value, always divide the total amount by the number of units given - here, total distance divided by total litres.

Question 11. The capacity of a tin of oil is 16.5 litres. How many tins will be required to hold 478.5 litres of oil?
Answer:
Capacity of 1 tin of oil = 16.5 litres
So, number of tins required to hold 478.5 litres of oil = \( \frac{478.5}{16.5} = \frac{4785}{165} = \frac{4785 \div 15}{165 \div 15} = \frac{319}{11} = 29 \)
So, 29 oil tins will be needed to hold 478.5 litres of oil.

Exam Tip: Divide the total quantity by the capacity of one container to find the number of containers needed - simplify the fraction step by step to keep the numbers manageable.

Question 12. The total weight of 37 bags of sugar is 3644.5 kg. Find the weight of each bag.
Answer:
Weight of 37 bags of sugar = 3644.5 kg
So, weight of 1 bag of sugar = \( \frac{3644.5}{37} = 98.5 \text{ kg} \)
So, each bag of sugar weighs 98.5 kg.

Exam Tip: To find the weight of one item from a total, divide the total weight by the number of items - a direct long division problem.

Question 13. The total capacity of 69 buckets of water is 586.5 litres. Find the capacity of each bucket.
Answer:
Capacity of 69 buckets of water = 586.5 litres
So, capacity of 1 such bucket = \( \frac{586.5}{69} = 8.5 \text{ litres} \)
So, the capacity of each water bucket is 8.5 litres.

Exam Tip: Same idea as before: divide the combined capacity by the number of buckets to get the capacity of a single bucket.

Question 14. Monica has 46 m of cloth. How many pieces, each of length 1.15 m, can she cut from it?
Answer:
Length of one piece of cloth = 1.15 m
So, number of pieces she can get from 46 m of cloth = \( \frac{46}{1.15} = \frac{46 \times 100}{1.15 \times 100} = \frac{4600}{115} = 40 \)
So, Monica has 40 pieces of cloth, each of length 1.15 m.

Exam Tip: Multiply both the total length and the piece length by 100 first to remove the decimal, which makes the division much simpler.

Question 15. Mr. Soni bought some bags of cement with a total weight of 1792.8 kg. If each bag weighs 49.8 kg, how many bags did he buy?
Answer:
Total weight of all the bags of cement = 1792.8 kg
Weight of each bag = 49.8 kg
So, number of bags = \( \frac{\text{Total weight}}{\text{Weight of each bag}} = \frac{1792.8}{49.8} = \frac{17928}{498} = 36 \)
So, Mr. Soni bought 36 bags of cement.

Exam Tip: Set up the division as total weight over unit weight, then clear the decimals by multiplying both parts by the same power of 10 before dividing.

Question 16. The thickness of a pile of plywood pieces is 1.89 m. If the thickness of one piece of plywood is 0.35 cm, find the number of pieces of plywood in the pile.
Answer:
Thickness of the pile of plywood pieces = \( 1.89 \text{ m} = 189 \text{ cm} \)
Thickness of one piece of plywood = 0.35 cm
So, the number of plywood pieces needed = \( \frac{189}{0.35} = \frac{189 \times 100}{0.35 \times 100} = \frac{18900}{35} = 540 \)
So, 540 pieces of plywood are needed to make a pile 1.89 m high.

Exam Tip: Convert every measurement to the same unit before dividing - here, changing the pile height from metres to centimetres first avoids errors.

Question 17. The product of two decimals is 261.36. If one of the decimals is 17.6, find the other.
Answer:
Product of the two decimals = 261.36
One of the decimals = 17.6
So, the other decimal = \( 261.36 \div 17.6 = \frac{2613.6}{176} = 14.85 \)
So, the other decimal is 14.85.

Exam Tip: To find a missing factor, divide the product by the factor you already know - the same approach used for whole numbers, just carried out with decimals.

Exercise 3E

Question 1. Expressed as a fraction in its lowest terms, 0.06 equals:
(a) \( \frac{3}{5} \)
(b) \( \frac{3}{50} \)
(c) \( \frac{6}{50} \)
(d) \( \frac{3}{500} \)
\( 0.06 = \frac{6}{100} = \frac{3}{50} \)
Answer: (b) \( \frac{3}{50} \)
In simple words: Write the digits after the point over 100, then cancel to the simplest form.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 2. Expressed as a mixed fraction, 1.04 equals:
(a) \( 1\frac{4}{100} \)
(b) \( 1\frac{1}{4} \)
(c) \( 1\frac{1}{25} \)
(d) \( 1\frac{2}{25} \)
\( 1.04 = \frac{104}{100} = \frac{26}{25} = 1\frac{1}{25} \)
Answer: (c) \( 1\frac{1}{25} \)
In simple words: Reduce the fraction part fully before writing it as a mixed number.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 3. Expressed as a decimal, \( 2\frac{2}{25} \) equals:
(a) 2.8
(b) 2.08
(c) 2.008
(d) 2.25
\( 2\frac{2}{25} = \frac{52}{25} \). On dividing, \( \frac{52}{25} = 2.08 \)
Answer: (b) 2.08
In simple words: Turn the mixed number into an improper fraction first, then divide the numerator by the denominator.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 5. 70 g expressed in kg is:
(a) 0.7 kg
(b) 0.07 kg
(c) 0.007 kg
(d) 7 kg
\( 70 \text{ g} = \frac{70}{1000} \text{ kg} = \frac{7}{100} \text{ kg} = 0.07 \text{ kg} \)
Answer: (b) 0.07 kg
In simple words: Remember 1000 g make 1 kg, so divide the gram value by 1000 to convert.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 6. 5 kg 6 g expressed in kg is:
(a) 5.06 kg
(b) 5.6 kg
(c) 5.006 kg
(d) 5.0006 kg
\( 5 \text{ kg } 6 \text{ g} = (5 \times 1000) \text{ g} + 6 \text{ g} = 5006 \text{ g} = \frac{5006}{1000} \text{ kg} = 5.006 \text{ kg} \)
Answer: (c) 5.006 kg
In simple words: Convert everything into grams first, then divide by 1000 to express the answer in kg.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 7. 2 km 5 m expressed in km is:
(a) 2.05 km
(b) 2.5 km
(c) 2.005 km
(d) 2.0005 km
\( 2 \text{ km } 5 \text{ m} = (2 \times 1000) \text{ m} + 5 \text{ m} = 2005 \text{ m} = \frac{2005}{1000} \text{ km} = 2.005 \text{ km} \)
Answer: (c) 2.005 km
In simple words: Convert everything into metres first, then divide by 1000 to express the answer in km.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 8. Which number should be subtracted from 1.007 to get 0.7?
(a) 0.37
(b) 0.037
(c) 0.307
(d) 3.07
Turning the decimals into like decimals gives 1.007 and 0.700. Placing the larger one on top and subtracting gives \( 1.007 - 0.700 = 0.307 \)
Answer: (c) 0.307
In simple words: This is a "what should be subtracted" question, so work it out as the bigger number minus the smaller one.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 9. If \( 0.1 - x = 0.03 \), then x equals:
(a) 0.7
(b) 0.07
(c) 0.007
(d) 0.13
We have \( 0.1 - x = 0.03 \), so \( x = 0.1 - 0.03 \). Turning them into like decimals gives 0.10 and 0.03, so \( x = 0.07 \)
Answer: (b) 0.07
In simple words: Rearrange the equation so x is alone, then simply subtract the two decimals given.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 10. If \( 3.07 + x = 3.5 \), then x equals:
(a) 4.3
(b) 0.043
(c) 0.43
(d) 0.34
We have \( 3.07 + x = 3.5 \), so \( x = 3.5 - 3.07 \). Turning them into like decimals gives 3.50 and 3.07, so \( x = 0.43 \)
Answer: (c) 0.43
In simple words: Move the known decimal to the other side of the equation, turning it into a straightforward subtraction.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 11. \( 0.23 \times 0.3 \) equals:
(a) 0.69
(b) 0.0069
(c) 0.069
(d) 6.9
\( 23 \times 3 = 69 \). The total decimal places in the given numbers = 3, so \( 0.23 \times 0.3 = 0.069 \)
Answer: (c) 0.069
In simple words: Multiply the digits first, then count the decimal places from both numbers together before fixing the point.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 12. \( 0.02 \times 30 \) equals:
(a) 6
(b) 0.6
(c) 0.06
(d) 0.006
\( 2 \times 30 = 60 \). The total decimal places = 2, so \( 0.02 \times 30 = 0.60 = 0.6 \)
Answer: (b) 0.6
In simple words: Even when one number is a whole number, still count only the decimal places from the decimal number involved.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 13. \( 0.25 \times 0.8 \) equals:
(a) 2
(b) 0.2
(c) 0.02
(d) 20
\( 25 \times 8 = 200 \). The total decimal places = 3, so \( 0.25 \times 0.8 = 0.200 = 0.2 \)
Answer: (b) 0.2
In simple words: After placing the decimal point, drop any unnecessary trailing zeros to reach the simplest form of the answer.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 14. \( 0.4 \times 0.4 \times 0.4 \) equals:
(a) 0.64
(b) 0.0064
(c) 0.064
(d) 6.4
\( 4 \times 4 \times 4 = 64 \). The total decimal places = 3, so \( 0.4 \times 0.4 \times 0.4 = 0.064 \)
Answer: (c) 0.064
In simple words: This is the same as \( (0.4)^3 \) - count the decimal places across all three copies of the number.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 15. \( 1.1 \times 0.1 \times 0.01 \) equals:
(a) 0.011
(b) 0.0011
(c) 0.11
(d) 0.00011
\( 11 \times 1 \times 1 = 11 \). The total decimal places = 4, so \( 1.1 \times 0.1 \times 0.01 = 0.0011 \)
Answer: (b) 0.0011
In simple words: With very small decimals, be extra careful counting the total number of decimal places - a single miscount changes the answer a lot.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 16. \( 2.08 \div 0.16 \) equals:
(a) 13
(b) 1.3
(c) 130
(d) 0.13
\( 2.08 \div 0.16 = \frac{2.08 \times 100}{0.16 \times 100} = \frac{208}{16} = 13 \)
Answer: (a) 13
In simple words: Multiply both numbers by 100 to remove the decimals, then simplify to an ordinary division.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 17. \( 1.02 \div 6 \) equals:
(a) 1.7
(b) 0.17
(c) 0.017
(d) 17
\( 1.02 \div 6 = \frac{1.02 \times 100}{6 \times 100} = \frac{102}{600} = \frac{17}{100} = 0.17 \)
Answer: (b) 0.17
In simple words: Dividing a decimal by a whole number works exactly like ordinary long division, once the place value is tracked correctly.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 18. \( 30.94 \div 0.7 \) equals:
(a) 44.2
(b) 4.42
(c) 442
(d) 4.24
\( 30.94 \div 0.7 = \frac{30.94 \times 100}{0.7 \times 100} = \frac{3094}{70} = 44.2 \)
Answer: (a) 44.2
In simple words: Clearing the decimal from the divisor first makes the division much simpler to carry out.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 19. \( 2.73 \div 1.3 \) equals:
(a) 21
(b) 2.1
(c) 0.21
(d) 1.2
\( 2.73 \div 1.3 = \frac{2.73 \times 100}{1.3 \times 100} = \frac{273}{130} = \frac{21}{10} = 2.1 \)
Answer: (b) 2.1
In simple words: Simplify the resulting fraction fully before converting it back to decimal form.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 20. \( 89.1 \div 2.2 \) equals:
(a) 40.5
(b) 4.05
(c) 405
(d) 45.0
\( 89.1 \div 2.2 = \frac{89.1 \times 10}{2.2 \times 10} = \frac{891}{22} = 40.5 \)
Answer: (a) 40.5
In simple words: When both numbers have only one decimal place, multiplying both by 10 is enough to clear the decimals.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.

Question 21. \( 0.5 \times 0.05 \) equals:
(a) 0.25
(b) 0.0025
(c) 0.025
(d) 2.5
\( 5 \times 5 = 25 \). The total decimal places = 3, so \( 0.5 \times 0.05 = 0.025 \)
Answer: (c) 0.025
In simple words: Count one decimal place from the first number and two from the second, giving three places in total for the answer.

Exam Tip: Work out the value first on rough paper, then match it to the closest option - this avoids being misled by close-looking distractors.