Chapter 3 exercise answers: A Peek Beyond the Point

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(a) 3/10 + 3 4/100

1. 3/10 + 3 4/100: add the wholes, tenths and hundredths separately.
2. Wholes 3, tenths 3, hundredths 4, so the sum is 3 3/10 4/100 = 3 34/100.

Answer3 34/100 = 3.34

(b) 9 5/10 7/100 + 2 1/10 3/100

1. Wholes: 9 + 2 = 11. Tenths: 5 + 1 = 6. Hundredths: 7 + 3 = 10.
2. 10 hundredths make 1 tenth, so 11 6/10 10/100 = 11 7/10.

Answer11 7/10 = 11.7

(c) 15 6/10 4/100 + 14 3/10 6/100

1. Wholes: 15 + 14 = 29. Tenths: 6 + 3 = 9. Hundredths: 4 + 6 = 10.
2. 10 hundredths make 1 tenth, giving 29 10/10, and 10 tenths make 1 whole, giving 30.

Answer30

(d) 7 7/100 − 4 4/100

1. Wholes: 7 − 4 = 3. Hundredths: 7 − 4 = 3.

Answer3 3/100 = 3.03

(e) 8 6/100 − 5 3/100

1. Wholes: 8 − 5 = 3. Hundredths: 6 − 3 = 3.

Answer3 3/100 = 3.03

(f) 12 6/100 2/100 − 9/10 9/100

1. The book prints 12 6/100 2/100 − 9/10 9/100. The first number has two hundredths parts and no tenths part, unlike every other number in this exercise, which is written as wholes, then tenths, then hundredths. So 6/100 is almost certainly a misprint for 6/10. NCERT's answer key reads it as 12 6/10 2/100.
2. Read as 12 6/10 2/100: that is 12.62, and 9/10 9/100 is 0.99.
3. 12.62 − 0.99: borrow 1 whole, so 11 16/10 2/100 − 9/10 9/100. Borrow 1 tenth: 11 15/10 12/100 − 9/10 9/100 = 11 6/10 3/100 = 11 63/100.
4. Read exactly as printed: 12 + 6/100 + 2/100 = 12 8/100 = 12.08.
5. 12.08 − 0.99 = 11.09.

AnswerRead as 12 6/10 2/100 (NCERT's answer key): 11 63/100 = 11.63. Read exactly as printed, 12 6/100 2/100: 11 9/100 = 11.09.

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(a) 5.3 + 2.6

1. Line up the decimal points.
2. Add tenths: 3 + 6 = 9. Add units: 5 + 2 = 7.
3. 5.3 + 2.6 = 7.9.

Answer7.9

(b) 18 + 8.8

1. Write 18 as 18.0.
2. Add: 18.0 + 8.8 = 26.8.

Answer26.8

(c) 2.15 + 5.26

1. Line up the points: 2.15 and 5.26.
2. Hundredths: 5 + 6 = 11, write 1, carry 1 to tenths.
3. Tenths: 1 + 2 + 1(carried) = 4. Units: 2 + 5 = 7.
4. 2.15 + 5.26 = 7.41.

Answer7.41

(d) 9.01 + 9.10

1. 9.01 + 9.10.
2. Hundredths: 1 + 0 = 1. Tenths: 0 + 1 = 1. Units: 9 + 9 = 18.
3. The sum is 18.11.

Answer18.11

(e) 29.19 + 9.91

1. 29.19 + 9.91.
2. Hundredths: 9 + 1 = 10, write 0, carry 1.
3. Tenths: 1 + 9 + 1(carried) = 11, write 1, carry 1.
4. Units: 29 + 9 + 1(carried) = 39.
5. The sum is 39.10, that is, 39.1.

Answer39.1

(f) 0.934 + 0.6

1. Write 0.6 as 0.600 to line up with 0.934.
2. Add thousandths, hundredths, tenths, units in turn: 0.934 + 0.600 = 1.534.

Answer1.534

(g) 0.75 + 0.03

1. 0.75 + 0.03.
2. Hundredths: 5 + 3 = 8. Tenths: 7 + 0 = 7.
3. The sum is 0.78.

Answer0.78

(h) 6.236 + 0.487

1. 6.236 + 0.487.
2. Thousandths: 6 + 7 = 13, write 3, carry 1.
3. Hundredths: 3 + 8 + 1(carried) = 12, write 2, carry 1.
4. Tenths: 2 + 4 + 1(carried) = 7. Units: 6 + 0 = 6.
5. The sum is 6.723.

Answer6.723

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(a) 5.6 − 2.3

1. Tenths: 6 − 3 = 3. Units: 5 − 2 = 3.
2. 5.6 − 2.3 = 3.3.

Answer3.3

(b) 18 − 8.8

1. Write 18 as 18.0.
2. 18.0 − 8.8: tenths 0 − 8 will not work, so borrow a unit: 17 units and 10 tenths.
3. 10 − 8 = 2 tenths; 17 − 8 = 9 units.
4. The answer is 9.2.

Answer9.2

(c) 10.4 − 4.5

1. 10.4 − 4.5: tenths 4 − 5 will not work, so borrow a unit: 9 units and 14 tenths.
2. 14 − 5 = 9 tenths; 9 − 4 = 5 units.
3. The answer is 5.9.

Answer5.9

(d) 17 − 16.198

1. Write 17 as 17.000 to match the thousandths in 16.198.
2. 17.000 − 16.198 needs borrowing right through units, tenths and hundredths.
3. It is the same as (17.000 − 16.000) − 0.198 = 1.000 − 0.198 = 0.802.

Answer0.802

(e) 17 − 0.05

1. Write 17 as 17.00.
2. 17.00 − 0.05: borrow so that 17.00 becomes 16 units, 9 tenths and 10 hundredths.
3. 10 − 5 = 5 hundredths; 9 − 0 = 9 tenths; 16 − 0 = 16 units.
4. The answer is 16.95.

Answer16.95

(f) 34.505 − 18.1

1. Write 18.1 as 18.100 to line up with 34.505.
2. Thousandths 5 − 0 = 5. Hundredths 0 − 0 = 0.
3. Tenths 5 − 1 = 4. Units 34 − 18 = 16.
4. The answer is 16.405.

Answer16.405

(g) 9.9 − 9.09

1. Write 9.9 as 9.90 to line up with 9.09.
2. Hundredths 0 − 9 will not work, so borrow a tenth: 9.90 becomes 9 units, 8 tenths, 10 hundredths.
3. 10 − 9 = 1 hundredth; 8 − 0 = 8 tenths; 9 − 9 = 0 units.
4. The answer is 0.81.

Answer0.81

(h) 6.236 − 0.487

1. Line up the points: 6.236 − 0.487.
2. Thousandths: 6 − 7 will not work, so break 1 hundredth into 10 thousandths. The 3 hundredths become 2 hundredths, and the 6 thousandths become 16. 16 − 7 = 9 thousandths.
3. Hundredths: 2 − 8 will not work, so break 1 tenth into 10 hundredths. The 2 tenths become 1 tenth, and the 2 hundredths become 12. 12 − 8 = 4 hundredths.
4. Tenths: 1 − 4 will not work, so break 1 unit into 10 tenths. The 6 units become 5 units, and the 1 tenth becomes 11. 11 − 4 = 7 tenths.
5. Units: 5 − 0 = 5. The answer is 5.749.

Answer5.749

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(a) 5/100

1. 5/100 means 5 hundredths.
2. Put the 5 in the hundredths place, with a 0 in the tenths place to hold it there: 0.05.

Answer0.05

(b) 16/1000

1. 16/1000 means 16 thousandths.
2. 16 thousandths is 1 hundredth and 6 thousandths, so write 0 in the tenths place, 1 in the hundredths place, 6 in the thousandths place: 0.016.

Answer0.016

(c) 12/10

1. 12/10 means 12 tenths.
2. 10 of those tenths make 1 whole unit, leaving 2 tenths: 1.2.

Answer1.2

(d) 254/1000

1. 254/1000 means 254 thousandths.
2. 254 thousandths is 2 tenths, 5 hundredths and 4 thousandths: 0.254.

Answer0.254

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(a) 0.34

1. 0.34 has 3 in the tenths place, 4 in the hundredths place and nothing in the thousandths place.
2. So 0.34 = 3/10 + 4/100 + 0/1000.

Answer0.34 = 3 tenths + 4 hundredths + 0 thousandths

(b) 1.02

1. 1.02 has 1 unit, 0 tenths, 2 hundredths and 0 thousandths: 1.02 = 1 + 2/100.
2. The sum must use only tenths, hundredths and thousandths, so change the 1 unit into tenths: 1 unit = 10 tenths.
3. So 1.02 = 10/10 + 2/100 + 0/1000.

Answer1.02 = 10 tenths + 2 hundredths + 0 thousandths (that is, 1 + 2/100)

(c) 0.8

1. 0.8 has 8 in the tenths place and nothing after it.
2. So 0.8 = 8/10 + 0/100 + 0/1000.

Answer0.8 = 8 tenths + 0 hundredths + 0 thousandths

(d) 0.362

1. 0.362 has 3 in the tenths place, 6 in the hundredths place and 2 in the thousandths place.
2. So 0.362 = 3/10 + 6/100 + 2/1000.

Answer0.362 = 3 tenths + 6 hundredths + 2 thousandths

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1. The line runs from 6.4 to 6.6, a gap of 0.2, and it is cut into 8 equal small parts by the marks shown.
2. So each small part is 0.2 ÷ 8 = 0.025.
3. Point a is the mark 2 parts after 6.4: 6.4 + 2 × 0.025 = 6.45.
4. Point c is the mark 5 parts after 6.4 (one part after the mid-mark 6.5): 6.4 + 5 × 0.025 = 6.525.
5. Point b is the mark 6 parts after 6.4 (two parts after 6.5): 6.4 + 6 × 0.025 = 6.55.

Answera = 6.45, c = 6.525, b = 6.55

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(a) 11.01, 1.011, 1.101, 11.10, 1.01

1. Compare the units digit first: 11.01 and 11.10 both start with 11, which beats the numbers starting with 1.
2. Between 11.01 and 11.10, compare tenths: 1 tenth beats 0 tenths, so 11.10 comes first, then 11.01.
3. Among 1.101, 1.011, 1.01, compare tenths: 1.101 has 1 tenth, the other two have 0 tenths, so 1.101 comes next.
4. Write 1.01 as 1.010 to compare with 1.011 at the thousandths place: 1.011 is bigger by 1 thousandth.
5. Descending order: 11.10, 11.01, 1.101, 1.011, 1.01.

Answer11.10, 11.01, 1.101, 1.011, 1.01

(b) 2.567, 2.675, 2.768, 2.499, 2.698

1. All five start with 2, so compare tenths: 2.768 and 2.698 have 7 and 6 tenths; 2.675 and 2.499 have 6 and 4.
2. 2.768 (7 tenths) is the biggest.
3. Among the rest, 2.698 and 2.675 both have 6 tenths; hundredths 9 beats 7, so 2.698 comes next, then 2.675.
4. 2.567 (5 tenths) comes next, and 2.499 (4 tenths) is the smallest.
5. Descending order: 2.768, 2.698, 2.675, 2.567, 2.499.

Answer2.768, 2.698, 2.675, 2.567, 2.499

(c) 4.678 g, 4.595 g, 4.600 g, 4.656 g, 4.666 g

1. All five begin 4.6.. except 4.595 g, which has fewer tenths (5) than the rest (6), so it is the smallest.
2. Among 4.600, 4.656, 4.666 and 4.678, compare hundredths: 0, 5, 6, 7 — so 4.678 g is biggest, then 4.666 g, then 4.656 g, then 4.600 g.
3. Descending order: 4.678 g, 4.666 g, 4.656 g, 4.600 g, 4.595 g.

Answer4.678 g, 4.666 g, 4.656 g, 4.600 g, 4.595 g

(d) 33.13 m, 33.31 m, 33.133 m, 33.331 m, 33.313 m

1. All start with 33, so look at the tenths: 33.31, 33.331 and 33.313 have 3 tenths; 33.13 and 33.133 have 1 tenth.
2. So 33.13 m and 33.133 m are the two smallest.
3. Among 33.31, 33.331, 33.313 (all 3 tenths), compare hundredths: 33.31 has 1 hundredth, the other two have 3.
4. So 33.31 m is smaller than both 33.331 m and 33.313 m.
5. Between 33.331 m and 33.313 m, compare thousandths: 1 versus 3, so 33.331 m is bigger.
6. Between 33.13 m and 33.133 m, 33.133 m has an extra thousandth, so it is bigger.
7. Descending order: 33.331 m, 33.313 m, 33.31 m, 33.133 m, 33.13 m.

Answer33.331 m, 33.313 m, 33.31 m, 33.133 m, 33.13 m

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(a) the decimal number closest to 30

1. Use each of the digits 1, 4, 0, 8, 6 exactly once, some before the decimal point and the rest after it (the same rule the chapter uses for its "closest to 25" puzzle).
2. The whole-number part decides most of the value, so it should be a 2-digit number near 30. From these digits the nearest are 18 (below 30) and 40 (above 30).
3. Below 30: the largest number starting 18. is 18.640, which is 30 − 18.640 = 11.36 away.
4. Above 30: the smallest number starting 40. puts the leftover 1, 6, 8 in increasing order, giving 40.168, which is 40.168 − 30 = 10.168 away.
5. A 1-digit whole part (at most 8.641) or a 3-digit one (at least 104.68) is much farther away.
6. Since 10.168 is less than 11.36, 40.168 is the closest.

Answer40.168

(b) the smallest possible decimal number between 100 and 1000

1. Between 100 and 1000 means the whole-number part has exactly 3 digits.
2. For the smallest one, the first digit is the smallest digit that is not 0, which is 1; then 0; then 4. The whole-number part is 104.
3. Put the leftover 6 and 8 after the point, smaller first: 104.68.
4. Any other 3-digit whole-number part is at least 106, so 104.68 is the smallest.

Answer104.68

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1. Compare 0.9, which has only 1 digit after the point, with 0.12345, which has 5 digits after the point.
2. Compare tenths first: 0.9 has 9 tenths, 0.12345 has only 1 tenth, so 0.9 is bigger even though it has fewer digits.
3. So a number is not decided by how many digits it has, but by the value of its digits, starting from the left.

AnswerNo — the number of digits does not decide the size; what matters is the value of the digits starting from the leftmost place.

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1. Write all five weights with the same number of decimal places: 0.25, 0.30, 0.50, 0.20, 0.05 kg.
2. Add the hundredths: 5 + 0 + 0 + 0 + 5 = 10, so write 0 and carry 1 to the tenths.
3. Add the tenths with the carry: 2 + 3 + 5 + 2 + 0 + 1(carried) = 13, so write 3 and carry 1 to the units.
4. There is 1 unit from the carry, so the total is 1.30 kg.

Answer1.3 kg

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1. First add the milk from the first three days: 3.79 + 4.2 + 4.25.
2. Line up the decimals (4.2 is 4.20): hundredths 9+0+5=14, write 4 carry 1; tenths 7+2+2+1(carried)=12, write 2 carry 1; units 3+4+4+1(carried)=12.
3. The first three days come to 12.24 L.
4. The last three days is the 6-day total minus the first three days: 25.00 − 12.24.
5. Borrowing across the units and tenths, 25.00 − 12.24 = 12.76.
6. So the last three days' supply is 12.76 L.

Answer12.76 L

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1. February's weight (34.50 kg) is less than January's (35.75 kg), so Tinku has lost weight.
2. Find how much: 35.75 − 34.50.
3. Hundredths 5−0=5; tenths 7−5=2; units 35−34=1.
4. The change is a loss of 1.25 kg.

AnswerHe lost weight; the change is −1.25 kg.

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1. Look at the steps: 5.5 → 6.4 is + 0.9; 6.4 → 6.39 is − 0.01; 6.39 → 7.29 is + 0.9; 7.29 → 7.28 is − 0.01.
2. So the rule is: add 0.9, then take away 0.01, again and again.
3. By this rule the next two terms after 7.28 should be 7.28 + 0.9 = 8.18 and 8.18 − 0.01 = 8.17. The book prints 6.18 and 6.17, a drop of 1.1 that no other step makes. NCERT's answer key writes the pattern with 8.18, 8.17, so 6.18, 6.17 is most likely a misprint.
4. Continuing the corrected pattern: 8.17 + 0.9 = 9.07, then 9.07 − 0.01 = 9.06.
5. If you continue from the printed 6.17 instead: 6.17 + 0.9 = 7.07, then 7.07 − 0.01 = 7.06.

AnswerWith the pattern read as … 8.18, 8.17 (NCERT's answer key): 9.07, 9.06. Continuing from the printed 6.17: 7.07, 7.06.

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1. 1 kilometre = 1000 metres.
2. 1 metre = 1000 millimetres.
3. So 1 kilometre = 1000 × 1000 millimetres = 1,000,000 millimetres.

Answer1,000,000 millimetres

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1. 45 paise is 45 hundredths of a rupee, that is, ₹0.45.
2. 1 lakh people means 1,00,000 people.
3. Total fee = ₹0.45 × 1,00,000.
4. 0.45 × 100000 = 45 × 1000 = 45,000.
5. The total insurance fee paid is ₹45,000.

Answer₹45,000

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(a) 10/1000 or 1/10

1. 10/1000 = 0.010, and 1/10 = 0.100.
2. 0.100 is bigger than 0.010, so 1/10 is greater.

Answer1/10 (= 0.1) is greater

(b) One-hundredth or 90 thousandths

1. One-hundredth = 1/100 = 0.01.
2. 90 thousandths = 90/1000 = 0.090, which is the same as 9/100.
3. 0.090 is bigger than 0.010, so 90 thousandths is greater.

Answer90 thousandths (= 9/100 = 0.09) is greater

(c) One-thousandth or 90 hundredths

1. One-thousandth = 1/1000 = 0.001.
2. 90 hundredths = 90/100 = 0.90, which is the same as 9/10.
3. 0.90 is far bigger than 0.001, so 90 hundredths is greater.

Answer90 hundredths (= 9/10 = 0.9) is greater

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(b) 12 tens and 12 tenths

1. 12 tens = 120.
2. 12 tenths = 1.2.
3. Add: 120 + 1.2 = 121.2.

Answer121.2

(c) 10 tens, 10 ones, 10 tenths, and 10 hundredths

1. 10 tens = 100; 10 ones = 10; 10 tenths = 1; 10 hundredths = 0.1.
2. Add: 100 + 10 + 1 + 0.1 = 111.1.

Answer111.1

(d) 25 tens, 25 ones, 25 tenths, and 25 hundredths

1. 25 tens = 250; 25 ones = 25; 25 tenths = 2.5; 25 hundredths = 0.25.
2. Add: 250 + 25 + 2.5 + 0.25 = 277.75.

Answer277.75

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1. Each number has 1 digit before the point and 3 after, so the two numbers use 8 digits, and no digit from 0-9 may be used twice.
2. The closest any sum can be to 10.5 is exactly 10.5, so try to hit it exactly.
3. Choose whole-number digits that add to 10, for example 4 and 6.
4. Choose decimal parts that add to 0.500 without reusing a digit, for example 0.102 and 0.398 (digits 1, 0, 2 and 3, 9, 8).
5. Check: 4.102 + 6.398 = 10.500 = 10.5 exactly, and the digits 4, 1, 0, 2, 6, 3, 9, 8 are all different, so no answer can be closer.
6. This puzzle has many correct answers; another is 1.032 + 9.468 = 10.5. Any pair of numbers with no repeated digit whose sum is exactly 10.5 is right.

Answer4.102 + 6.398 = 10.5 (one of many correct answers)

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(a) 1/2

1. 1/2 = 5/10, since multiplying top and bottom by 5 gives a tenths fraction.
2. 5/10 = 0.5.

Answer0.5

(b) 3/2

1. 3/2 = 15/10, multiplying top and bottom by 5.
2. 15/10 is 1 whole and 5 tenths, so 1.5.

Answer1.5

(c) 1/4

1. 1/4 = 25/100, multiplying top and bottom by 25.
2. 25/100 = 0.25.

Answer0.25

(d) 3/4

1. 3/4 = 75/100, multiplying top and bottom by 25.
2. 75/100 = 0.75.

Answer0.75

(e) 1/5

1. 1/5 = 2/10, multiplying top and bottom by 2.
2. 2/10 = 0.2.

Answer0.2

(f) 4/5

1. 4/5 = 8/10, multiplying top and bottom by 2.
2. 8/10 = 0.8.

Answer0.8

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