Mastery Guide: Place Value, Decimal Arithmetic, Fraction Conversions & Percentage Applications

(a) $0.4\ \underline{\hspace{20pt}}\ 0.5$: $4$ tenths $< 5$ tenths $\implies \mathbf{0.4 < 0.5}$

(b) $1.3\ \underline{\hspace{20pt}}\ 1.6$: Whole numbers equal, tenths $3 < 6 \implies \mathbf{1.3 < 1.6}$

(c) $34.56\ \underline{\hspace{20pt}}\ 35.66$: Whole numbers $34 < 35 \implies \mathbf{34.56 < 35.66}$

(d) $6.67\ \underline{\hspace{20pt}}\ 6.69$: Tenths equal, hundredths $7 < 9 \implies \mathbf{6.67 < 6.69}$

(e) $0.45\ \underline{\hspace{20pt}}\ 0.45$: Both numbers identical $\implies \mathbf{0.45 = 0.45}$

(f) $23.12\ \underline{\hspace{20pt}}\ 51.31$: Whole numbers $23 < 51 \implies \mathbf{23.12 < 51.31}$

(g) $71.2\ \underline{\hspace{20pt}}\ 71.02$: Pad zero: $71.20$ vs $71.02$. Tenths $2 > 0 \implies \mathbf{71.2 > 71.02}$

(h) $6.06\ \underline{\hspace{20pt}}\ 6.1$: Pad zero: $6.06$ vs $6.10$. Tenths $0 < 1 \implies \mathbf{6.06 < 6.1}$

(i) $4.8\ \underline{\hspace{20pt}}\ 4.80$: Trailing zeros do not change value $\implies \mathbf{4.8 = 4.80}$

(a) $0.23, 0.45, 0.12, 0.67$: $\implies \mathbf{0.67, 0.45, 0.23, 0.12}$

(b) $2.4, 2.7, 2.1, 2.9$: $\implies \mathbf{2.9, 2.7, 2.4, 2.1}$

(c) $14.56, 14.65, 14.12, 14.89$: $\implies \mathbf{14.89, 14.65, 14.56, 14.12}$

(d) $3.12, 3.21, 3.02, 3.20$: $\implies \mathbf{3.21, 3.20, 3.12, 3.02}$

(a) $0.8, 0.2, 0.5, 0.9$: $\implies \mathbf{0.2, 0.5, 0.8, 0.9}$

(b) $5.43, 5.34, 5.41, 5.14$: $\implies \mathbf{5.14, 5.34, 5.41, 5.43}$

(c) $11.02, 11.20, 11.12, 11.22$: $\implies \mathbf{11.02, 11.12, 11.20, 11.22}$

(d) $6.78, 6.87, 6.08, 6.80$: $\implies \mathbf{6.08, 6.78, 6.80, 6.87}$

(a) $4.131 + 8.3$: $4.131 + 8.300 = \mathbf{12.431}$

(b) $3.211 + 1.860$: $3.211 + 1.860 = \mathbf{5.071}$

(c) $12.45 + 6.321$: $12.450 + 6.321 = \mathbf{18.771}$

(d) $7.892 + 4.108$: $7.892 + 4.108 = \mathbf{12.000 = 12}$

(e) $15.6 + 9.875$: $15.600 + 9.875 = \mathbf{25.475}$

(f) $0.456 + 0.789$: $0.456 + 0.789 = \mathbf{1.245}$

(g) $23.14 + 16.9$: $23.14 + 16.90 = \mathbf{40.04}$

(h) $8.005 + 3.995$: $8.005 + 3.995 = \mathbf{12.000 = 12}$

(i) $45.123 + 12.877$: $45.123 + 12.877 = \mathbf{58.000 = 58}$

(j) $1.234 + 5.678$: $1.234 + 5.678 = \mathbf{6.912}$

(k) $67.8 + 23.456$: $67.800 + 23.456 = \mathbf{91.256}$

(l) $9.999 + 0.001$: $9.999 + 0.001 = \mathbf{10.000 = 10}$

(a) $9.410 - 2.392$: $9.410 - 2.392 = \mathbf{7.018}$

(b) $11.45 - 2.86$: $11.45 - 2.86 = \mathbf{8.59}$

(c) $15.678 - 9.456$: $15.678 - 9.456 = \mathbf{6.222}$

(d) $8.5 - 3.25$: $8.50 - 3.25 = \mathbf{5.25}$

(e) $20.005 - 12.789$: $20.005 - 12.789 = \mathbf{7.216}$

(f) $14.12 - 6.895$: $14.120 - 6.895 = \mathbf{7.225}$

(g) $7.8 - 4.356$: $7.800 - 4.356 = \mathbf{3.444}$

(h) $50.0 - 24.678$: $50.000 - 24.678 = \mathbf{25.322}$

(i) $6.789 - 1.234$: $6.789 - 1.234 = \mathbf{5.555}$

(j) $18.45 - 9.876$: $18.450 - 9.876 = \mathbf{8.574}$

(k) $10.01 - 4.567$: $10.010 - 4.567 = \mathbf{5.443}$

(l) $32.1 - 15.678$: $32.100 - 15.678 = \mathbf{16.422}$

(a) How much sugar did she use in total?
$$\text{Total Sugar} = 50.55 + 28.50 = \mathbf{79.05\text{ g}}$$

(b) How much more sugar was used for the cake than cupcakes?
$$\text{Difference} = 50.55 - 28.50 = \mathbf{22.05\text{ g}}$$

(a) $2.34 \times 10$: $\mathbf{23.4}$

(b) $5.678 \times 100$: $\mathbf{567.8}$

(c) $0.45 \times 1000$: $\mathbf{450}$

(d) $12.3 \times 10$: $\mathbf{123}$

(e) $0.009 \times 100$: $\mathbf{0.9}$

(f) $7.812 \times 1000$: $\mathbf{7812}$

(g) $65.4 \times 100$: $\mathbf{6540}$

(h) $0.123 \times 10$: $\mathbf{1.23}$

(i) $3.4 \times 1000$: $\mathbf{3400}$

(j) $98.76 \times 10$: $\mathbf{987.6}$

(k) $1.005 \times 100$: $\mathbf{100.5}$

(l) $0.06 \times 1000$: $\mathbf{60}$

(a) $4.23 \times 12$: $423 \times 12 = 5076 \implies \mathbf{50.76}$

(b) $6.15 \times 24$: $615 \times 24 = 14760 \implies \mathbf{147.60 = 147.6}$

(c) $12.5 \times 15$: $125 \times 15 = 1875 \implies \mathbf{187.5}$

(d) $3.456 \times 11$: $3456 \times 11 = 38016 \implies \mathbf{38.016}$

(e) $8.05 \times 32$: $805 \times 32 = 25760 \implies \mathbf{257.60 = 257.6}$

(f) $1.78 \times 45$: $178 \times 45 = 8010 \implies \mathbf{80.10 = 80.1}$

(g) $9.21 \times 18$: $921 \times 18 = 16578 \implies \mathbf{165.78}$

(h) $0.85 \times 25$: $85 \times 25 = 2125 \implies \mathbf{21.25}$

(i) $14.2 \times 50$: $142 \times 50 = 7100 \implies \mathbf{710.0 = 710}$

(j) $5.112 \times 16$: $5112 \times 16 = 81792 \implies \mathbf{81.792}$

(k) $7.4 \times 35$: $74 \times 35 = 2590 \implies \mathbf{259.0 = 259}$

(l) $2.65 \times 42$: $265 \times 42 = 11130 \implies \mathbf{111.30 = 111.3}$

(a) $2.3 \times 1.4$: $23 \times 14 = 322 \implies \mathbf{3.22}$

(b) $4.15 \times 2.3$: $415 \times 23 = 9545 \implies \mathbf{9.545}$

(c) $0.5 \times 0.7$: $5 \times 7 = 35 \implies \mathbf{0.35}$

(d) $6.12 \times 3.4$: $612 \times 34 = 20808 \implies \mathbf{20.808}$

(e) $1.25 \times 0.8$: $125 \times 8 = 1000 \implies \mathbf{1.000 = 1}$

(f) $7.8 \times 2.15$: $78 \times 215 = 16770 \implies \mathbf{16.770 = 16.77}$

(g) $0.45 \times 1.2$: $45 \times 12 = 540 \implies \mathbf{0.540 = 0.54}$

(h) $9.1 \times 4.2$: $91 \times 42 = 3822 \implies \mathbf{38.22}$

(i) $3.15 \times 2.45$: $315 \times 245 = 77175 \implies \mathbf{7.7175}$

(j) $0.12 \times 0.34$: $12 \times 34 = 408 \implies \mathbf{0.0408}$

(k) $5.6 \times 1.8$: $56 \times 18 = 1008 \implies \mathbf{10.08}$

(l) $8.25 \times 0.6$: $825 \times 6 = 4950 \implies \mathbf{4.950 = 4.95}$

• Cost of $10\text{ m} = 25.5 \times 10 = \mathbf{\text{Rs. } 255}$

• Cost of $100\text{ m} = 25.5 \times 100 = \mathbf{\text{Rs. } 2550}$

• Cost of $1000\text{ m} = 25.5 \times 1000 = \mathbf{\text{Rs. } 25500}$

(a) $45.6 \div 10$: $\mathbf{4.56}$

(b) $789.2 \div 100$: $\mathbf{7.892}$

(c) $123.4 \div 1000$: $\mathbf{0.1234}$

(d) $6.8 \div 10$: $\mathbf{0.68}$

(e) $54.1 \div 100$: $\mathbf{0.541}$

(f) $9.5 \div 1000$: $\mathbf{0.0095}$

(g) $345.0 \div 10$: $\mathbf{34.5}$

(h) $12.0 \div 100$: $\mathbf{0.12}$

(i) $8.0 \div 1000$: $\mathbf{0.008}$

(a) $24.8 \div 4$: $\mathbf{6.2}$

(b) $45.65 \div 5$: $\mathbf{9.13}$

(c) $18.9 \div 9$: $\mathbf{2.1}$

(d) $56.7 \div 7$: $\mathbf{8.1}$

(e) $3.75 \div 5$: $\mathbf{0.75}$

(f) $81.9 \div 3$: $\mathbf{27.3}$

(g) $14.4 \div 12$: $\mathbf{1.2}$

(h) $62.5 \div 25$: $\mathbf{2.5}$

(i) $9.45 \div 15$: $\mathbf{0.63}$

(a) $4.8 \div 0.6$: $\frac{48}{6} = \mathbf{8}$

(b) $12.5 \div 2.5$: $\frac{125}{25} = \mathbf{5}$

(c) $6.75 \div 1.5$: $\frac{67.5}{15} = \mathbf{4.5}$

(d) $14.4 \div 1.2$: $\frac{144}{12} = \mathbf{12}$

(e) $8.1 \div 0.9$: $\frac{81}{9} = \mathbf{9}$

(f) $15.75 \div 3.5$: $\frac{157.5}{35} = \mathbf{4.5}$

(g) $0.48 \div 0.08$: $\frac{48}{8} = \mathbf{6}$

(h) $2.25 \div 0.5$: $\frac{22.5}{5} = \mathbf{4.5}$

(i) $9.6 \div 3.2$: $\frac{96}{32} = \mathbf{3}$

(a) Each bottle holds $4.5\text{ litres}$?
$$\text{Number of bottles} = 67.5 \div 4.5 = \frac{675}{45} = \mathbf{15\text{ bottles}}$$

(b) Each bottle holds $2.5\text{ litres}$?
$$\text{Number of bottles} = 67.5 \div 2.5 = \frac{675}{25} = \mathbf{27\text{ bottles}}$$

(a) $\frac{45}{2}$: $45 \div 2 = \mathbf{22.5}$

(b) $\frac{19}{25}$: $\frac{19 \times 4}{100} = \mathbf{0.76}$

(c) $\frac{7}{40}$: $7 \div 40 = \mathbf{0.175}$

(d) $\frac{3}{8}$: $3 \div 8 = \mathbf{0.375}$

(e) $\frac{1}{4}$: $1 \div 4 = \mathbf{0.25}$

(f) $\frac{4}{5}$: $\frac{8}{10} = \mathbf{0.8}$

(g) $\frac{13}{20}$: $\frac{65}{100} = \mathbf{0.65}$

(h) $\frac{9}{50}$: $\frac{18}{100} = \mathbf{0.18}$

(i) $\frac{11}{4}$: $11 \div 4 = \mathbf{2.75}$

(j) $\frac{7}{8}$: $7 \div 8 = \mathbf{0.875}$

(k) $\frac{23}{5}$: $\frac{46}{10} = \mathbf{4.6}$

(l) $\frac{31}{100}$: $\mathbf{0.31}$

(i) $5.0$: $\frac{50}{10} = \mathbf{5\text{ (or }\frac{5}{1})}$

(ii) $2.1$: $\mathbf{\frac{21}{10}\text{ (or }2\frac{1}{10})}$

(iii) $24.0$: $\mathbf{24\text{ (or }\frac{24}{1})}$

(iv) $37.2$: $\frac{372}{10} = \mathbf{\frac{186}{5}\text{ (or }37\frac{1}{5})}$

(v) $75.0$: $\mathbf{75\text{ (or }\frac{75}{1})}$

(vi) $121.0$: $\mathbf{121\text{ (or }\frac{121}{1})}$

(vii) $55.5$: $\frac{555}{10} = \mathbf{\frac{111}{2}\text{ (or }55\frac{1}{2})}$

(viii) $625.0$: $\mathbf{625\text{ (or }\frac{625}{1})}$

(i) $9.02$: Tenths is $0 < 5 \implies \mathbf{9}$

(ii) $4.444$: Tenths is $4 < 5 \implies \mathbf{4}$

(iii) $7.89$: Tenths is $8 \ge 5 \implies \mathbf{8}$

(iv) $9.65$: Tenths is $6 \ge 5 \implies \mathbf{10}$

(v) $15.301$: Tenths is $3 < 5 \implies \mathbf{15}$

(a) $47.125$: 1 d.p.: $\mathbf{47.1}$ • 2 d.p.: $\mathbf{47.13}$

(b) $4.732$: 1 d.p.: $\mathbf{4.7}$ • 2 d.p.: $\mathbf{4.73}$

(c) $2.322$: 1 d.p.: $\mathbf{2.3}$ • 2 d.p.: $\mathbf{2.32}$

(d) $0.942$: 1 d.p.: $\mathbf{0.9}$ • 2 d.p.: $\mathbf{0.94}$

(e) $45.675$: 1 d.p.: $\mathbf{45.7}$ • 2 d.p.: $\mathbf{45.68}$

(f) $2.150$: 1 d.p.: $\mathbf{2.2}$ • 2 d.p.: $\mathbf{2.15}$

(g) $91.547$: 1 d.p.: $\mathbf{91.5}$ • 2 d.p.: $\mathbf{91.55}$

(h) $94.172$: 1 d.p.: $\mathbf{94.2}$ • 2 d.p.: $\mathbf{94.17}$

(i) $5.183$: 1 d.p.: $\mathbf{5.2}$ • 2 d.p.: $\mathbf{5.18}$

(j) $3.767$: 1 d.p.: $\mathbf{3.8}$ • 2 d.p.: $\mathbf{3.77}$

(k) $4.172$: 1 d.p.: $\mathbf{4.2}$ • 2 d.p.: $\mathbf{4.17}$

(l) $90.267$: 1 d.p.: $\mathbf{90.3}$ • 2 d.p.: $\mathbf{90.27}$

(a) $52.90 + 17.98$: $53 + 18 = \mathbf{71}$ (Actual: $70.88$)

(b) $630.1 + 280.9$: $630 + 281 = \mathbf{911}$ (Actual: $911.0$)

(c) $41.01 + 36.87$: $41 + 37 = \mathbf{78}$ (Actual: $77.88$)

(d) $307.2 + 357.6$: $307 + 358 = \mathbf{665}$ (Actual: $664.8$)

(e) $741.2 + 30.10$: $741 + 30 = \mathbf{771}$ (Actual: $771.30$)

(f) $845.1 + 396.9$: $845 + 397 = \mathbf{1242}$ (Actual: $1242.0$)

(g) $63.81 + 25.91$: $64 + 26 = \mathbf{90}$ (Actual: $89.72$)

(h) $21.35 + 83.05$: $21 + 83 = \mathbf{104}$ (Actual: $104.40$)

(i) $99.99 + 87.91$: $100 + 88 = \mathbf{188}$ (Actual: $187.90$)

(j) $943.6 + 834.6$: $944 + 835 = \mathbf{1779}$ (Actual: $1778.2$)

(k) $123.4 + 567.8$: $123 + 568 = \mathbf{691}$ (Actual: $691.2$)

(l) $737.8 + 721.2$: $738 + 721 = \mathbf{1459}$ (Actual: $1459.0$)

(a) $22.30 - 17.99$: $22 - 18 = \mathbf{4}$ (Actual: $4.31$)

(b) $78.92 - 69.11$: $79 - 69 = \mathbf{10}$ (Actual: $9.81$)

(c) $56.23 - 11.26$: $56 - 11 = \mathbf{45}$ (Actual: $44.97$)

(d) $234.6 - 159.8$: $235 - 160 = \mathbf{75}$ (Actual: $74.8$)

(e) $587.6 - 320.9$: $588 - 321 = \mathbf{267}$ (Actual: $266.7$)

(f) $402.3 - 292.1$: $402 - 292 = \mathbf{110}$ (Actual: $110.2$)

(g) $995.5 - 747.1$: $996 - 747 = \mathbf{249}$ (Actual: $248.4$)

(h) $673.1 - 430.5$: $673 - 431 = \mathbf{242}$ (Actual: $242.6$)

(i) $53.25 - 25.62$: $53 - 26 = \mathbf{27}$ (Actual: $27.63$)

(j) $544.1 - 41.45$: $544 - 41 = \mathbf{503}$ (Actual: $502.65$)

(k) $3.5 - 2.1$: $4 - 2 = \mathbf{2}$ (Actual: $1.4$)

(l) $9.355 - 7.316$: $9 - 7 = \mathbf{2}$ (Actual: $2.039$)

1. $0.25 + 3 \times 2.05 \div 0.5$:
Division: $2.05 \div 0.5 = 4.1$
Multiplication: $3 \times 4.1 = 12.3$
Addition: $0.25 + 12.3 = \mathbf{12.55}$

2. $2.5 \div 0.5 + 4 \times 2.5$:
Division: $2.5 \div 0.5 = 5$
Multiplication: $4 \times 2.5 = 10$
Addition: $5 + 10 = \mathbf{15}$

3. $14 + 2 \div 4 - 0.5 \times 3$:
Division: $2 \div 4 = 0.5$
Multiplication: $0.5 \times 3 = 1.5$
Addition & Subtraction: $14 + 0.5 - 1.5 = 14.5 - 1.5 = \mathbf{13}$

4. $9 + 2.5 \div 5 \times 0.3 - 1$:
Division: $2.5 \div 5 = 0.5$
Multiplication: $0.5 \times 0.3 = 0.15$
Addition & Subtraction: $9 + 0.15 - 1 = 9.15 - 1 = \mathbf{8.15}$

5. $0.8 \times 1.6 \div 0.04 + 2.95$:
Division: $1.6 \div 0.04 = 40$
Multiplication: $0.8 \times 40 = 32$
Addition: $32 + 2.95 = \mathbf{34.95}$

6. $13 \div 5.2 + 0.024 \times 8 + 0.3$:
Division: $13 \div 5.2 = 2.5$
Multiplication: $0.024 \times 8 = 0.192$
Addition: $2.5 + 0.192 + 0.3 = \mathbf{2.992}$

7. $\left(\frac{1}{5} + 1\right) \div \frac{2}{5} + 1.5 \times 3 - 4.2$:
Brackets: $\frac{1}{5} + 1 = 1.2$ (or $\frac{6}{5}$)
Division: $1.2 \div 0.4 = 3$
Multiplication: $1.5 \times 3 = 4.5$
Addition & Subtraction: $3 + 4.5 - 4.2 = 7.5 - 4.2 = \mathbf{3.3}$

8. $25 \times \frac{3}{5} + 9.5 - 5 \times 3 + (4.2 - 3.4)$:
Brackets: $4.2 - 3.4 = 0.8$
Multiplications: $25 \times \frac{3}{5} = 15$, and $5 \times 3 = 15$
Combine: $15 + 9.5 - 15 + 0.8 = 9.5 + 0.8 = \mathbf{10.3}$

(a) $\frac{4}{100}$: $\mathbf{4\%}$

(b) $\frac{35}{100}$: $\mathbf{35\%}$

(c) $\frac{76}{100}$: $\mathbf{76\%}$

(d) $\frac{12}{100}$: $\mathbf{12\%}$

(e) $\frac{28}{100}$: $\mathbf{28\%}$

(f) $\frac{47}{100}$: $\mathbf{47\%}$

(g) $\frac{45}{100}$: $\mathbf{45\%}$

(h) $\frac{66}{100}$: $\mathbf{66\%}$

(i) $\frac{89}{100}$: $\mathbf{89\%}$

(j) $\frac{1}{100}$: $\mathbf{1\%}$

(a) $13\%$: $\mathbf{\frac{13}{100}}$

(b) $24\%$: $\frac{24}{100} = \mathbf{\frac{6}{25}}$

(c) $46\%$: $\frac{46}{100} = \mathbf{\frac{23}{50}}$

(d) $55\%$: $\frac{55}{100} = \mathbf{\frac{11}{20}}$

(e) $68\%$: $\frac{68}{100} = \mathbf{\frac{17}{25}}$

(f) $72\%$: $\frac{72}{100} = \mathbf{\frac{18}{25}}$

(g) $87\%$: $\mathbf{\frac{87}{100}}$

(h) $98\%$: $\frac{98}{100} = \mathbf{\frac{49}{50}}$

(i) $11\%$: $\mathbf{\frac{11}{100}}$

(j) $10\%$: $\frac{10}{100} = \mathbf{\frac{1}{10}}$

(a) $15\%$: $\mathbf{0.15}$

(b) $26\%$: $\mathbf{0.26}$

(c) $47\%$: $\mathbf{0.47}$

(d) $52\%$: $\mathbf{0.52}$

(e) $63\%$: $\mathbf{0.63}$

(f) $74\%$: $\mathbf{0.74}$

(g) $85\%$: $\mathbf{0.85}$

(h) $96\%$: $\mathbf{0.96}$

(i) $17\%$: $\mathbf{0.17}$

(j) $18\%$: $\mathbf{0.18}$

(a) $\frac{1}{10}$: $\frac{1}{10} \times 100\% = \mathbf{10\%}$

(b) $\frac{2}{5}$: $\frac{2}{5} \times 100\% = \mathbf{40\%}$

(c) $\frac{4}{10}$: $\frac{4}{10} \times 100\% = \mathbf{40\%}$

(d) $\frac{9}{25}$: $\frac{9}{25} \times 100\% = \mathbf{36\%}$

(e) $\frac{3}{20}$: $\frac{3}{20} \times 100\% = \mathbf{15\%}$

(f) $\frac{21}{50}$: $\frac{21}{50} \times 100\% = \mathbf{42\%}$

(g) $\frac{1}{5}$: $\frac{1}{5} \times 100\% = \mathbf{20\%}$

(h) $\frac{7}{20}$: $\frac{7}{20} \times 100\% = \mathbf{35\%}$

(i) $\frac{9}{10}$: $\frac{9}{10} \times 100\% = \mathbf{90\%}$

(j) $\frac{6}{25}$: $\frac{6}{25} \times 100\% = \mathbf{24\%}$

(a) $0.06$: $0.06 \times 100\% = \mathbf{6\%}$

(b) $0.14$: $0.14 \times 100\% = \mathbf{14\%}$

(c) $0.23$: $0.23 \times 100\% = \mathbf{23\%}$

(d) $0.34$: $0.34 \times 100\% = \mathbf{34\%}$

(e) $0.43$: $0.43 \times 100\% = \mathbf{43\%}$

(f) $0.55$: $0.55 \times 100\% = \mathbf{55\%}$

(g) $0.63$: $0.63 \times 100\% = \mathbf{63\%}$

(h) $0.71$: $0.71 \times 100\% = \mathbf{71\%}$

(i) $0.98$: $0.98 \times 100\% = \mathbf{98\%}$

(j) $0.3$: $0.30 \times 100\% = \mathbf{30\%}$

(a) The number of absent students:
$$\text{Absent} = \frac{5}{100} \times 40 = \frac{200}{100} = \mathbf{2\text{ students}}$$

(b) The number of present students:
$$\text{Present} = 40 - 2 = \mathbf{38\text{ students}}$$

• Fraction form: $\frac{12}{100} = \mathbf{\frac{3}{25}}$

• Decimal form: $\mathbf{0.12}$

• Decimal form: $0.60 = \mathbf{0.6}$

• Fraction form: $\frac{60}{100} = \mathbf{\frac{3}{5}}$

• (Bonus volume calculation: $\frac{3}{5} \times 85 = 51\text{ litres empty}$)

• Percentage form: $\frac{1}{5} \times 100\% = \mathbf{20\%}$

• Decimal form: $1 \div 5 = \mathbf{0.2}$

• Fraction form: $\frac{160}{200} = \mathbf{\frac{4}{5}}$

• Decimal form: $160 \div 200 = \mathbf{0.8}$

• Percentage form: $\frac{4}{5} \times 100\% = \mathbf{80\%}$

(a) Putting $\underline{\hspace{30pt}}$ at the right of a decimal does not affect its value:
Correct Answer: ii) $0$

(b) When multiplying a decimal by $100$, we move the decimal point $2$ places to the $\underline{\hspace{30pt}}$:
Correct Answer: iv) right

(c) We represent the percentage by the symbol $\underline{\hspace{30pt}}$:
Correct Answer: iv) $\%$

(d) $20\%$ of $540$ is $\underline{\hspace{30pt}}$:
Calculation: $\frac{20}{100} \times 540 = 2 \times 54 = 108$
Correct Answer: ii) $108$

(e) The percentage is a special kind of fraction whose denominator is always $\underline{\hspace{30pt}}$:
Correct Answer: iii) $100$

(a) $0.5\ \underline{\hspace{20pt}}\ 0.8$: $\mathbf{0.5 < 0.8}$

(b) $1.8\ \underline{\hspace{20pt}}\ 1.4$: $\mathbf{1.8 > 1.4}$

(c) $45.67\ \underline{\hspace{20pt}}\ 45.77$: $\mathbf{45.67 < 45.77}$

(d) $7.78\ \underline{\hspace{20pt}}\ 7.70$: $\mathbf{7.78 > 7.70}$

(e) $1.56\ \underline{\hspace{20pt}}\ 1.56$: $\mathbf{1.56 = 1.56}$

(f) $34.23\ \underline{\hspace{20pt}}\ 62.42$: $\mathbf{34.23 < 62.42}$

(a) $5.242 + 9.003$: $\mathbf{14.245}$

(b) $3.622 + 22.971$: $\mathbf{26.593}$

(c) $4.32 + 90.16$: $\mathbf{94.48}$

(d) $13.12 + 86.57$: $\mathbf{99.69}$

(e) $58.57 + 6.118$: $58.570 + 6.118 = \mathbf{64.688}$

(f) $10.561 + 27.16$: $10.561 + 27.160 = \mathbf{37.721}$

(g) $92.93 - 31.33$: $\mathbf{61.60}$

(h) $8.25 - 4.97$: $\mathbf{3.28}$

(i) $4.63 - 1.21$: $\mathbf{3.42}$

(j) $22.92 - 2.001$: $22.920 - 2.001 = \mathbf{20.919}$

(k) $6.119 - 1.55$: $6.119 - 1.550 = \mathbf{4.569}$

(l) $20.36 - 6.211$: $20.360 - 6.211 = \mathbf{14.149}$

(a) How much tape was there in both of the rolls altogether?
$$\text{Total Tape} = 16.38 + 56.82 = \mathbf{73.20\text{ m}\text{ (or }73.2\text{ m)}}$$

(b) Which roll has more tape and how much?
$$\text{The second roll has more tape by } 56.82 - 16.38 = \mathbf{40.44\text{ metres}}$$

(a) How many pieces of cloth will be obtained?
$$\text{Number of pieces} = 43.5 \div 1.5 = \frac{435}{15} = \mathbf{29\text{ pieces}}$$

(b) What will be the total length of $12$ pieces of cloth of length $1.5\text{ metres}$?
$$\text{Total length} = 12 \times 1.5 = \mathbf{18.0\text{ metres}\text{ (or }18\text{ m)}}$$

(a) $32.855 \times 10$: $\mathbf{328.55}$

(b) $4.39 \times 100$: $\mathbf{439}$

(c) $5.98 \times 1000$: $\mathbf{5980}$

(d) $6.54 \times 21$: $654 \times 21 = 13734 \implies \mathbf{137.34}$

(e) $4.14 \times 43$: $414 \times 43 = 17802 \implies \mathbf{178.02}$

(f) $7.17 \times 6.5$: $717 \times 65 = 46605 \implies \mathbf{46.605}$

(g) $69.7 \times 2.31$: $697 \times 231 = 161007 \implies \mathbf{161.007}$

(h) $2.19 \times 4.87$: $219 \times 487 = 106653 \implies \mathbf{10.6653}$

(i) $4.13 \times 6.12$: $413 \times 612 = 252756 \implies \mathbf{25.2756}$

(j) $82.6 \div 100$: $\mathbf{0.826}$

(k) $3.12 \div 1.3$: $\frac{31.2}{13} = \mathbf{2.4}$

(l) $4.21 \div 2.5$: $\frac{42.1}{25} = \mathbf{1.684}$

(a) $63.11 + 2.809$: $63 + 3 = \mathbf{66}$ (Actual: $65.919$)

(b) $74.1 + 3.9$: $74 + 4 = \mathbf{78}$ (Actual: $78.0$)

(c) $521.2 + 479.8$: $521 + 480 = \mathbf{1001}$ (Actual: $1001.0$)

(d) $74.92 - 36.02$: $75 - 36 = \mathbf{39}$ (Actual: $38.90$)

(e) $324.6 - 241.6$: $325 - 242 = \mathbf{83}$ (Actual: $83.0$)

(f) $888.8 - 479.2$: $889 - 479 = \mathbf{410}$ (Actual: $409.6$)

• Fraction form: $\frac{432}{600} = \frac{72}{100} = \mathbf{\frac{18}{25}}$

• Decimal form: $432 \div 600 = \mathbf{0.72}$

(a) What is the number of students who wear glasses?
$$\text{Students wearing glasses} = \frac{25}{100} \times 32 = \frac{1}{4} \times 32 = \mathbf{8\text{ students}}$$

(b) How many students do not wear glasses?
$$\text{Students not wearing glasses} = 32 - 8 = \mathbf{24\text{ students}}$$

(a) How many questions are based on "Numbers and Operations"?
$$\text{Questions} = \frac{60}{100} \times 20 = \frac{6}{10} \times 20 = \mathbf{12\text{ questions}}$$

(b) Write the number of remaining questions in fraction and decimal form:
Remaining questions $= 20 - 12 = 8\text{ questions}$ (which is $40\%$).
• Fraction form: $\frac{8}{20} = \mathbf{\frac{2}{5}}$
• Decimal form: $\frac{8}{20} = \mathbf{0.4}$