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0 + 2 = 2 

1 + 1 = 2 

-1 + 3 = 2 

-2 + 4 = 2 

-3 + 5 = 2 (and many more.)

(iv) Division

Loss of calory in 30 days = 4800

∴ Loss of calory in 1 day = \(\frac{4800}{30}\) = 160 calories 

∴ 160 calories lost per day.

To get the number to be subtracted

We have 7.1 – 0.713 = 6.387 

∴ The number to be subtracted = 6.387

(i) -275°C = (-275 + 273)K = -2 K 

(ii) 45°C = (45 + 273)K = 318 K 

(iii) -400°C = (-400 + 273)K = -127 K 

(iv) -273°C = (-273 + 273)K = 0 K

(-1) + a number = 10 

∴ The number = 10 + 1 = 11

Temperature in the first day = -5°C 

Temperature in the next day = 9°C 

∴ Increase in temperature = 9°C – (-5°C) 

= 9°C + (+5°C) = 14°C

-86945 – (94860) = -86945 + (Additive inverse of 94860) 

= -86945 + (-94860) 

= -1,81,805

(i) 6 protons and 6 electrons → (+6) + (-6) = 0 

(ii) 9 protons and 12 electrons → (+9) + (-12) 

= 9 - 12 = -3 ⇒ 3 electrons 

(iii) 4 protons and 8 electrons → (+4) + (-8) = +4 – 8 = -4 ⇒ 4 electrons 

(iv) 7 protons and 6 electrons → (+7) + (-6) = +1 = 1 proton

(i) 37.3 = 30 + 7 + \(\frac{3}{10}\)

= 3 x 10¹ + 7 x 10⁰ + 3 x 10^-1

(ii) 658.37 = 600 + 50 + 8 + \(\frac{3}{10}\) + \(\frac{7}{100}\)

= 6 x 10² + 5 x 10¹ + 8 x 100 + 3 x 10^-1 + 7 x 10^-2 

(iii) 237.6 = 200 + 30 + 7 + \(\frac{6}{10}\)

= 2 x 10² + 3 x 10¹ + 7 x 10⁰ + 6 x 10^-1 

(iv) 5678.358 = 5000 + 600 + 70 + 8 + \(\frac{3}{10}\) + \(\frac{5}{100}\) + \(\frac{8}{1000}\)

= 5 x 10³ + 6 x 10² + 7 x 10¹ + 8 x 10⁰ + 3 x 10^-1 + 5 x 10^-2 + 8 x 10^-3

(-9999) + (-2001) + (-5999) 

= -12,000 + (-5999) 

= -17,999

\(\frac{-72}{8}\) = - 9

(iii) 0.00900

Answer is (i) 20

Answer is (i) =

37.70 [=] 37.7

(-30) x (-70) x 15 = (+2100) x 15 = 31,500

For each incorrect question the score = -1 

In paper I, score for 25 incorrect questions – 25 x (-1) = -25 

In paper II, for 13 incorrect question the score 

= 13 x (-1) = -13 

The total marks get reduced = (-25) + (-13) = -38 

-38 marks will be reduced.

(i) False 

(ii) True 

(iii) False

(i) 35 x 22 = -770 

(ii) (-10) x 12 x (-9) = (-120) x (-9) = +1080 

(iii) (-9) x (-8) x (-7) x (-6) = (+72) x (-7) x (-6) = (-504) x (-6) = +3024 

(iv) (-25) x 0 x 45 x 90 = 0 x 45 x 90 = 0 x 90 = 0

(v) (-2) x (+50) x (-25) x 4 = (-100) x -25 x 4 = 2500 x 4 = 10,000

(i) \(\frac{3}{10}\) = 0.3

(ii) 3\(\frac{1}{2}\) = \(\frac{7}{2}\)

= \(\frac{7\times5}{2\times5}\)

= \(\frac{35}{10}\)

= 3.5

(iii) 3\(\frac{3}{5}\) = \(\frac{18}{5}\)

= \(\frac{18\times2}{5\times2}\)

= \(\frac{36}{10}\)

= 3.6

(iv) \(\frac{3}{2}\) = \(\frac{3\times5}{2\times5}\)

= \(\frac{15}{10}\)

= 1.5

(v) \(\frac{4}{5}\) = \(\frac{4\times2}{5\times2}\)

= \(\frac{8}{10}\)

= 0.8

(vi) \(\frac{99}{100}\) = 0.99

(vii) 3\(\frac{19}{25}\) = \(\frac{94}{25}\)

= \(\frac{94\times4}{25\times4}\)

= \(\frac{376}{100}\)

= 3.76

(i) 15 – 12 = 3 & 12 - 15 = 12 + (-15) = -3

15 - 12 > 12 - 15

(ii) -21 – 32 = (-21) + (-32) = -53 

Also -32 – (-21) = (-32) + (+21) = -11; -53 < -11

-21 - 32 < (-32) - (-21)

(i) 16 is an even interger. 

If negative integers are multiplied even number of times, the product is a positive integer. 

∴ 16 times a negative integer is a positive integer.

(ii) 29 times negative integer. 

If negative integers are multiplied odd number of times, the product is a negative integer. 29 is odd. 

∴ 29 times negative integers is a negative integer.

(i) -10 – 8 = -18 & -10 + 8 = – 2 

(ii) (-20) + 10 = -10 & (-20) – (-10) = -10 

(iii) -70 – 50 = (-70) + (-50) = -20 

(iv) 100 – (+100) = 0 & 100 – (-100) = 100 + (+100) = 200 

(v) -50 – 30 = -50 + (-30) = -80 

Also -100 + 20 = – 80

The whole number is equal in both the numbers. 

Now comparing the tenths place we have 3 > 0 

⇒ 3.03 < 3.30 

Smaller number is 3.03

(i) \(\frac{16}{1000}\) = 0.016

(ii) \(\frac{638}{10}\) = 63.8

(iii) \(\frac{1}{20}\) = \(\frac{1\times5}{20\times5}\) 

= \(\frac{5}{100}\)

= 0.05

(iv) \(\frac{3}{50}\) = \(\frac{3\times2}{50\times2}\)

= \(\frac{6}{100}\)

= 0.06

(i) \(\frac{23}{10000}\) = 0.0023

(ii) \(\frac{421}{100}\) = 4.21

(iii) \(\frac{37}{10}\) = 3.7

(i) 25.8 + 18.53. 

Using place value grid.

Therefore 25.8 + 18.53 = 44.33

(ii) 17.4 + 23.435 

Lets use the place value grid.

Therefore 17.4 + 23.435 = 40.835

2.57 [=] 2.570

2.1 x 3.2 = 6.72 and 

21 x 32 = 672. 

In both the cases the digits numbers are the same. 

But the place value differs.

(i) 2.125 = \(\frac{2125}{1000}\) 

= \(\frac{2125÷25}{1000÷25}\)

= \(\frac{85}{40}\)

= \(\frac{85÷5}{40÷5}\)

= \(\frac{17}{8}\)

(ii) 0.0005 = \(\frac{5}{1000}\)

= \(\frac{5÷5}{1000÷5}\)

= \(\frac{1}{2000}\)

(i) 0.0005 = \(\frac{5}{10000}\)

= \(\frac{5÷5}{10000÷5}\)

= \(\frac{1}{2000}\)

(ii) 6.24 = \(\frac{624}{100}\)

= \(\frac{624÷4}{100÷4}\)

= \(\frac{156}{25}\)

(i) 8 m 30 cm in metres 

8 m + \(\frac{30}{100}\) m = 8 m + 0.30 m 

= 8.30 m 

(ii) 24 km 200 m in kilometres 

24 km + \(\frac{200}{1000}\) km 

= 24 km + 0.200 km 

= 24.200 km

1.7, 1.9, 1.6, …

Comparing the numbers from left to right. 

Ascending order : 1.45, 1.54, 4.15, 4.51, 5.14, 5.41

3.75 = \(\frac{375}{100}\)

= \(\frac{15}{4}\)

5\(\frac{1}{5}\) = \(\frac{26}{5}\)

= \(\frac{26\times2}{5\times2}\)

= \(\frac{52}{10}\)

= 5.2

\(\frac{1}{4}\) = \(\frac{1\times25}{4\times25}\)

= \(\frac{25}{100}\) 

= 0.25

100 cm = 1 m; 

1 cm = \(\frac{1}{100}\) m 

(i) 1328 cm = \(\frac{1328}{100}\) m 

= 13.28 m 

(ii) 419 cm = \(\frac{419}{100}\) m 

= 4.19 m

1 m = \(\frac{1}{1000}\) km 

= 0.001 Km 

(i) 256 m = \(\frac{256}{1000}\) km 

= 0.256 km

(ii) 4567 m = \(\frac{4567}{1000}\) km 

= 4.567 km

(i) 2.34 = 2 + \(\frac{34}{100}\) 

= 2 + \(\frac{34÷2}{100÷2}\)

= 2 + \(\frac{17}{50}\)

= 2\(\frac{17}{50}\)

= \(\frac{117}{50}\)

(ii) 0.18 = 0 + \(\frac{18}{100}\) 

= \(\frac{18÷2}{100÷2}\)

= \(\frac{9}{50}\)

(iii) 3.56 = 3 + \(\frac{56}{100}\)

= 3 + \(\frac{56÷4}{100÷4}\)

= 3 + \(\frac{14}{25}\)

= 3\(\frac{14}{25}\)

= \(\frac{89}{25}\)

1 – (v) Rounded to tenth place

2 – (iv) Rounded to thousandth place

3 – (i) Rounded to hundredth place

4 – (iii) Rounded to nearest whole number

5 – (ii) Rounded to ten thousandth place

(a) 24.4003 

Rounding 24.4003 upto 3 places of decimal means rounding to the nearest thousandths place. Underlining the digit in the thousandths place of 24.4003 gives 24.4003. In 24.4003 the digit next to the thousandths value is 3 which is less than 5.

∴ The underlined digit remains the same. So the rounded value of 24.4003 upto 3 places of decimal is 24.400.

(b) 1251.2345 

Rounding 1251.2345 upto 3 places of decimal means rounding to the nearest thousandths place. Underlining the digit in the thousandths place of 1251.2345 gives 1251.2345, the digit next to the thousandths place value is 5 and so we add 1 to the underlined digit. So the rounded value of 1251.2345 upto 3 places of decimal is 1251.235.

(c) 61.00203 

Rounding 61.00203 upto 3 places of decimal means rounding to the nearest thousandths place. Underlining the digit in the thousandth place of 61.00203 gives 61.00203. In 61.00203, the digit next to the thousandths place value is 0, which is less than 5. 

Hence the underlined digit remains the same. So the rounded value of 61.00203 upto 3 places of decimal is 61.002.

(i) 123.37 

Rounding 123.37 upto one places of decimal means round to the nearest tenths place. 

Underling the digit in the tenths place of 123.37 gives 123.37. Since the digit next to the tenth place value is 7 which is greater than 5, we add 1 to the underlined digit to get 123.4. Hence the rounded value of 123.37 upto one places of decimal is 123.4. 

(ii) 19.99 

Rounding 19.99 upto one places of decimal means round to the nearest tenth place. 

Underling the digit in the tenths place of 19.99 gives 19.99. Since the digit next to the tenth place value is 9 which is greater than 5, we add 1 to the underlined digit to get 20. 

Hence the rounded value of 19.99 upto one places of decimal is 20.0.

(iii) 910.546 

Rounding 910.546 upto one places of decimal means round to the nearest tenths place underlining the digit in the tenths place of 910.546 gives 910.546. Since the digit next to the tenth place value is 4, which is less than 5 the underlined digit remains the same. Hence the rounded value of 910.546 upto one places of decimal is 910.5.

Answer is (i) 4.5

(i) 2.35, 2.53, 5.32, 3.52, 3.25 

Comparing the whole number parts of all the numbers 5 is the greatest and 5 > 3 > 2. 

∴ Greatest number is 5.32 

Next 3.52 and 3.25 are equal in their whole number. 

So comparing their digits in tenths place, we get 5 > 2 

So 3.52 > 3.25 

Now comparing 2.35 and 2.53 their whole number parts also equal.

∴ Comparing the digit in tenths place we get 

2.53 > 2.35 … (2) 

Ascending order : 

2.35 < 2.53 < 3.25 < 3.52 < 5.32

(ii) 123.45, 123.54, 125.43, 125.34, 125.3 

Comparing the whole number parts we have 123 is the smallest number and two numbers 123.45 and 123.54 have same whole number part. 

So in 123.45 and 123.54 comparing their digits in the tenths place we get 4 < 5 

∴ 123.45 < 123.54 … (1)

 Now comparing the remaining numbers 125.43, 125.34, 125.3 they all have the same whole number part. 

Comparing the numbers in the tenths place we have 3 < 4 

∴ 125.43 is the greatest … (2) 

Also tenths place value 3 = 3 in 125.34 and 125.3 

Again comparing the hundredths place value in 

125.34 and 125.3, we get

125.3 < 125.34 …(3) 

From (1), (2) and (3) we have, 

123.45 < 123.54 < 125.3 < 125.34 < 125.43

According to the problem {[(I + 2) x 5] – 10} ÷ 4 = 15 

{[(I + 2) x 5] – 10} = 15 x 4 = 60 

I + 2 = \(\frac{70}{5}\) = 14 

(I + 2) x 5 = 60 + 10 = 70 

I = 14 – 2; I = 12

Underlining the digit to be rounded 110.929. Since the digit next to the underlined digit is 2 which is less than 5. 

∴ The underlined digit 9 remains the same. 

Hence the rounded number is 110.9

(i) 5 mm 

1 mm = \(\frac{1}{10}\) cm 

= 0.1 cm 

5 mm = \(\frac{5}{10}\)

= 0.5 cm 

(ii) 9 mm 

1 mm = \(\frac{1}{10}\) cm 

= 0.1 cm 

9 mm = \(\frac{9}{10}\) cm 

= 0.9 cm 

(iii) 42 mm 

1 mm = \(\frac{1}{10}\) cm 

= 0.1 cm 

42 mm = \(\frac{42}{10}\) cm 

= 4.2 cm 

(iv) 8 cm 9 mm 

1 mm = \(\frac{1}{10}\) cm 

= 0.1 cm 

8 cm 9 mm = 8 cm + \(\frac{9}{10}\) cm 

= 8.9 cm 

(v) 375 mm

1 mm = \(\frac{1}{10}\) cm 

= 0.1 cm 

375 mm = \(\frac{375}{10}\) cm 

= 37.5 cm

(i) 992; tenths place 

Underlining the digit to be rounded 5.992. Since the digit next to the underlined digit is 9 greater than 5, we add 1 to the underlined digit. 

Hence the rounded number is 6.0. 

(ii) 21.805; hundredth place 

Underlining the digit to be rounded 21.805 since the digit next to the underlined digit is 5, we add 1 to the underlined digit. 

Hence the rounded number is 21.81.

(iii) 35.0014; thousandth place 

Underlining the digit to be rounded 35.0014. Since the digit next to the underlined digit is 4 less than 5 the underlined digit remains the same. 

Hence the rounded number is 35.001.

(i) -1 

(ii) -5 

(iii) -36 

(iv) 1