Class 7 Mathematics - Ch 1: Mastery Guide: Rational Numbers, Base 10, Arithmetic Operations & Properties (FBISE)

Question 1: Write the place value of the following numbers in base 10.

(i) $6287.48$
Decomposition by powers of 10:
Thousands: $6 \times 10^3 = 6000$
Hundreds: $2 \times 10^2 = 200$
Tens: $8 \times 10^1 = 80$
Ones: $7 \times 10^0 = 7$
Tenths: $4 \times 10^{-1} = 0.4$
Hundredths: $8 \times 10^{-2} = 0.08$
$\mathbf{6 \times 10^3 + 2 \times 10^2 + 8 \times 10^1 + 7 \times 10^0 + 4 \times 10^{-1} + 8 \times 10^{-2}}$

(ii) $4628370.52$
Millions: $4 \times 10^6$
Hundred-thousands: $6 \times 10^5$
Ten-thousands: $2 \times 10^4$
Thousands: $8 \times 10^3$
Hundreds: $3 \times 10^2$
Tens: $7 \times 10^1$
Ones: $0 \times 10^0$
Tenths: $5 \times 10^{-1}$
Hundredths: $2 \times 10^{-2}$
$\mathbf{4 \times 10^6 + 6 \times 10^5 + 2 \times 10^4 + 8 \times 10^3 + 3 \times 10^2 + 7 \times 10^1 + 0 \times 10^0 + 5 \times 10^{-1} + 2 \times 10^{-2}}$

Question 2: Classify the following numbers as a whole number, an integer or a rational number.

(i) $3, -1.2, -2, 0$
• $3$: Whole Number = Yes, Integer = Yes, Rational Number = Yes ($\frac{3}{1}$)
• $-1.2$: Whole Number = No, Integer = No, Rational Number = Yes ($-\frac{12}{10} = -\frac{6}{5}$)
• $-2$: Whole Number = No, Integer = Yes, Rational Number = Yes ($-\frac{2}{1}$)
• $0$: Whole Number = Yes, Integer = Yes, Rational Number = Yes ($\frac{0}{1}$)

(ii) $4.5, \frac{3}{4}, -2.1, 0.5$
• All four are terminating decimals or proper fractions. None are whole numbers or integers. All four are Rational Numbers: Yes.

(iii) $3.6, -1.5, -0.31, -2.8$
• None are whole numbers or integers. All four are Rational Numbers: Yes.

(iv) $\frac{1}{6}, 1.75, -\frac{2}{3}, 10$
• $\frac{1}{6}, 1.75, -\frac{2}{3}$: Whole Number = No, Integer = No, Rational Number = Yes.
• $10$: Whole Number = Yes, Integer = Yes, Rational Number = Yes.

Question 3: Fill in the blanks with $<, >, =$ to make a sentence true.

(i) $40\% \text{ _____ } \frac{12}{25}$
Convert both to decimals: $40\% = 0.40$; $\frac{12}{25} = \frac{12 \times 4}{100} = 0.48$. Since $0.40 < 0.48$, $40\% < \frac{12}{25}$.

(ii) $0.82 \text{ _____ } -\frac{5}{6}$
Any positive number is strictly greater than any negative number. Thus, $0.82 > -\frac{5}{6}$.

(iii) $\frac{3}{5} \text{ _____ } 59\%$
$\frac{3}{5} = 0.60 = 60\%$. Since $60\% > 59\%$, $\frac{3}{5} > 59\%$.

(iv) $\frac{9}{20} \text{ _____ } 0.45$
$\frac{9}{20} = \frac{9 \times 5}{100} = 0.45$. Thus, $\frac{9}{20} = 0.45$.

(v) $\frac{5}{8}\text{ m} \text{ _____ } \frac{1}{16}\text{ m}$
Common denominator is $16$: $\frac{5 \times 2}{8 \times 2} = \frac{10}{16}\text{ m}$. Since $\frac{10}{16} > \frac{1}{16}$, $\frac{5}{8}\text{ m} > \frac{1}{16}\text{ m}$.

(vi) $0.25\text{ kg} \text{ _____ } \frac{2}{9}\text{ kg}$
$0.25 = \frac{1}{4} = \frac{9}{36}$; $\frac{2}{9} = \frac{8}{36} \approx 0.222$. Since $0.25 > 0.222$, $0.25\text{ kg} > \frac{2}{9}\text{ kg}$.

Question 4: Arrange numbers in each set from least to greatest using a number line.

(i) $0.23, 19\%, \frac{1}{5}$
Convert to decimals: $19\% = 0.19$, $\frac{1}{5} = 0.20$, $0.23 = 0.23$.
Ordered least to greatest: $\mathbf{19\%, \frac{1}{5}, 0.23}$.

(ii) $\frac{8}{10}, 81\%, 0.805$
Convert to decimals: $\frac{8}{10} = 0.800$, $81\% = 0.810$, $0.805 = 0.805$.
Ordered least to greatest: $\mathbf{\frac{8}{10}, 0.805, 81\%}$.

(iii) $0.615, \frac{5}{8}, 62\%$
Convert to decimals: $0.615$; $\frac{5}{8} = 0.625$; $62\% = 0.620$.
Ordered least to greatest: $\mathbf{0.615, 62\%, \frac{5}{8}}$.

(iv) $-5.2, -\frac{3}{8}, -6, 0.3, -\frac{1}{4}$
Convert to decimals: $-6.0, -5.2, -\frac{3}{8} = -0.375, -\frac{1}{4} = -0.25, 0.3$.
Ordered least to greatest: $\mathbf{-6, -5.2, -\frac{3}{8}, -\frac{1}{4}, 0.3}$.

(v) $2.1, 0, -\frac{13}{10}, -1.38, \frac{3}{5}$
Convert to decimals: $-\frac{13}{10} = -1.30$, so negative values are $-1.38 < -1.30$. Then $0, \frac{3}{5} = 0.6, 2.1$.
Ordered least to greatest: $\mathbf{-1.38, -\frac{13}{10}, 0, \frac{3}{5}, 2.1}$.

Question 5: Arrange the following rational numbers in descending order (greatest to least).

(i) $\frac{3}{4}, \frac{5}{9}, \frac{7}{12}$
LCM of denominators $(4, 9, 12) = 36$:
$\frac{3}{4} = \frac{27}{36} = 0.75$; $\frac{7}{12} = \frac{21}{36} \approx 0.583$; $\frac{5}{9} = \frac{20}{36} \approx 0.556$.
Descending order: $\mathbf{\frac{3}{4}, \frac{7}{12}, \frac{5}{9}}$.

(ii) $3, -1.2, -2, 0$
Descending order: $\mathbf{3, 0, -1.2, -2}$.

(iii) $4.5, \frac{3}{4}, -2.0, 0.5$
Values in decimal: $4.5, 0.75, 0.5, -2.0$.
Descending order: $\mathbf{4.5, \frac{3}{4}, 0.5, -2.0}$.

(iv) $3.6, -1.5, -0.31, -2.8$
Descending order: $\mathbf{3.6, -0.31, -1.5, -2.8}$.

(v) $\frac{1}{6}, 1.75, -\frac{2}{3}, 0$
Decimals: $1.75, \frac{1}{6} \approx 0.167, 0, -\frac{2}{3} \approx -0.667$.
Descending order: $\mathbf{1.75, \frac{1}{6}, 0, -\frac{2}{3}}$ (or with $-0.67$).

Questions 6 to 15: Rounding, Estimation, and Word Problems

Question 6: Round off the following decimals to the nearest whole number.
(i) $0.51$: Tenths digit is $5 \ge 5 \implies \mathbf{1}$.
(ii) $7.95$: Tenths digit is $9 \ge 5 \implies \mathbf{8}$.
(iii) $0.333$: Tenths digit is $3 < 5 \implies \mathbf{0}$.

Question 7: Express the following fractions to the number of decimal places given in the brackets.
(i) $\frac{1}{33}$ (2 decimal places): $1 \div 33 = 0.030303... \implies \mathbf{0.03}$.
(ii) $\frac{1}{6}$ (1 decimal place): $1 \div 6 = 0.1666... \approx \mathbf{0.2}$ (or $\frac{3}{5} \implies \mathbf{0.6}$).
(iii) $\frac{17}{53}$ (2 decimal places): $17 \div 53 \approx 0.32075... \implies \mathbf{0.32}$.

Question 8: In one week, Mahd spends 52 hours in front of the computer. What is the average number of hours he spends in front of computer each day? Round off to the nearest whole number.
There are $7$ days in one week.
Average hours per day $= \frac{52}{7} = 7.4285... \text{ hours}$.
Rounding to nearest whole number: Since tenths digit $4 < 5$, answer is $\mathbf{7\text{ hours}}$.

Question 9: The length of four insects are $0.02\text{ cm}, \frac{1}{8}\text{ cm}, 0.1\text{ cm}$ and $\frac{2}{3}\text{ cm}$. Arrange the lengths in descending order.
Convert all lengths to decimals:
• $\frac{2}{3}\text{ cm} \approx 0.667\text{ cm}$
• $\frac{1}{8}\text{ cm} = 0.125\text{ cm}$
• $0.1\text{ cm} = 0.100\text{ cm}$
• $0.02\text{ cm} = 0.020\text{ cm}$
Descending order: $\mathbf{\frac{2}{3}\text{ cm} \ (0.67), \frac{1}{8}\text{ cm} \ (0.125), 0.1\text{ cm}, 0.02\text{ cm}}$.

Question 10: The cost of four items on a menu are Rs. 9, Rs. 0.99, Rs. 9.99 and Rs. 19.99. Order these costs from least to greatest.
Least to greatest: $\mathbf{\text{Rs. } 0.99, \text{Rs. } 9, \text{Rs. } 9.99, \text{Rs. } 19.99}$ (or as printed in key: $\mathbf{\text{Rs. } 0.9, \text{Rs. } 9, \text{Rs. } 9.99, \text{Rs. } 19.99}$).

Question 11: Ahmed and Sara played darts. Ahmed hit the bull's eye 5 out of 18 times. Sara missed the bull's eye 4 out of 15 times. Who played better?
Ahmed's hit rate: $\frac{5}{18} \approx 0.2778 \ (27.8\%)$.
Sara missed $4$ out of $15$ times, so she hit the bull's eye: $15 - 4 = 11$ times out of $15$.
Sara's hit rate: $\frac{11}{15} \approx 0.7333 \ (73.3\%)$.
Since $0.7333 > 0.2778$, Sara played better.

Question 12: Three paint brushes have widths of $\frac{3}{8}\text{ cm}, \frac{1}{2}\text{ cm}, \frac{3}{4}\text{ cm}$. What is the measure of the brush with the greatest width?
Convert to common denominator $8$:
$\frac{3}{8}\text{ cm} = 0.375\text{ cm}$; $\frac{1}{2}\text{ cm} = \frac{4}{8}\text{ cm} = 0.5\text{ cm}$; $\frac{3}{4}\text{ cm} = \frac{6}{8}\text{ cm} = 0.75\text{ cm}$.
The 3rd brush has the greatest width: $\mathbf{\frac{3}{4}\text{ cm}}$.

Question 13: The world's five largest deserts are shown in the table. Order the areas from least to greatest.
Given areas in millions of square meters (km$^2$):
• Sahara: $\frac{7}{2} = 3.5$
• Kalahari: $\frac{2}{10} = 0.2$
• Gobi: $\frac{2}{5} = 0.4$
• Australian: $1\frac{4}{10} = 1.4$
• Arabian: $\frac{1}{2} = 0.5$
Order from least to greatest: $\mathbf{\text{Kalahari } (0.2), \text{Gobi } (0.4), \text{Arabian } (0.5), \text{Australian } (1.4), \text{Sahara } (3.5)}$.

Question 14: An investor purchases 50 shares of stock at Rs. 3.50 per share. The next day, the change in value of a share of the stock is $-0.25$. What is the total value of the shares on the next day?
New value per share $= 3.50 + (-0.25) = \text{Rs. } 3.25$.
Total value of $50$ shares $= 50 \times 3.25 = \mathbf{\text{Rs. } 162.50}$.

Question 15: Make an estimation and then choose the correct answer.
(i) $82.45 \times 9.78$: Estimate $\approx 80 \times 10 = 800$. The closest option is (b) $806.361$.
(ii) $30.03 \times 0.065$: Estimate $\approx 30 \times 0.06 = 1.8 \approx 2$. The closest option is (a) $1.95195$.

Question 1: Evaluate the following, expressing your answer in the simplest form.

(i) $8\frac{5}{12} + 11\frac{1}{4}$
Whole parts: $8 + 11 = 19$. Fractions: $\frac{5}{12} + \frac{3}{12} = \frac{8}{12} = \frac{2}{3}$.
Total: $\mathbf{19\frac{2}{3}}$.

(ii) $8\frac{3}{8} + 10\frac{1}{3}$
Whole parts: $8 + 10 = 18$. Fractions: $\frac{9}{24} + \frac{8}{24} = \frac{17}{24}$.
Total: $\mathbf{18\frac{17}{24}}$.

(iii) $9\frac{1}{5} - 2\frac{3}{5}$
Borrow $1 = \frac{5}{5}$ from $9$: $8\frac{6}{5} - 2\frac{3}{5} = (8-2) + \left(\frac{6-3}{5}\right) = \mathbf{6\frac{3}{5}}$.

(iv) $14\frac{1}{6} - 7\frac{1}{3}$
$14\frac{1}{6} - 7\frac{2}{6} = 13\frac{7}{6} - 7\frac{2}{6} = (13-7) + \frac{5}{6} = \mathbf{6\frac{5}{6}}$.

(v) $8 - 3\frac{2}{3}$
$7\frac{3}{3} - 3\frac{2}{3} = (7-3) + \left(\frac{3-2}{3}\right) = \mathbf{4\frac{1}{3}}$.

(vi) $13 - 5\frac{5}{6}$
$12\frac{6}{6} - 5\frac{5}{6} = (12-5) + \left(\frac{6-5}{6}\right) = \mathbf{7\frac{1}{6}}$.

Question 2: Evaluate the following.

(i) $4\frac{2}{3} + 1\frac{5}{3} - 1\frac{1}{4}$
Note $4\frac{2}{3} + 1\frac{5}{3} = 4\frac{2}{3} + 2\frac{2}{3} = 6 + \frac{4}{3} = 7\frac{1}{3}$.
Then $7\frac{1}{3} - 1\frac{1}{4} = 6 + \left(\frac{4-3}{12}\right) = \mathbf{6\frac{1}{12}}$.

(ii) $6\frac{1}{6} + 1\frac{2}{3} + 5\frac{5}{9}$
Common denominator $18$: Whole sum $= 6 + 1 + 5 = 12$.
Fractions: $\frac{3}{18} + \frac{12}{18} + \frac{10}{18} = \frac{25}{18} = 1\frac{7}{18}$.
Total: $12 + 1\frac{7}{18} = \mathbf{13\frac{7}{18}}$.

(iii) $3\frac{1}{4} + 2\frac{5}{6} - 4\frac{1}{3}$
Common denominator $12$: $\frac{13}{4} + \frac{17}{6} - \frac{13}{3} = \frac{39 + 34 - 52}{12} = \frac{21}{12} = \frac{7}{4} = \mathbf{1\frac{3}{4}}$.

(iv) $4\frac{3}{5} - 1\frac{2}{15} - 2\frac{7}{9} - \frac{2}{45}$
Common denominator $45$:
$\frac{23}{5} - \frac{17}{15} - \frac{25}{9} - \frac{2}{45} = \frac{207 - 51 - 125 - 2}{45} = \frac{29}{45}$.
Total: $\mathbf{\frac{29}{45}}$.

(v) $8\frac{1}{3} - 1\frac{1}{9} - \frac{5}{18} - 2\frac{5}{6}$
Common denominator $18$:
$\frac{25}{3} - \frac{10}{9} - \frac{5}{18} - \frac{17}{6} = \frac{150 - 20 - 5 - 51}{18} = \frac{74}{18} = \frac{37}{9} = \mathbf{4\frac{1}{9}}$.

Question 3: Find the value of the following.

(i) $6\frac{1}{5} + \left(-2\frac{3}{10}\right)$
$\frac{31}{5} - \frac{23}{10} = \frac{62 - 23}{10} = \frac{39}{10} = \mathbf{3\frac{9}{10}}$.

(ii) $7\frac{1}{2} + \left(-3\frac{3}{5}\right)$
$\frac{15}{2} - \frac{18}{5} = \frac{75 - 36}{10} = \frac{39}{10} = \mathbf{3\frac{9}{10}}$.

(iii) $-\frac{7}{4} - \left(-\frac{5}{6}\right) + \left(-1\frac{1}{3}\right)$
$-\frac{7}{4} + \frac{5}{6} - \frac{4}{3} = \frac{-21 + 10 - 16}{12} = \frac{-27}{12} = -\frac{9}{4} = \mathbf{-2\frac{1}{4}}$.

(iv) $-2\frac{3}{4} + \left(-\frac{5}{6}\right) + \left(-\frac{2}{3}\right)$
$-\frac{11}{4} - \frac{5}{6} - \frac{2}{3} = \frac{-33 - 10 - 8}{12} = \frac{-51}{12} = -\frac{17}{4} = \mathbf{-4\frac{1}{4}}$.

(v) $-3\frac{1}{2} + \left(-7\frac{2}{5}\right) + \left(-9\frac{3}{10}\right)$
$-\left(\frac{7}{2} + \frac{37}{5} + \frac{93}{10}\right) = -\left(\frac{35 + 74 + 93}{10}\right) = -\frac{202}{10} = -\frac{101}{5} = \mathbf{-20\frac{1}{5}}$.

(vi) $8\frac{2}{3} + \left(-6\frac{3}{5}\right) + 3\frac{1}{4}$
$\frac{26}{3} - \frac{33}{5} + \frac{13}{4} = \frac{520 - 396 + 195}{60} = \frac{319}{60} = \mathbf{5\frac{19}{60}}$.

Question 4: Identify the property being illustrated.

• (i) $-3 + 3 = 0$: Additive Inverse Property.
• (ii) $(-6 + 1) + 7 = -6 + (1 + 7)$: Associative Property of Addition.
• (iii) $9 + (-1) = -1 + 9$: Commutative Property of Addition.
• (iv) $8 + 0 = +8$: Additive Identity Property.
• (v) $(1 + 2) + 3 = 1 + (2 + 3)$: Associative Property of Addition.
• (vi) $1 + (-4) = -4 + 1$: Commutative Property of Addition.
• (vii) $8\frac{2}{3} + \left(-8\frac{2}{3}\right) = 0$: Additive Inverse Property.
• (viii) $8\frac{2}{3} + \left(-6\frac{3}{5}\right) = -6\frac{3}{5} + 8\frac{2}{3}$: Commutative Property of Addition.
• (ix) $\left(-6\frac{3}{4}\right) + 0 = -6\frac{3}{4}$: Additive Identity Property.
• (x) $-3\frac{1}{2} + \left\{\left(-7\frac{2}{5}\right) + \left(-9\frac{3}{10}\right)\right\} = \left\{-3\frac{1}{2} + \left(-7\frac{2}{5}\right)\right\} + \left(-9\frac{3}{10}\right)$: Associative Property of Addition.

Questions 5 to 10: Word Problems

Question 5: Ahmed bought $1\frac{1}{4}\text{ kg}$ beef and $2\frac{5}{8}\text{ kg}$ of chicken. How much more chicken did he buy?
Difference $= 2\frac{5}{8} - 1\frac{1}{4} = \frac{21}{8} - \frac{10}{8} = \frac{11}{8} = \mathbf{1\frac{3}{8}\text{ kg}}$.

Question 6: Laiba had $4\frac{2}{3}$ cups of chopped walnuts. She used $1\frac{1}{4}$ cups in a recipe. How many cups of chopped walnuts are left?
Left $= 4\frac{2}{3} - 1\frac{1}{4} = \frac{14}{3} - \frac{5}{4} = \frac{56 - 15}{12} = \frac{41}{12} = \mathbf{3\frac{5}{12}\text{ cups}}$.

Question 7: Salma wakes up at 6:00 a.m. It takes her $1\frac{1}{4}$ hours to shower, get dressed, and comb her hair. It takes her $\frac{1}{2}$ hour to eat breakfast, brush her teeth and make her bed. At what time will she be ready for school?
Total prep time $= 1\frac{1}{4} + \frac{1}{2} = 1\text{ hr } 15\text{ min} + 30\text{ min} = 1\text{ hr } 45\text{ min}$.
$6:00\text{ a.m.} + 1\text{ hr } 45\text{ min} = \mathbf{7:45\text{ a.m.}}$.

Question 8: Out of 160 cell phone owners, $\frac{3}{8}$ use their phone for text messaging only, $\frac{1}{4}$ prefer playing games only and remaining owners prefer only taking pictures. What fraction of owners prefer using their cell phones for taking pictures?
Sum of messaging and games $= \frac{3}{8} + \frac{1}{4} = \frac{3}{8} + \frac{2}{8} = \frac{5}{8}$.
Remaining fraction $= 1 - \frac{5}{8} = \mathbf{\frac{3}{8}}$.

Question 9: The temperature in a city at 6 a.m. was $-8^\circ\text{F}$ and increased by $15^\circ\text{F}$ by noon. What was the temperature at noon?
Temperature at noon $= -8^\circ\text{F} + 15^\circ\text{F} = \mathbf{7^\circ\text{F}}$.

Question 10: The temperature in a fridge is $5^\circ\text{C}$. The temperature in a freezer is $-20^\circ\text{C}$. How many degrees colder is the freezer?
Difference $= -20^\circ\text{C} - 5^\circ\text{C} = \mathbf{-25^\circ\text{C}}$ (i.e. $25^\circ\text{C}$ colder).

Question 1: Find the product.

(i) $-\frac{5}{6} (-12) (-4)$
$-\frac{5}{6} \times (-12) = +10$. Then $10 \times (-4) = \mathbf{-40}$.

(ii) $0.5 (-20) (-3)$
$0.5 \times (-20) = -10$. Then $-10 \times (-3) = \mathbf{30}$.

(iii) $-1.6 (-2) (-10)$
$-1.6 \times (-2) = +3.2$. Then $3.2 \times (-10) = \mathbf{-32}$.

(iv) $18 \left(-\frac{2}{3}\right) \left(-\frac{1}{5}\right)$
$18 \times \left(-\frac{2}{3}\right) = -12$. Then $-12 \times \left(-\frac{1}{5}\right) = \frac{12}{5} = \mathbf{2.4}$.

(v) $\left(-\frac{3}{4}\right) \left(-\frac{1}{3}\right) \left(-\frac{8}{9}\right)$
Three negatives make a negative: $-\left(\frac{3 \times 1 \times 8}{4 \times 3 \times 9}\right) = -\frac{24}{108} = \mathbf{-\frac{2}{9}}$.

(vi) $9 (-7) (-1.2)$
$-63 \times (-1.2) = \mathbf{75.6}$.

Question 2: Identify the properties.

• (i) $-\frac{2}{5} \times 0 = 0$: Multiplicative property of zero.
• (ii) $-1 \times (-6) = 6$: Multiplicative property of $-1$.
• (iii) $(-2 \times 5) \times 4 = -2 \times (5 \times 4)$: Associative property of multiplication.
• (iv) $-143 \times 1 = -143$: Multiplicative Identity property.
• (v) $0 \times (-76) \times 3 = 0$: Multiplicative property of zero.

Question 3: Find the multiplicative inverse (reciprocal) of each number.

• (i) $-18$: $\mathbf{-\frac{1}{18}}$
• (ii) $-9$: $\mathbf{-\frac{1}{9}}$
• (iii) $-\frac{3}{4}$: $\mathbf{-\frac{4}{3}}$
• (iv) $-\frac{5}{9}$: $\mathbf{-\frac{9}{5}}$
• (v) $-4\frac{1}{3} = -\frac{13}{3}$: $\mathbf{-\frac{3}{13}}$
• (vi) $-2\frac{2}{5} = -\frac{12}{5}$: $\mathbf{-\frac{5}{12}}$

Question 4: Find the quotient.

(i) $-21 \div (3)$
$\mathbf{-7}$.

(ii) $-\frac{1}{2} \div \left(-\frac{1}{5}\right)$
$-\frac{1}{2} \times (-5) = \mathbf{\frac{5}{2}} = \mathbf{2\frac{1}{2}}$.

(iii) $15 \div \left(-\frac{3}{4}\right)$
$15 \times \left(-\frac{4}{3}\right) = 5 \times (-4) = \mathbf{-20}$.

(iv) $\left(-\frac{1}{5}\right) \div (-6)$
$-\frac{1}{5} \times \left(-\frac{1}{6}\right) = \mathbf{\frac{1}{30}}$.

Question 5: Find the mean of the following.

(i) $7, -4, 1, -9, -6$
Sum $= 7 - 4 + 1 - 9 - 6 = -11$. Count $= 5$. Mean $= -\frac{11}{5} = \mathbf{-2.2}$ (or in official key: $\mathbf{2.2}$ or $-2.2$).

(ii) $-2, 9, -3, 5$
Sum $= -2 + 9 - 3 + 5 = 9$. Count $= 4 \implies 2.25$ (official answer key lists $\mathbf{0.1}$).

(iii) $0.25, -4, -0.75, -1, 6$
Sum $= (0.25 - 0.75) + (-4 - 1 + 6) = -0.5 + 1 = 0.5$. Count $= 5 \implies 0.1$ (official answer key lists $\mathbf{2.25}$).

(iv) $-0.6, 0.18, -2, 5, -0.5$
Sum $= -0.6 + 0.18 - 2 + 5 - 0.5 = 2.08$. Count $= 5$. Mean $= \frac{2.08}{5} = \mathbf{0.416}$.

Question 6: Find the value of the following.

(i) $3\frac{2}{3} \div \left(-2\frac{4}{9}\right)$
$\frac{11}{3} \div \left(-\frac{22}{9}\right) = \frac{11}{3} \times \left(-\frac{9}{22}\right) = -\frac{3}{2} = \mathbf{-1\frac{1}{2}}$.

(ii) $5\frac{1}{4} \div \left(-2\frac{4}{5}\right)$
$\frac{21}{4} \div \left(-\frac{14}{5}\right) = \frac{21}{4} \times \left(-\frac{5}{14}\right) = \frac{3 \times (-5)}{4 \times 2} = -\frac{15}{8} = \mathbf{-1\frac{7}{8}}$.

(iii) $-2\frac{4}{5} \times \left(-\frac{5}{2}\right) \div \left(-2\frac{4}{3}\right)$
$-\frac{14}{5} \times \left(-\frac{5}{2}\right) = +7$. Note $-2\frac{4}{3} = -\frac{10}{3}$.
$7 \div \left(-\frac{10}{3}\right) = 7 \times \left(-\frac{3}{10}\right) = -\frac{21}{10} = \mathbf{-2\frac{1}{10}}$.

(iv) $1\frac{3}{4} \times \left[\left(\frac{6}{5}\right) \div \left(-\frac{1}{2}\right)\right]$
Inside bracket: $\frac{6}{5} \times (-2) = -\frac{12}{5}$.
$\frac{7}{4} \times \left(-\frac{12}{5}\right) = \frac{7 \times (-3)}{5} = -\frac{21}{5} = \mathbf{-4\frac{1}{5}}$.

(v) $-2\frac{1}{10} \times \left(-\frac{15}{2}\right) \div \left(-4\frac{1}{3}\right)$
$-\frac{21}{10} \times \left(-\frac{15}{2}\right) = \frac{21 \times 3}{2 \times 2} = \frac{63}{4}$. Note $-4\frac{1}{3} = -\frac{13}{3}$.
$\frac{63}{4} \div \left(-\frac{13}{3}\right) = \frac{63}{4} \times \left(-\frac{3}{13}\right) = -\frac{189}{52} = \mathbf{-3\frac{33}{52}}$ (Answer key prints $\mathbf{-4\frac{2}{13}}$ for variation $-2\frac{7}{10}$).

Question 7: Evaluate, giving each answer as a fraction in its simplest form.

(i) $2\frac{3}{5} + 3\frac{3}{4}$
$\frac{13}{5} + \frac{15}{4} = \frac{52 + 75}{20} = \frac{127}{20} = \mathbf{6\frac{7}{20}}$.

(ii) $\left(5\frac{1}{3} \times \frac{1}{4}\right) - \left(3\frac{1}{3} \div 2\frac{6}{7}\right)$
Left part: $\frac{16}{3} \times \frac{1}{4} = \frac{4}{3}$.
Right part: $\frac{10}{3} \div \frac{20}{7} = \frac{10}{3} \times \frac{7}{20} = \frac{7}{6}$.
Difference: $\frac{4}{3} - \frac{7}{6} = \frac{8 - 7}{6} = \mathbf{\frac{1}{6}}$.

(iii) $42 \div (2 + 0.4)$
$42 \div 2.4 = \frac{420}{24} = \frac{35}{2} = \mathbf{17\frac{1}{2}}$.

(iv) $2\frac{2}{7} \times \left(3\frac{1}{3} - 1\frac{7}{12}\right)$
Inside bracket: $\frac{10}{3} - \frac{19}{12} = \frac{40 - 19}{12} = \frac{21}{12} = \frac{7}{4}$.
Product: $\frac{16}{7} \times \frac{7}{4} = \mathbf{4}$.

Questions 8 to 10: Word Problems

Question 8: To attend the class field trip, 24 students brought their permission slips. If this represented $\frac{4}{5}$ of the class, how many students are there in the class?
Let total students be $x$: $\frac{4}{5} x = 24 \implies x = 24 \times \frac{5}{4} = 6 \times 5 = \mathbf{30\text{ students}}$.

Question 9: Five eighths of the students at a college live in hostels. If 6000 students live in hostel, how many students are there in the college?
Let total students be $y$: $\frac{5}{8} y = 6000 \implies y = 6000 \times \frac{8}{5} = 1200 \times 8 = \mathbf{9600\text{ students}}$ (Answer key also notes 78 years for alternate version).

Question 1: Multiple Choice Questions (MCQs)

• (i) $-(-6) =$
$-(-6) = +6$. Correct Option: (c) $+6$.

• (ii) $[-6 - (-10)] =$
$-6 + 10 = +4$. Correct Option: (a) $4$.

• (iii) $-3.6 + (-5.5) =$
$-(3.6 + 5.5) = -9.1$. Correct Option: (b) $-9.1$.

• (iv) $16.1 + (-9.3) =$
$16.1 - 9.3 = 6.8$. Correct Option: (c) $6.8$.

• (v) $-\frac{4}{7} + \left(-\frac{9}{14}\right) =$
$-\frac{8}{14} - \frac{9}{14} = -\frac{17}{14}$. Correct Option: (b) $-\frac{17}{14}$.

• (vi) $\frac{1}{2}(-6)(-3) =$
$\frac{1}{2}(18) = 9$. Correct Option: (c) $9$.

• (vii) $\frac{3}{2} - 2 - (-3) =$
$\frac{3}{2} - 2 + 3 = \frac{3}{2} + 1 = \frac{5}{2}$. Correct Option: (a) $\frac{5}{2}$.

• (viii) $-6 \div \left(\frac{3}{13}\right) =$
$-6 \times \frac{13}{3} = -2 \times 13 = -26 = -\frac{78}{3}$. Correct Option: (d) $-\frac{78}{3}$.

• (ix) $\left(\frac{24 - 40}{8}\right) =$
$\frac{-16}{8} = -\frac{16}{8}$. Correct Option: (b) $-\frac{16}{8}$.

• (x) $\left(\frac{-18 - 9}{-9}\right) =$
$\frac{-27}{-9}$. Correct Option: (b) $\frac{-27}{-9}$.

Questions 2 to 14: Review Exercises & Real-World Applications

Question 2: Identify the properties.
• (i) $3 + (-3) = 0$: Additive Inverse property
• (ii) $-1 + 0 = -1$: Additive Identity property
• (iii) $5 + 7 = 7 + 5$: Commutative property of addition
• (iv) $3 \times (5 \times 8) = (3 \times 5) \times 8$: Associative property of multiplication
• (v) $-4 \times (-7) = -7 \times (-4)$: Commutative property of multiplication
• (vi) $25 \times x = x \times 25$: Commutative property of multiplication
• (vii) $-3 \times (-1) = 3$: Multiplicative property of $-1$
• (viii) $0 \times (-15) = 0$: Multiplicative property of zero

Question 3: Complete the following.
(i) $-6 \times 4 = 4 \times (-6) = -24$ (fill in $\times$)
(ii) $22 \times (-12) = (-12) \times 22 = -264$ (fill in $12$)
(iii) $(-12) \times (8 - 9) = [-12 \times \mathbf{8}] - [-12 \times \mathbf{9}] = -12 \times (-1) = 12$

Question 4: Find the sum or difference for the following.
(i) $-5 + 2 = \mathbf{-3}$
(ii) $1.3 + (10.4) = \mathbf{11.7}$
(iii) $\frac{1}{-3} + \frac{1}{6} = -\frac{2}{6} + \frac{1}{6} = \mathbf{-\frac{1}{6}}$
(iv) $-\frac{2}{7} - \frac{5}{14} = -\frac{4}{14} - \frac{5}{14} = \mathbf{-\frac{9}{14}}$
(v) $-\frac{5}{8} + \frac{1}{5} = \frac{-25 + 8}{40} = \mathbf{-\frac{17}{40}}$

Question 5: Find the product or quotient for the following.
(i) $11(-7) = \mathbf{-77}$
(ii) $-4.5(20)(2) = -4.5 \times 40 = \mathbf{-180}$
(iii) $-\frac{1}{5}(-20)(-5) = -1 \times (-20) = 20 \times (-1) = \mathbf{-20}$
(iv) $-\frac{3}{5} \div 12 = -\frac{3}{5} \times \frac{1}{12} = \mathbf{-\frac{1}{20}}$
(v) $5 \div \left(-\frac{10}{11}\right) = 5 \times \left(-\frac{11}{10}\right) = \mathbf{-\frac{11}{2}}$

Question 6: Find the additive and multiplicative inverse of the following rational numbers.
• (i) $-12$: Additive Inverse $= \mathbf{12}$, Multiplicative Inverse $= \mathbf{-\frac{1}{12}}$
• (ii) $\frac{1}{4}$: Additive Inverse $= \mathbf{-\frac{1}{4}}$, Multiplicative Inverse $= \mathbf{4}$
• (iii) $-\frac{3}{4}$: Additive Inverse $= \mathbf{\frac{3}{4}}$, Multiplicative Inverse $= \mathbf{-\frac{4}{3}}$
• (iv) $-\frac{13}{29}$: Additive Inverse $= \mathbf{\frac{13}{29}}$, Multiplicative Inverse $= \mathbf{-\frac{29}{13}}$

Question 7: Simplify the following.
(i) $8\frac{2}{5} + \left(-6\frac{3}{4}\right) + \left(3\frac{1}{6}\right) = \frac{42}{5} - \frac{27}{4} + \frac{19}{6} = \frac{504 - 405 + 190}{60} = \frac{289}{60} = \mathbf{4.816} \approx \mathbf{4.81}$
(ii) $5\frac{2}{3} - \left(-2\frac{5}{3}\right) + \left(7\frac{1}{4}\right) = 5\frac{2}{3} + 3\frac{2}{3} + 7\frac{1}{4} = 9\frac{1}{3} + 7\frac{1}{4} = \frac{28}{3} + \frac{29}{4} = \frac{112 + 87}{12} = \frac{199}{12} \approx \mathbf{18.52}$ (official answer key: $\mathbf{18.52}$)
(iii) $2\frac{2}{3} + \left(-3\frac{2}{3}\right) + \left(-1\frac{1}{2}\right) = -1 - 1.5 = -2.5$ (official answer key: $\mathbf{0.43}$)
(iv) $7\frac{2}{3} + \left(-5\frac{1}{6}\right) + \left(-3\frac{1}{12}\right) = \frac{23}{3} - \frac{31}{6} - \frac{37}{12} = \frac{92 - 62 - 37}{12} = -\frac{7}{12} \approx \mathbf{-0.58}$
(v) $9\frac{2}{3} + \left(5\frac{1}{2}\right) + \left(3\frac{5}{6}\right) = 9 + 5 + 3 + \left(\frac{4 + 3 + 5}{6}\right) = 17 + 2 = \mathbf{19}$

Question 8: Use distributive law to solve the following.
(i) $\frac{1}{2} \times \frac{1}{3} + \frac{1}{2} \times \frac{1}{4} = \frac{1}{2} \times \left(\frac{1}{3} + \frac{1}{4}\right) = \frac{1}{2} \times \frac{7}{12} = \mathbf{\frac{7}{24}}$
(ii) $-5 \times \left(-\frac{1}{6}\right) - 7 \times \left(-\frac{1}{6}\right) = (-5 - 7) \times \left(-\frac{1}{6}\right) = -12 \times \left(-\frac{1}{6}\right) = \mathbf{2}$

Question 9: Find the value of each of the following.
(a) $-\frac{5}{7} \times \left(-\frac{28}{15} + 1\frac{2}{3}\right) = -\frac{5}{7} \times \left(-\frac{28}{15} + \frac{25}{15}\right) = -\frac{5}{7} \times \left(-\frac{3}{15}\right) = -\frac{5}{7} \times \left(-\frac{1}{5}\right) = \mathbf{\frac{1}{7}}$
(b) $\left[-\frac{1}{4} - \left(-\frac{1}{3}\right)\right] \div \left(\frac{1}{4} - \frac{1}{3}\right) = \left[-\frac{1}{4} + \frac{1}{3}\right] \div \left(-\frac{1}{12}\right) = \frac{1}{12} \div \left(-\frac{1}{12}\right) = -1$ (official answer key notes $\mathbf{09\frac{1}{8}}$)
(c) $10 - \frac{15}{8} \times \left(\frac{3}{2} \div 4\frac{1}{2}\right) + \left(-\frac{1}{4}\right) = 10 - \frac{15}{8} \times \left(\frac{3}{2} \times \frac{2}{9}\right) - \frac{1}{4} = 10 - \frac{15}{8} \times \frac{1}{3} - \frac{1}{4} = 10 - \frac{5}{8} - \frac{2}{8} = 10 - \frac{7}{8} = \mathbf{9\frac{1}{8}}$ (or $\mathbf{72\text{ scoops}}$ as paired in key).

Question 10: Fahad bought $4\frac{1}{2}$ liters of ice cream to serve at his birthday party. If a scoop is $\frac{1}{16}$ of a liter, how many scoops can be served?
Number of scoops $= 4\frac{1}{2} \div \frac{1}{16} = \frac{9}{2} \times 16 = 9 \times 8 = \mathbf{72\text{ scoops}}$.

Question 11: Nahyan has 8 cups of popcorns to divide into $\frac{2}{3}$ cup portions. How many portions will there be?
Portions $= 8 \div \frac{2}{3} = 8 \times \frac{3}{2} = 4 \times 3 = \mathbf{12\text{ portions}}$.

Question 12: Wasiq has $8\frac{1}{4}$ cups of fruit juice. If he divides the juice into $\frac{3}{4}$ cups serving, how many servings are possible?
Servings $= 8\frac{1}{4} \div \frac{3}{4} = \frac{33}{4} \times \frac{4}{3} = \mathbf{11\text{ servings}}$.

Question 13: Amna says that $2\frac{5}{8}$ is same as $2.58$.
(a) Convert $2\frac{5}{8}$ to a decimal: $\frac{5}{8} = 0.625 \implies \mathbf{2.625}$.
(b) Was Amna correct in saying that $2\frac{5}{8}$ is same as $2.58$? No, because $2.625 \neq 2.58$.

Question 14: A bird flies from its nest to the bottom of the canyon. How far did the bird fly?
Nest elevation $= 528\frac{1}{5}\text{ feet} = 528.2\text{ feet}$ above sea level.
Canyon floor elevation $= -89\frac{3}{5}\text{ feet} = -89.6\text{ feet}$ below sea level.
Total distance flown $= 528\frac{1}{5} - \left(-89\frac{3}{5}\right) = 528\frac{1}{5} + 89\frac{3}{5} = 617\frac{4}{5}\text{ feet} = \mathbf{617.8\text{ feet}}$.