4. Another Peek Beyond the Point
Answers to all Figure it Out and in-text questions on multiplying and dividing decimals and the leap year calendar. CBSE / NCERT Class 7 Ganita Prakash Part 2.
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4.1 A Quick Recap of Decimals
310 = 0.3; 4100 = 0.04; 671000 = 0.067; 457100 = 4.57; 71100 = 0.71; 43100 = 0.43; 9100 = 0.09
Cinnamon 50 g = 501000 kg = 0.05 kg
Cumin 100 g = 1001000 kg = 0.1 kg
Cardamom 25 g = 251000 kg = 0.025 kg
Pepper 250 g = 2501000 kg = 0.25 kg
| Fraction | Expanding the numerator | Sum | Decimal |
|---|---|---|---|
| 84710000 | 80010000 + 4010000 + 710000 = 8100 + 41000 + 710000 | 0.08 + 0.004 + 0.0007 | 0.0847 |
| 173100 | 100100 + 70100 + 3100 = 1 + 710 + 3100 | 1 + 0.7 + 0.03 | 1.73 |
| 231000 | 201000 + 31000 = 2100 + 31000 | 0.02 + 0.003 | 0.023 |
Move the decimal point to the left by as many places as there are zeroes in the divisor.
12310 = 12.3; 24100 = 0.24; 6781000 = 0.678
4.2 Decimal Multiplication
Yes. For example, 2.5 × 0.4 = 1 and 1.25 × 0.8 = 1.
Yes. For example, 0.5 × 4 = 2 and 2.5 × 6 = 15.
5.96 has 2 decimal places and 24.8 has 1. So, the product has 3: 147.808.
5.8 × 1.24 = 5810 × 124100 = 71921000 = 7.192.
1. Find the products in tenths, hundredths, …
(a) 6 × 4 tenths = 24 tenths = 2.4
(b) 7 × 0.3 = 7 × 3 tenths = 21 tenths = 2.1
(c) 9 × 5 hundredths = 45 hundredths = 0.45
2. Find the products.
(a) 27.34 × 6 = 164.04
(b) 4.23 × 3.7 = 15.651
(c) 0.432 × 0.23 = 0.09936
3. Cloth for 3 shirts
1.65 × 3 = 4.95 m
4. Meenu’s total
4 × 15.50 + 3 × 2.75 = 62 + 8.25 = ₹70.25
5. Height of 36 coins
36 × 1.45 mm = 52.2 mm = 5.22 cm
6. Price of 2.250 kg oranges at ₹56.50 per kg
56.5 × 2.25 = ₹127.125 (about ₹127.13).
Yes, we can write 56.5 and 2.25. Zeros at the end of the decimal part do not change the value, so the product is the same.
7. Dwarakanath’s profit
Profit on one notebook = 30 − 23.6 = ₹6.4. For 50 notebooks: 6.4 × 50 = ₹320.
8. Given 18 × 12 = 216
(a) 21.6 (b) 2.16 (c) 2.16 (d) 0.0216 (e) 0.000216 (f) 21.6
The product is less than 1 in (d) and (e).
9. In which is the product less than 1?
(b) 0.7 × 0.6 = 0.42 and (d) 0.07 × 0.06 = 0.0042. Both numbers are less than 1, so the product is less than 1.
10. Complete the table.
| Number | × 10 | × 100 | × 1000 |
|---|---|---|---|
| 5.7 | 57 | 570 | 5700 |
| 23.02 | 230.2 | 2302 | 23020 |
| 0.92 | 9.2 | 92 | 920 |
| 0.306 | 3.06 | 30.6 | 306 |
| 24.67 | 246.7 | 2467 | 24670 |
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4.3 Decimal Division
0.039 m = 3.9 cm = 39 mm.
| Decimal | ÷ 10 | ÷ 100 | ÷ 1000 | ÷ 10000 |
|---|---|---|---|---|
| 18.7 | 1.87 | 0.187 | 0.0187 | 0.00187 |
| 21.1 | 2.11 | 0.211 | 0.0211 | 0.00211 |
| 0.13 | 0.013 | 0.0013 | 0.00013 | 0.000013 |
| 2.146 | 0.2146 | 0.02146 | 0.002146 | 0.0002146 |
| 0.0058 | 0.00058 | 0.000058 | 0.0000058 | 0.00000058 |
1. Find the quotient by making the denominator 10, 100 or 1000.
(a) 185 = 3610 = 3.6
(b) 4154 = 10375100 = 103.75
(c) 12172 = 608510 = 608.5
(d) 48278 = 6033751000 = 603.375
2. Choose the correct answer.
(a) 15264 = (iii) 381.5
(b) 35678 = (iii) 445.875
3. Quotients
(a) 33 (b) 3.3 (c) 0.33 (d) 0.033
4. Quotients
(a) 15.75 (b) 1.575 (c) 0.1575 (d) 0.01575 (e) 0.001575
10 ÷ 9 = 1.111…; 100 ÷ 11 = 9.0909…
When we divide by 7, the remainder can only be 1, 2, 3, 4, 5 or 6. So, a remainder must come again, and from then on the digits repeat: 1 ÷ 7 = 0.142857 142857…
× 1 = 142857, × 2 = 285714, × 3 = 428571, × 4 = 571428, × 5 = 714285, × 6 = 857142. The same digits come in the same cyclic order.
× 7 = 999999.
1 ÷ 17 = 0.0588235294117647… The repeating block 0588235294117647 is another cyclic number.
| Divisor | Example | Quotient |
|---|---|---|
| Greater than 1 | 128 ÷ 4 = 32 | less than the dividend |
| Equal to 1 | 128 ÷ 1 = 128 | equal to the dividend |
| Between 0 and 1 | 128 ÷ 0.4 = 320 | greater than the dividend |
1. Decimal form
(a) 25 = 0.4
(b) 134 = 3.25
(c) 450 = 0.08
(d) 58 = 0.625
2. Quotients
(a) 24.86 ÷ 1.2 = 248.6 ÷ 12 = 20.7166… (about 20.72)
(b) 5.728 ÷ 1.52 = 572.8 ÷ 152 = 3.768… (about 3.77)
3. Using 156 × 12 = 1872
(a) 18.72 (b) 156 (c) 1.2 (d) 0.01872
4. Fill in.
(a) 25 ÷ 1000 = 0.025
(b) 25 ÷ 0.1 = 250
(c) 25 ÷ 10 = 2.5
(d) 25 ÷ 10 = 25 × 0.1
(e) 25 ÷ 0.10 = 25 × 10
(f) 25 ÷ 0.01 = 25 × 100
5. Quotients
(a) 1.64 (b) 16.4 (c) 164
Yes. 24.6 ÷ 1.5 = 16.4, which is the same as 2.46 ÷ 0.15. Multiplying both numbers by 10 does not change the quotient.
6. A 4 m block cut into 5 equal pieces
4 ÷ 5 = 0.8 m
7. Side of a 12-sided regular polygon with perimeter 208.8 cm
208.8 ÷ 12 = 17.4 cm
8. 3 litres shared by 8 friends
3 ÷ 8 = 0.375 L = 375 mL each
9. Distance per litre
234.45 ÷ 12.6 = 18.607… km, about 18.6 km per litre
10. 13.5 kg flour among 15 students
13.5 ÷ 15 = 0.9 kg each
12×2×2×2×2 = 0.03125 and 15×5×5×5×5 = 0.00032.
The digits of one are the powers of the other: 1 ÷ 32 has 3125 and 1 ÷ 3125 has 32.
Reason: 2 × 5 = 10. So, 132 = 3125100000 and 13125 = 32100000.
4.4 Look Before You Leap!
February gets the extra day (29 days).
25 of the 100 years are divisible by 4. Days = 100 × 365 + 25 × 1 = 36,525.
Another expression: 75 × 365 + 25 × 366 = 36,525.
Leap years: divisible by 400 → 25; divisible by 4 but not by 100 → 2500 − 100 = 2400. Total = 2425 leap years.
Calendar days = 10,000 × 365 + 2425 = 36,52,425.
Actual days = 10,000 × 365.2422 = 36,52,422.
The calendar has about 3 extra days. One fix: do not add the leap day in years divisible by 4000. This removes 2 more days in 10,000 years.
1. Which is cheaper per gram?
Chikki: 70.5 ÷ 210 ≈ ₹0.336 per gram. Chips: 33.25 ÷ 110 ≈ ₹0.302 per gram.
So, the potato chips are cheaper per gram.
2. Decimal at the arrow
First line: 3.1 to 3.2 in 10 parts, each 0.01. The arrow is at 3.16.
Second line: 2.15 to 2.17 in 10 parts, each 0.002. The arrow is at 2.162.
3. Shyamala’s profit
Cost = 3 × 30 = ₹90. Sale = 35 × 5 = ₹175. Profit = ₹85.
4. Books on a 160 cm shelf (2.5 cm each)
160 ÷ 2.5 = 64 books. So, not all 80 books fit (16 are left out). No space is left on the shelf.
5. Fill in the blanks.
5.5 km = 5500 m; 35 cm = 0.35 m; 14.5 cm = 145 mm;
68 g = 0.068 kg; 9.02 m = 9020 mm; 125.5 ml = 0.1255 l
6. Sridharacharya’s problem
6.25 ÷ 2.5 = 2.5.
60.25 ÷ 3.5 = 602.5 ÷ 35 = 17.214… (= 17314).
7. Fill the boxes in 2 ways.
(a) 0.6 × 4 = 2.4; 1.2 × 2 = 2.4; 0.8 × 3 = 2.4
(b) 2.9 × 5 = 14.5; 1.45 × 10 = 14.5; 0.5 × 29 = 14.5
8. Given 756 ÷ 36 = 21
(a) 21 (b) 21 (c) 2100 (d) 0.21 (e) 2100 (f) 21
9. Complete the a ÷ b table.
| b \ a | 1517 | 151.7 | 15.17 | 1.517 | 15170 |
|---|---|---|---|---|---|
| 37 | 41 | 4.1 | 0.41 | 0.041 | 410 |
| 3.7 | 410 | 41 | 4.1 | 0.41 | 4100 |
| 0.37 | 4100 | 410 | 41 | 4.1 | 41000 |
| 0.037 | 41000 | 4100 | 410 | 41 | 4,10,000 |
| 370 | 4.1 | 0.41 | 0.041 | 0.0041 | 41 |
10. Using 2, 4, 5, 8, 0 in __ . _ × _ . _
(a) Maximum: 82.0 × 5.4 = 442.8
(b) Minimum: 45.8 × 0.2 = 9.16
(c) Greater than 150: 28.0 × 5.4 = 151.2 (also 84.5 × 2.0 = 169)
(d) Nearest to 100: 20.5 × 4.8 = 98.4
(e) Nearest to 5: 45.8 × 0.2 = 9.16 (no product can be smaller)
11. Sort in increasing order.
(f) 7.9682 < (c) 30.75… < (a) 230.9… < (e) 245.05 < (d) 260.0… < (b) 1952.6…
Multiplying by a number less than 1 makes it smaller; dividing by it makes it bigger.
Puzzle: Hidato
Puzzle 1 (5 × 5):
| 25 | 23 | 1 | 2 | 3 |
| 24 | 20 | 22 | 5 | 4 |
| 19 | 14 | 21 | 8 | 6 |
| 18 | 13 | 15 | 7 | 9 |
| 17 | 16 | 12 | 11 | 10 |
Puzzle 3 (1 to 56; blank cells are not part of the grid):
| 8 | 9 | ||||||||
| 7 | 10 | ||||||||
| 4 | 6 | 11 | 12 | 13 | 14 | ||||
| 5 | 3 | 2 | 1 | 17 | 15 | ||||
| 45 | 46 | 47 | 48 | 31 | 16 | 18 | 19 | ||
| 44 | 43 | 49 | 32 | 30 | 28 | 27 | 20 | ||
| 55 | 54 | 42 | 51 | 50 | 33 | 29 | 26 | 24 | 21 |
| 56 | 53 | 52 | 41 | 36 | 35 | 34 | 25 | 23 | 22 |
| 40 | 37 | ||||||||
| 39 | 38 |
Puzzle 2 has 37 cells but asks for the numbers 1 to 39, so it cannot be completed as printed. Please check this puzzle in your book.