A Quick Recap of Decimals (Page 67 – 68)

Q. Jonali and Pallabi play a game. Jonali says a fraction and Pallabi gives the equivalent decimal. Write Pallabi’s answer in the blank spaces.

Solution:
$\frac{3}{10}$ = 0.3
$\frac{4}{100}$ = 0.04
$\frac{67}{1000}$ = 0.067
$\frac{457}{100}$ = 4.57
$\frac{71}{100}$ = 0.71
$\frac{43}{100}$ = 0.43
$\frac{9}{100}$ = 0.09

Q. Jonali goes to the market to buy spices. She purchases 50 g of Cinnamon, 100 g of Cumin seeds, 25 g of Cardamom and 250 g of Pepper. Express each of the quantities in kilograms by writing them in terms of fractions as well as decimals.
Solution:
1000 g = 1 kg
1 g = $\frac{1}{1000}$ kg
Cinnamon purchased = 50 g = $\frac{50}{1000}$ kg = $\frac{5}{100}$ kg = 0.05 kg

Cumin seeds purchased = 100 g = $\frac{100}{1000}$ kg = $\frac{1}{10}$ kg = 0.1 kg

Cardamom purchased = 25 g = $\frac{25}{1000}$ kg = 0.025 kg

Pepper purchased = 250 g = $\frac{250}{1000}$ kg = $\frac{25}{100}$ kg = 0.25 kg

Q. Write the following fractions as a sum of fractions and also as decimals:

Solution:

Figure it Out (Page 73)

1. Recall that a tenth is 0.1, a hundredth is 0.01, and so on. Find the following products in tenths, hundredths and so on:
(a)  6 × 4 tenths = 24 tenths
(b)  7 × 0.3
(c)  9 × 5 hundredths
Solution:
(a) 6 × 4 tenths = 24 tenths

(b) 0.3 = 3 tenths
7 × 3 tenths = 21 tenths

(c) 9 × 5 hundredths = 45 hundredths

2. Find the products:
(a)  27.34 × 6
(b)  4.23 × 3.7
(c)  0.432 × 0.23
Solution:
(a) 27.34 × 6
2734 × 6 = 16404
Decimal places in 27.34 = 2
Decimal places in 6 = 0
Total decimal places in the product = 2 + 0 = 2
So,
27.34 × 6 = 164.04.

(b) 4.23 × 3.7
423 × 37 = 15651
Decimal places in 4.23 = 2
Decimal places in 3.7 = 1
Total decimal places in the product = 2 + 1 = 3
So,
4.23 × 3.7 = 15.651.

(c) 0.432 × 0.23
432 × 23 = 9936
Decimal places in 0.432 = 3
Decimal places in 0.23 = 2
Total decimal places in the product = 3 + 2 = 5
So,
0.432 × 0.23 = 0.09936.

3. Thejus needs 1.65 m of cloth for a shirt. How many metres of cloth are needed for 3 shirts?
Solution:
Cloth required for a shirt = 1.65 m
Cloth required for 3 shirts = 3 × 1.65
= 3 × $\frac{165}{100}$
= $\frac{495}{100}$ = 4.95 m.

4. Meenu bought 4 notebooks and 3 erasers. The cost of each book was ₹15.50 and each eraser was ₹2.75. How much did she spend in all?
Solution:
Cost of each notebook = ₹15.50
Cost of each eraser = ₹2.75
Money spent on 4 notebooks = 4 × 15.50
= 4 × $\frac{1550}{100}$
= $\frac{6200}{100}$ = ₹62.
Money spent on 3 erasers = 3 × 2.75
= 3 × $\frac{275}{100}$
= $\frac{825}{100}$ = ₹8.25.
Therefore, total money spent = ₹62 + ₹8.25 = ₹70.25

5. The thickness of a rupee coin is 1.45 mm. What is the total height of the cylinder formed by placing 36 rupee coins one over the other? Write the answer in centimeters.
Solution:
Thickness of a rupee coin = 1.45 mm
Total height of the cylinder formed by placing 36 rupee coins one over the other = 36 × 1.45
= 36 × $\frac{145}{100}$
= $\frac{5220}{100}$ = 52.20 mm = $\frac{52.20}{10}$ cm = 5.220 cm.

6. The price of 1 kg of oranges is ₹56.50. What is the price of 2.250 kg of oranges? Can we write 56.50 as 56.5 and 2.250 as 2.25 and multiply? Will we get the same product? Why?
Solution:
(i) Price of 1 kg orange = ₹56.50
Price of 2.250 kg oranges = 2.250 × 56.50
= $\frac{2250}{1000}$ × $\frac{5650}{100}$
= $\frac{12712500}{100000}$
= $\frac{127125}{1000}$ = ₹127.125

(ii) Yes, we can write 56.50 as 56.5 and 2.250 as 2.25.
Now multiply,
56.5 × 2.25 = $\frac{565}{10}$ × $\frac{225}{100}$
= $\frac{127125}{1000}$ = ₹127.125

(iii) Yes, we get the same product. It is because trailing zeros after the decimal point do not change the value of a number.

7. Dwarakanath purchases notebooks at a wholesale price of ₹23.6 per piece and sells each notebook at ₹30/-. How much profit does he make if he sells 50 books in a week?
Solution:
Profit made on selling each notebook = ₹30 – ₹23.6 = ₹6.4
Profit made on selling 50 books = 50 × 6.4
= 50 × $\frac{64}{10}$
= 5 × 64 = ₹320.

8. Given that 18 × 12 = 216, find the products:
(a) 18 × 1.2
(b) 18 × 0.12
(c) 1.8 × 1.2
(d) 0.18 × 0.12
(e) 0.018 × 0.012
(f) 1.8 × 12
In which of the cases above is the product less than 1?
Solution:
(a) 18 × 1.2
18 × 12 = 216
Decimal places in 18 = 0
Decimal places in 1.2 = 1
Total decimal places in the product = 0 + 1 = 1
So,
18 × 1.2 = 21.6

(b) 18 × 0.12
18 × 12 = 216
Decimal places in 18 = 0
Decimal places in 0.12 = 2
Total decimal places in the product = 0 + 2 = 2
So,
18 × 0.12 = 2.16

(c) 1.8 × 1.2
18 × 12 = 216
Decimal places in 1.8 = 1
Decimal places in 1.2 = 1
Total decimal places in the product = 1 + 1 = 2
So,
1.8 × 1.2 = 2.16

(d) 0.18 × 0.12
18 × 12 = 216
Decimal places in 0.18 = 2
Decimal places in 0.12 = 2
Total decimal places in the product = 2 + 2 = 4
So,
0.18 × 0.12 = 0.0216

(e) 0.018 × 0.012
18 × 12 = 216
Decimal places in 0.018 = 3
Decimal places in 0.012 = 3
Total decimal places in the product = 3 + 3 = 6
So,
0.018 × 0.012 = 0.000216

(f) 1.8 × 12
18 × 12 = 216
Decimal places in 1.8 = 1
Decimal places in 12 = 0
Total decimal places in the product = 1 + 0 = 1
So,
1.8 × 12 = 21.6

Hence, the product is less than 1 in the cases (d) and (e).

9. In which of the following multiplications is the product less than 1? Can you find the answer without actually doing the multiplications?
(a) 7 × 0.6
(b) 0.7 × 0.6
(c) 0.7 × 6
(d) 0.07 × 0.06
Solution:
(a) 7 × 0.6
Here, one number (0.6) is between 0 and 1, and the other number (7) is greater than 1.
So, the product is less than 7 but greater than 0.7.
Hence, the product is greater than 1.

(b) 0.7 × 0.6
Here, both numbers, 0.7 and 0.6, are between 0 and 1.
So, the product is less than both numbers.
Since both numbers are less than 1, their product is less than 1.

(c) 0.7 × 6
Here, one number (0.7) is between 0 and 1, and the other number (6) is greater than 1.
So, the product is less than 6 but greater than 0.7.
Hence, the product is greater than 1.

(d) 0.07 × 0.06
Here, both numbers, 0.07 and 0.06, are between 0 and 1.
So, the product is less than both numbers.
Since both numbers are less than 1, their product is less than 1.

Therefore, the product is less than 1 in the following cases:
(b) 0.7 × 0.6 and (d) 0.07 × 0.06.

10. Multiplying the following numbers by 10, 100 and 1000 to complete the table.

Solution:

Figure it Out (Page 83)

1. Find the quotient by converting the denominator into 1, 10, 100 or 1000 and verify the solution by the long division method (division by place value).
(a) $\frac{18}{5}$
(b) $\frac{415}{4}$
(c) $\frac{1217}{2}$
(d) $\frac{4827}{8}$
Solution:
(a) $\frac{18}{5}$
Multiplying numerator and denominator by 2,
$\frac{18\times 2}{5\times 2}$ = $\frac{36}{10}$ = 3.6

Verification:
$\frac{18}{5}$ = 18 ÷ 5
18 = 1 Ten + 8 Ones
Step 1:
Regroup 1 Ten into 10 Ones.
10 Ones + 8 Ones = 18 Ones.
18 Ones ÷ 5 = 3 Ones and 3 Ones remain.

Step 2:
Regroup 3 Ones into 30 Tenths.
30 Tenths ÷ 5 = 6 Tenths.
So, the quotient is 3.6.

(b) $\frac{415}{4}$
Multiplying numerator and denominator by 25,
$\frac{415\times 25}{4\times 25}$ = $\frac{10375}{100}$ = 103.75

Verification:
$\frac{415}{4}$ = 415 ÷ 4
415 = 4 Hundreds + 1 Ten + 5 Ones
Step 1:
Regroup 4 Hundreds into 40 Tens.
40 Tens + 1 Ten = 41 Tens.
41 Tens ÷ 4 = 10 Tens and 1 Ten remain.

Step 2:
Regroup 1 Ten into 10 Ones.
10 Ones + 5 Ones = 15 Ones.
15 Ones ÷ 4 = 3 Ones and 3 Ones remain.

Step 3:
Regroup 3 Ones into 30 Tenths.
30 Tenths ÷ 4 = 7 Tenths and 2 Tenths remain.

Step 4:
Regroup 2 Tenths into 20 Hundredths.
20 Hundredths ÷ 4 = 5 Hundredths.
So, the quotient is 103.75.

(c) $\frac{1217}{2}$
Multiplying numerator and denominator by 5,
$\frac{1217\times 5}{2\times 5}$ = $\frac{6085}{10}$ = 608.5

Verification:
$\frac{1217}{2}$ = 1217 ÷ 2
1217 = 1 Thousands + 2 Hundreds + 1 Tens + 7 Ones

Step 1:
Regroup 1 Thousand into 10 Hundreds.
10 Hundreds + 2 Hundreds = 12 Hundreds.
12 Hundreds ÷ 2 = 6 Hundreds.

Step 2:
Bring down 1 Ten.
1 Ten ÷ 2 = 0 Tens and 1 Ten remain.

Step 3:
Regroup 1 Ten into 10 Ones.
10 Ones + 7 Ones = 17 Ones.
17 Ones ÷ 2 = 8 Ones and 1 One remain.

Step 4:
Regroup 1 One into 10 Tenths.
10 Tenths ÷ 2 = 5 Tenths.
So, the quotient is 608.5.

(d) $\frac{4827}{8}$
Multiply numerator and denominator by 125,
$\frac{4827\times 125}{8\times 125}$ = $\frac{603375}{1000}$ = 603.375

Verification:
$\frac{4827}{8}$ = 4827 ÷ 8

4827 = 4 Thousands + 8 Hundreds + 2 Tens + 7 Ones

Step 1:
Regroup 4 Thousands into 40 Hundreds.
40 Hundreds + 8 Hundreds = 48 Hundreds.
48 Hundreds ÷ 8 = 6 Hundreds.

Step 2:
Bring down 2 Tens.
2 Tens ÷ 8 = 0 Tens and 2 Tens remain.

Step 3:
Regroup 2 Tens into 20 Ones.
20 Ones + 7 Ones = 27 Ones.
27 Ones ÷ 8 = 3 Ones and 3 Ones remain.

Step 4:
Regroup 3 Ones into 30 Tenths.
30 Tenths ÷ 8 = 3 Tenths and 6 Tenths remain.

Step 5:
Regroup 6 Tenths into 60 Hundredths.
60 Hundredths ÷ 8 = 7 Hundredths and 4 Hundredths remain.

Step 6:
Regroup 4 Hundredths into 40 Thousandths.
40 Thousandths ÷ 8 = 5 Thousandths.
So, the quotient is 603.375.

2. Choose the correct answer:
(a) $\frac{1526}{4}$ =
(i) 38.15
(ii) 380.15
(iii) 381.5
(iv) 381.05
(b) $\frac{3567}{8}$ =
(i) 4458.75
(ii) 44.5875
(iii) 445.875
(iv) 4458.75
Solution:
(a) $\frac{1526}{4}$
Multiplying numerator and denominator by 25,
= $\frac{1526\times 25}{4\times 25}$
= $\frac{38150}{100}$ = $\frac{3815}{10}$ = 381.5.

(b) $\frac{3567}{8}$
Multiplying numerator and denominator by 125,
= $\frac{3567\times 125}{8\times 125}$
= $\frac{445875}{1000}$ = 445.875.

3. What is the quotient?
(a) 132 ÷ 4 =
(b) 13.2 ÷ 4 =
(c) 1.32 ÷ 4 =
(d) 0.132 ÷ 4 =
Solution:
(a) 132 ÷ 4 =
132 = 1 Hundreds + 3 Tens + 2 Ones
Step 1: Regroup 1 Hundreds into 10 Tens.
10 Tens + 3 Tens = 13 Tens.
13 Tens ÷ 4 = 3 Tens and 1 Tens remain.

Step 2: Regroup 1 Tens into 10 Ones. 10 Ones + 2 Ones = 12 Ones.
12 Ones ÷ 4 = 3 Ones.
So, the quotient is 33.

(b) 13.2 ÷ 4 =
13.2 = 1 Tens + 3 Ones + 2 Tenths
Step 1: Regroup 1 Tens into 10 Ones.
10 Ones + 3 Ones = 13 Ones.
13 Ones ÷ 4 = 3 Ones and 1 Ones remain.

Step 2: Regroup 1 Ones into 10 Tenths. 10 Tenths + 2 Tenths = 12 Tenths.
12 Tenths ÷ 4 = 3 Tenths.
So, the quotient is 3.3.

(c) 1.32 ÷ 4 =
1.32 = 1 Ones + 3 Tenths + 2 Hundredths
Step 1: Regroup 1 Ones into 10 Tenths.
10 Tenths + 3 Tenths = 13 Tenths.
13 Tenths ÷ 4 = 3 Tenths and 1 Tenths remain.

Step 2: Regroup 1 Tenths into 10 Hundredths.
10 Hundredths + 2 Hundredths = 12 Hundredths.
12 Hundredths ÷ 4 = 3 Hundredths.
So, the quotient is 0.33.

(d) 0.132 ÷ 4 =
0.132 = 1 Tenths + 3 Hundredths + 2 Thousandths
Step 1: Regroup 1 Tenths into 10 Hundredths.
10 Hundredths + 3 Hundredths = 13 Hundredths.
13 Hundredths ÷ 4 = 3 Hundredths and 1 Hundredths remain.

Step 2: Regroup 1 Hundredths into 10 Thousandths.
10 Thousandths + 2 Thousandths = 12 Thousandths.
12 Thousandths ÷ 4 = 3 Thousandths.
So, the quotient is 0.033.

4. What is the quotient?
(a) 126 ÷ 8 =
(b) 12.6 ÷ 8 =
(c) 1.26 ÷ 8 =
(d) 0.126 ÷ 8 =
(e) 0.0126 ÷ 8 =
Solution:
(a) 126 ÷ 8 =
126 = 1 Hundreds + 2 Tens + 6 Ones
Step 1: Regroup 1 Hundreds into 10 Tens.
10 Tens + 2 Tens = 12 Tens.
12 Tens ÷ 8 = 1 Ten and 4 Tens remain.

Step 2: Regroup 4 Tens into 40 Ones.
40 Ones + 6 Ones = 46 Ones.
46 Ones ÷ 8 = 5 Ones and 6 Ones remain.

Step 3: Regroup 6 Ones into 60 Tenths.
60 Tenths ÷ 8 = 7 Tenths and 4 Tenths remain.

Step 4: Regroup 4 Tenths into 40 Hundredths.
40 Hundredths ÷ 8 = 5 Hundredths.
So, the quotient is 15.75.

(b) 12.6 ÷ 8 =
12.6 = 1 Tens + 2 Ones + 6 Tenths
Step 1:
Regroup 1 Tens into 10 Ones.
10 Ones + 2 Ones = 12 Ones.
12 Ones ÷ 8 = 1 One and 4 Ones remain.

Step 2:
Regroup 4 Ones into 40 Tenths.
40 Tenths + 6 Tenths = 46 Tenths.
46 Tenths ÷ 8 = 5 Tenths and 6 Tenths remain.

Step 3:
Regroup 6 Tenths into 60 Hundredths.
60 Hundredths ÷ 8 = 7 Hundredths and 4 Hundredths remain.

Step 4:
Regroup 4 Hundredths into 40 Thousandths.
40 Thousandths ÷ 8 = 5 Thousandths.
So, the quotient is 1.575.

(c) 1.26 ÷ 8 =
1.26 = 1 One + 2 Tenths + 6 Hundredths

Step 1:
Regroup 1 One into 10 Tenths.
10 Tenths + 2 Tenths = 12 Tenths.
12 Tenths ÷ 8 = 1 Tenth and 4 Tenths remain.

Step 2:
Regroup 4 Tenths into 40 Hundredths.
40 Hundredths + 6 Hundredths = 46 Hundredths.
46 Hundredths ÷ 8 = 5 Hundredths and 6 Hundredths remain.

Step 3:
Regroup 6 Hundredths into 60 Thousandths.
60 Thousandths ÷ 8 = 7 Thousandths and 4 Thousandths remain.

Step 4:
Regroup 4 Thousandths into 40 Ten-thousandths.
40 Ten-thousandths ÷ 8 = 5 Ten-thousandths.
So, the quotient is 0.1575.

(d) 0.126 ÷ 8
0.126 = 1 Tenth + 2 Hundredths + 6 Thousandths

Step 1:
Regroup 1 Tenth into 10 Hundredths.
10 Hundredths + 2 Hundredths = 12 Hundredths.
12 Hundredths ÷ 8 = 1 Hundredth and 4 Hundredths remain.

Step 2:
Regroup 4 Hundredths into 40 Thousandths.
40 Thousandths + 6 Thousandths = 46 Thousandths.
46 Thousandths ÷ 8 = 5 Thousandths and 6 Thousandths remain.

Step 3:
Regroup 6 Thousandths into 60 Ten-thousandths.
60 Ten-thousandths ÷ 8 = 7 Ten-thousandths and 4 Ten-thousandths remain.

Step 4:
Regroup 4 Ten-thousandths into 40 Hundred-thousandths.
40 Hundred-thousandths ÷ 8 = 5 Hundred-thousandths.
So, the quotient is 0.01575.

(e) 0.0126 ÷ 8 =
0.0126 = 1 Hundredth + 2 Thousandths + 6 Ten-thousandths

Step 1:
Regroup 1 Hundredth into 10 Thousandths.
10 Thousandths + 2 Thousandths = 12 Thousandths.
12 Thousandths ÷ 8 = 1 Thousandth and 4 Thousandths remain.

Step 2:
Regroup 4 Thousandths into 40 Ten-thousandths.
40 Ten-thousandths + 6 Ten-thousandths = 46 Ten-thousandths.
46 Ten-thousandths ÷ 8 = 5 Ten-thousandths and 6 Ten-thousandths remain.

Step 3:
Regroup 6 Ten-thousandths into 60 Hundred-thousandths.
60 Hundred-thousandths ÷ 8 = 7 Hundred-thousandths and 4 Hundred-thousandths remain.

Step 4:
Regroup 4 Hundred-thousandths into 40 Millionths.
40 Millionths ÷ 8 = 5 Millionths.
So, the quotient is 0.001575.

Figure it Out (Page 83)

1. Express the following fractions in decimal form:
(a) $\frac{2}{5}$
(b) $\frac{13}{4}$
(c) $\frac{4}{50}$
(d) $\frac{5}{8}$
Solution:
(a) $\frac{2}{5}$
Multiplying numerator and denominator by 2,
= $\frac{2\times 2}{5\times 2}$ = $\frac{4}{10}$ = 0.4

(b) $\frac{13}{4}$
Multiplying numerator and denominator by 25,
= $\frac{13\times 25}{4\times 25}$ = $\frac{325}{100}$ = 3.25

(c) $\frac{4}{50}$
Multiplying numerator and denominator by 2,
= $\frac{4\times 2}{50\times 2}$ = $\frac{8}{100}$ = 0.08

(d) $\frac{5}{8}$
Multiplying numerator and denominator by 125,
= $\frac{5\times 125}{8\times 125}$ = $\frac{625}{1000}$ = 0.625

2. Find the quotients:
(a)  24.86 ÷ 1.2
(b)  5.728 ÷ 1.52
Solution:
(a) 24.86 ÷ 1.2
= $\frac{2486}{100}$ ÷ $\frac{12}{10}$
= $\frac{2486}{100}$ × $\frac{10}{12}$
= $\frac{2486}{10}$ × $\frac{1}{12}$
= $\frac{2486}{120}$ = 20.7166…

So, the quotient is 20.7167.

(b) 5.728 ÷ 1.52
= $\frac{5728}{1000}$ ÷ $\frac{152}{100}$
= $\frac{5728}{1000}$ × $\frac{100}{152}$
= $\frac{5728}{10}$ × $\frac{1}{152}$
= $\frac{5728}{1520}$ = 3.768…

So, the quotient is 3.77.

3. Evaluate the following using the information 156 × 12 = 1872.
(a) 15.6 × 1.2 = __________
(b) 187.2 ÷ 1.2 = __________
(c) 18.72 ÷ 15.6 = __________
(d) 0.156 × 0.12 = __________
Solution:
(a) 15.6 × 1.2 = __________
15.6 × 1.2 = $\frac{156}{10}$ × $\frac{12}{10}$
= $\frac{156\times 12}{100}$
= $\frac{1872}{100}$ = 18.72

(b) 187.2 ÷ 1.2 = __________
187.2 ÷ 1.2 = $\frac{1872}{10}$ ÷ $\frac{12}{10}$
= $\frac{1872}{10}$ × $\frac{10}{12}$
= $\frac{1872}{12}$ = 156

(c) 18.72 ÷ 15.6 = __________
18.72 ÷ 15.6 = $\frac{1872}{100}$ ÷ $\frac{156}{10}$
= $\frac{1872}{100}$ × $\frac{10}{156}$
= $\frac{1872}{10}$ × $\frac{1}{156}$
= $\frac{1872}{10\times 156}$
= $\frac{12}{10}$ = 1.2

(d) 0.156 × 0.12 = __________
0.156 × 0.12 = $\frac{156}{1000}$ × $\frac{12}{100}$
= $\frac{156\times 12}{100000}$
= $\frac{1872}{100000}$ = 0.01872

4. Evaluate the following:
(a) 25 ÷ ______ = 0.025
(b) 25 ÷ ______ = 250
(c) 25 ÷ ______ = 2.5
(d) 25 ÷ 10 = 25 × _____
(e) 25 ÷ 0.10 = 25 × ______
(f) 25 ÷ 0.01 = 25 × ______
Solution:
(a) 25 ÷ ______ = 0.025
25 ÷ 1000 = 0.025

(b) 25 ÷ ______ = 250
25 ÷ 0.1 = 250

(c) 25 ÷ ______ = 2.5
25 ÷ 10 = 2.5

(d) 25 ÷ 10 = 25 × _____
25 ÷ 10 = 25 × 0.1

(e) 25 ÷ 0.10 = 25 × ______
25 ÷ 0.10 = 25 × 10

(f) 25 ÷ 0.01 = 25 × ______
25 ÷ 0.01 = 25 × 100

5. Find the quotient:
(a) 2.46 ÷ 1.5 =
(b) 2.46 ÷ 0.15 =
(c) 2.46 ÷ 0.015 =
Solution:
(a) 2.46 ÷ 1.5 = $\frac{246}{100}$ ÷ $\frac{15}{10}$
= $\frac{246}{100}$ × $\frac{10}{15}$
= $\frac{246}{10}$ × $\frac{1}{15}$
= $\frac{246}{150}$ = 1.64

(b) 2.46 ÷ 0.15 = $\frac{246}{100}$ ÷ $\frac{15}{100}$
= $\frac{246}{100}$ × $\frac{100}{15}$
= $\frac{246}{15}$ = 16.4

(c) 2.46 ÷ 0.015 = $\frac{246}{100}$ ÷ $\frac{15}{1000}$
= $\frac{246}{100}$ × $\frac{1000}{15}$
= 246 × $\frac{10}{15}$
= $\frac{2460}{15}$ = 164

6. A 4 m long wooden block has to be cut into 5 pieces of equal length. What is the length of each piece?
Solution:
Length of wooden block = 4 m
Number of equal pieces = 5
Length of each piece = 4 ÷ 5 = 0.8 m

7. If the perimeter of a regular polygon with 12 sides is 208.8 cm, what is the length of its side?
Solution:
Perimeter of the regular polygon = 208.8 cm
Number of sides = 12
Length of each side = 208.8 ÷ 12
= $\frac{2088}{10}$ ÷ 12
= $\frac{2088}{10}$ × $\frac{1}{12}$
= $\frac{2088}{120}$ = 17.4 cm

8. 3 litres of watermelon juice is shared among 8 friends equally. How much watermelon juice will each get? Express the quantity of juice in millilitres.
Solution:
Total watermelon juice = 3 litres
Number of friends = 8
Quantity of juice each will get = 3 ÷ 8 = 0.375 litres
= 0.375 × 1000 = 375 ml
Therefore, each friend will get 375 millilitres of watermelon juice.

9. A car covers 234.45 km using 12.6 litres of petrol. What is the distance travelled per litre?
Solution:
Distance covered = 234.45 km
Petrol used = 12.6 litres
Distance travelled per litre = 234.45 ÷ 12.6
= $\frac{23445}{100}$ ÷ $\frac{126}{10}$
= $\frac{23445}{100}$ × $\frac{10}{126}$
= $\frac{23445}{1260}$ = 18.607 km

10. 13.5 kg of flour (aata) was distributed equally among 15 students. How much flour did each student receive?
Solution:
Total flour = 13.5 kg
Number of students = 15
Flour received by each student = 13.5 ÷ 15
= $\frac{135}{10}$ ÷ 15
= $\frac{135}{10}$ × $\frac{1}{15}$
= $\frac{135}{150}$ = 0.9 kg
Therefore, each student received 0.9 kg of flour.

Figure it Out (Page 83)

1. A 210 gram packet of peanut chikki costs ₹70.5, while a 110 gram packet of potato chips costs ₹33.25. Which is cheaper?
Solution:
Peanut chikki
Weight = 210 g
Cost = ₹70.5
Cost per gram of peanut chikki = 70.5 ÷ 210 = 0.335

Potato chips
Weight = 110 g
Cost = ₹33.25
Cost per gram of potato chips = 33.25 ÷ 110 = 0.302

Since, 0.302 < 0.335
Therefore, potato chips are cheaper.

2. Write the decimal number at the arrow mark:

Solution:

3. Shyamala bought 3 kg bananas at ₹30/- per kg. She counted 35 bananas in all. She sells each banana for ₹5/-. How much profit does she make selling all the bananas?

Solution:
Cost of 1 kg bananas = ₹30
Cost of 3 kg bananas = 3 × 30 = ₹90
Number of bananas bought = 35
Selling price of each banana = ₹5
Total selling price = 5 × 35 = ₹135
Profit = 175 − 90 = ₹85
Therefore, Shyamala makes a profit of ₹85 by selling all the bananas.

4. A teacher placed textbooks that are 2.5 cm thick on a bookshelf. The teacher wanted to place 80 textbooks on the shelf. The bookshelf is 160 cm long. How many books could be placed on the shelf? Was there any space left? If yes, how much?
Solution:
Thickness of one textbook = 2.5 cm
Length of the bookshelf = 160 cm
Number of books that can be placed on the shelf = 160 ÷ 2.5 = 64
Space occupied by 64 textbooks = 64 × 2.5 = 160 cm
Space left on the shelf = 160 − 160 = 0 cm
Therefore, 64 textbooks could be placed on the shelf and no space was left.

5. Fill in the following blanks appropriately:

Solution:
(a) 5.5 km = _________ m
5.5 km = 5.5 × 1000 m = $\frac{55}{10}$ × 1000 m = 5500 m.

(b) 35 cm = ________ m
1 m = 100 cm
or 1 cm = $\frac{1}{100}$ m
35 cm = 35 × $\frac{1}{100}$ m = $\frac{35}{100}$ m = 0.35 m.

(c) 14.5 cm = _______ mm
1 cm = 10 mm
14.5 cm = 14.5 × 10 mm = $\frac{145}{10}$ × 10 mm = 145 mm.

(d) 68 g = ________ kg
1 kg = 1000 g
or 1 g = $\frac{1}{1000}$ kg
68 g = 68 × $\frac{1}{1000}$ kg = $\frac{68}{1000}$ kg = 0.068 kg.

(e) 9.02 m = ________ mm
1 m = 100 cm = 1000 × 10 mm = 1000 mm
9.02 = 9.02 × 1000 mm = $\frac{902}{100}$ × 1000 mm = 9020 mm.

(f) 125.5 ml = _______ L
1 L = 1000 ml
or 1 ml = $\frac{1}{1000}$ L
125.5 ml = 125.5 × $\frac{1}{1000}$ L = $\frac{1255}{10}$ × $\frac{1}{1000}$ L = $\frac{1255}{10000}$ L = 0.1255 L.

6. The following problem was set by Sridharacharya in his book, Patiganita. “6$\frac{1}{4}$ is divided by 2$\frac{1}{2}$, and 60$\frac{1}{4}$ is divided by 3$\frac{1}{2}$. Tell the quotients separately.” Can you try to solve it by converting the fractions into decimals?
Solution:
(i) 6$\frac{1}{4}$ ÷ 2$\frac{1}{2}$
6$\frac{1}{4}$ = $\frac{25}{4}$ = 6.25
2$\frac{1}{2}$ = $\frac{5}{2}$ = 2.5
So,
6.25 ÷ 2.5
= $\frac{625}{100}$ ÷ $\frac{25}{10}$
= $\frac{625}{100}$ × $\frac{10}{25}$
= $\frac{25}{10}$ = 2.5

(ii) 60$\frac{1}{4}$ ÷ 3$\frac{1}{2}$
60$\frac{1}{4}$ = $\frac{241}{4}$ = 60.25
3$\frac{1}{2}$ = $\frac{7}{4}$ = 3.5
So,
60.25 ÷ 3.5
= $\frac{6025}{100}$ ÷ $\frac{35}{10}$
= $\frac{6025}{100}$ × $\frac{10}{35}$
= $\frac{6025}{350}$ = 17.21
Thus, the quotients are 2.5 and 17.21 respectively.

7. Fill the boxes in at least 2 different ways:
(a) ☐ × ☐ = 2.4
(b) ☐ × ☐ = 14.5
Solution:
(a) ☐ × ☐ = 2.4
Two different ways:
0.6 × 4 = 2.4
1.2 × 2 = 2.4

(b) ☐ × ☐ = 14.5
Two different ways:
1.45 × 10 = 14.5
2.9 × 5 = 14.5

8. Find the following quotients given that 756 ÷ 36 = 21:
(a) 75.6 ÷ 3.6
(b) 7.56 ÷ 0.36
(c) 756 ÷ 0.36
(d) 75.6 ÷ 360
(e) 7560 ÷ 3.6
(f) 7.56 ÷ 0.36
Solution:
(a) 75.6 ÷ 3.6
Multiply both numbers by 10,
75.6 ÷ 3.6 = 756 ÷ 36 = 21.

(b) 7.56 ÷ 0.36
Multiply both numbers by 100,
7.56 ÷ 0.36 = 756 ÷ 36 = 21.

(c) 756 ÷ 0.36
Multiply both numbers by 100,
756 ÷ 0.36 = 75600 ÷ 36

Since 756 ÷ 36 = 21,
75600 ÷ 36 = 2100.

(d) 75.6 ÷ 360
Multiply both numbers by 10,
75.6 ÷ 360 = 756 ÷ 3600

Divide both by 36,
756 ÷ 3600 = 21 ÷ 100 = 0.21.

(e) 7560 ÷ 3.6
Multiply both numbers by 10,
7560 ÷ 3.6 = 75600 ÷ 36 = 2100.

(f) 7.56 ÷ 0.36
Multiply both numbers by 100,
7.56 ÷ 0.36 = 756 ÷ 36 = 21.

9. Find the missing cells if each cell represents a ÷ b:

Solution:

10. Using the digits 2, 4, 5, 8, and 0 fill the boxes ☐☐.☐ × ☐.☐ to get the:
(a) maximum product
(b) minimum product
(c) product greater than 150
(d) product nearest to 100
(e) product nearest to 5
Solution:
(a) Maximum product:
Placing largest digits 8 and 5 at tens and ones place to keep both numbers as large as possible.
82.0 × 5.4 = 442.8

(b) Minimum product:
Putting smallest digits in the tens and ones place to keep both numbers as small as possible.
45.8 × 0.2 = 9.16

(c) Product greater than 150:
54.0 × 2.8 = 151.2

(d) Product nearest to 100:
20.5 × 4.8 = 98.4

(e) Product nearest to 5:
45.8 × 0.2 = 9.16

11. Sort the following expressions in increasing order:
(a) 245.05 × 0.942368
(b) 245.05 × 7.9682
(c) 245.05 ÷ 7.9682
(d) 245.05 ÷ 0.942368
(e) 245.05
(f) 7.9682
Solution:
0.942368 < 1
7.9682 > 1
(a) 245.05 × 0.942368 → multiplication by a number less than 1 → less than 245.05

(b) 245.05 × 7.9682 → multiplication by a number greater than 1 → largest

(c) 245.05 ÷ 7.9682 → division by a number greater than 1 → smallest

(d) 245.05 ÷ 0.942368 → division by a number less than 1 → greater than 245.05

(e) 245.05 → a large number

(f) 7.9682 → a small number itself

Increasing order:
245.05 ÷ 7.9682 < 7.9682 < 245.05 × 0.942368 < 245.05 < 245.05 ÷ 0.942368 < 245.05 × 7.968