\displaystyle \textbf{Question 1: }\text{Which of the following statements are true?}
\displaystyle \text{(i) }27:36=4.5:6
\displaystyle \text{(ii) }\frac{3}{4}:\frac{15}{16} =\frac{2}{3}:\frac{5}{6}
\displaystyle \text{(iii) }\text{Rs. }14:\text{Rs. }21 =2\text{ pens}:3\text{ pens}
\displaystyle \text{(iv) }6.5\text{ km}:2.6\text{ km} =\text{Rs. }60:\text{Rs. }24
\displaystyle \text{Answer:}
\displaystyle \text{(i) }27:36=\frac{27}{36}=\frac{3}{4}.
\displaystyle 4.5:6=\frac{4.5}{6}=\frac{3}{4}.
\displaystyle \therefore 27:36=4.5:6.
\displaystyle \text{Hence, the statement is true}.

\displaystyle \text{(ii) }\frac{3}{4}:\frac{15}{16} =\frac{3}{4}\times\frac{16}{15}=\frac{4}{5}.
\displaystyle \frac{2}{3}:\frac{5}{6} =\frac{2}{3}\times\frac{6}{5}=\frac{4}{5}.
\displaystyle \therefore \frac{3}{4}:\frac{15}{16} =\frac{2}{3}:\frac{5}{6}.
\displaystyle \text{Hence, the statement is true}.

\displaystyle \text{(iii) }\text{Rs. }14:\text{Rs. }21 =\frac{14}{21}=\frac{2}{3}.
\displaystyle 2\text{ pens}:3\text{ pens}=\frac{2}{3}.
\displaystyle \therefore \text{Rs. }14:\text{Rs. }21 =2\text{ pens}:3\text{ pens}.
\displaystyle \text{Hence, the statement is true}.

\displaystyle \text{(iv) }6.5\text{ km}:2.6\text{ km} =\frac{6.5}{2.6}=\frac{5}{2}.
\displaystyle \text{Rs. }60:\text{Rs. }24 =\frac{60}{24}=\frac{5}{2}.
\displaystyle \therefore 6.5\text{ km}:2.6\text{ km} =\text{Rs. }60:\text{Rs. }24.
\displaystyle \text{Hence, the statement is true}.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Check whether the following numbers are in proportion or not:}
\displaystyle \text{(i) }8,\ 12,\ 18,\ 24
\displaystyle \text{(ii) }6.4,\ 3.6,\ 4.8,\ 2.7
\displaystyle \text{(iii) }11\frac{1}{3},\ 9\frac{1}{3},\ 8\frac{1}{2},\ 7
\displaystyle \text{(iv) }0.36,\ 1.8,\ 6.4,\ 32
\displaystyle \text{(v) }\frac{3}{4},\ \frac{5}{6},\ \frac{7}{8},\ \frac{9}{10}
\displaystyle \text{Answer:}
\displaystyle \text{(i) }8:12=\frac{8}{12}=\frac{2}{3}.
\displaystyle 18:24=\frac{18}{24}=\frac{3}{4}.
\displaystyle \text{Since }\frac{2}{3}\ne\frac{3}{4},\quad 8:12\ne18:24.
\displaystyle \therefore 8,\ 12,\ 18,\ 24\text{ are not in proportion}.

\displaystyle \text{(ii) }6.4:3.6=\frac{6.4}{3.6} =\frac{64}{36}=\frac{16}{9}.
\displaystyle 4.8:2.7=\frac{4.8}{2.7} =\frac{48}{27}=\frac{16}{9}.
\displaystyle \therefore 6.4:3.6=4.8:2.7.
\displaystyle \therefore 6.4,\ 3.6,\ 4.8,\ 2.7\text{ are in proportion}.

\displaystyle \text{(iii) }11\frac{1}{3}:9\frac{1}{3} =\frac{34}{3}:\frac{28}{3}=\frac{34}{28}=\frac{17}{14}.
\displaystyle 8\frac{1}{2}:7=\frac{17}{2}:7 =\frac{17}{2}\times\frac{1}{7}=\frac{17}{14}.
\displaystyle \therefore 11\frac{1}{3}:9\frac{1}{3} =8\frac{1}{2}:7.
\displaystyle \therefore 11\frac{1}{3},\ 9\frac{1}{3},\ 8\frac{1}{2},\ 7\text{ are in proportion}.

\displaystyle \text{(iv) }0.36:1.8=\frac{0.36}{1.8} =\frac{36}{180}=\frac{1}{5}.
\displaystyle 6.4:32=\frac{6.4}{32} =\frac{64}{320}=\frac{1}{5}.
\displaystyle \therefore 0.36:1.8=6.4:32.
\displaystyle \therefore 0.36,\ 1.8,\ 6.4,\ 32\text{ are in proportion}.

\displaystyle \text{(v) }\frac{3}{4}:\frac{5}{6} =\frac{3}{4}\times\frac{6}{5}=\frac{9}{10}.
\displaystyle \frac{7}{8}:\frac{9}{10} =\frac{7}{8}\times\frac{10}{9}=\frac{35}{36}.
\displaystyle \text{Since }\frac{9}{10}\ne\frac{35}{36},\quad \frac{3}{4}:\frac{5}{6}\ne\frac{7}{8}:\frac{9}{10}.
\displaystyle \therefore \frac{3}{4},\ \frac{5}{6},\ \frac{7}{8},\ \frac{9}{10}\text{ are not in proportion}.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Find the value of }x\text{ in each of the following:}
\displaystyle \text{(i) }8:x::6:27
\displaystyle \text{(ii) }5.6:3.5::x:1.25
\displaystyle \text{(iii) }1\frac{4}{5}:2\frac{4}{5}::x:3\frac{1}{2}
\displaystyle \text{(iv) }\frac{2}{3}:\frac{4}{7}::1\frac{5}{6}:x
\displaystyle \text{Answer:}
\displaystyle \text{(i) }\frac{8}{x}=\frac{6}{27}.
\displaystyle 6x=8\times27.
\displaystyle \therefore x=\frac{8\times27}{6}=36.

\displaystyle \text{(ii) }\frac{5.6}{3.5}=\frac{x}{1.25}.
\displaystyle x=\frac{5.6\times1.25}{3.5}=2.
\displaystyle \therefore x=2.

\displaystyle \text{(iii) }1\frac{4}{5}:2\frac{4}{5} =x:3\frac{1}{2}.
\displaystyle \frac{9}{5}:\frac{14}{5}=x:\frac{7}{2}.
\displaystyle \frac{9}{14}=\frac{2x}{7}.
\displaystyle 18x=63.
\displaystyle \therefore x=\frac{63}{18}=\frac{7}{2}.

\displaystyle \text{(iv) }\frac{2}{3}:\frac{4}{7} =1\frac{5}{6}:x.
\displaystyle \frac{\frac{2}{3}}{\frac{4}{7}} =\frac{\frac{11}{6}}{x}.
\displaystyle \frac{7}{6}=\frac{11}{6x}.
\displaystyle 7x=11.
\displaystyle \therefore x=\frac{11}{7}.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Find the fourth proportional to:}
\displaystyle \text{(i) }2.8,\ 14\text{ and }3.5
\displaystyle \text{(ii) }3\frac{1}{3},\ 1\frac{2}{3} \text{ and }2\frac{1}{2}
\displaystyle \text{(iii) }1\frac{5}{7},\ 2\frac{3}{14} \text{ and }3\frac{3}{5}
\displaystyle \text{(iv) }1\frac{1}{5},\ 1\frac{3}{5} \text{ and }2.1
\displaystyle \text{Answer:}
\displaystyle \text{(i) Let the fourth proportional be }x.
\displaystyle 2.8:14=3.5:x.
\displaystyle \frac{2.8}{14}=\frac{3.5}{x}.
\displaystyle x=\frac{14\times3.5}{2.8}=17.5=\frac{35}{2}.
\displaystyle \therefore \text{the fourth proportional is }\frac{35}{2}.

\displaystyle \text{(ii) Let the fourth proportional be }x.
\displaystyle 3\frac{1}{3}:1\frac{2}{3}=2\frac{1}{2}:x.
\displaystyle \frac{10}{3}:\frac{5}{3}=\frac{5}{2}:x.
\displaystyle 2=\frac{5}{2x}.
\displaystyle 4x=5.
\displaystyle \therefore x=\frac{5}{4}.

\displaystyle \text{(iii) Let the fourth proportional be }x.
\displaystyle 1\frac{5}{7}:2\frac{3}{14}=3\frac{3}{5}:x.
\displaystyle \frac{12}{7}:\frac{31}{14}=\frac{18}{5}:x.
\displaystyle \frac{24}{31}=\frac{18}{5x}.
\displaystyle 120x=558.
\displaystyle \therefore x=\frac{558}{120}=\frac{93}{20}.

\displaystyle \text{(iv) Let the fourth proportional be }x.
\displaystyle 1\frac{1}{5}:1\frac{3}{5}=2.1:x.
\displaystyle \frac{6}{5}:\frac{8}{5}=\frac{21}{10}:x.
\displaystyle \frac{3}{4}=\frac{21}{10x}.
\displaystyle 30x=84.
\displaystyle \therefore x=\frac{84}{30}=2.8.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Find the third proportional to:}
\displaystyle \text{(i) }12,\ 16
\displaystyle \text{(ii) }4.5,\ 6
\displaystyle \text{(iii) }5\frac{1}{2},\ 16\frac{1}{2}
\displaystyle \text{(iv) }3\frac{1}{2},\ 8\frac{3}{4}
\displaystyle \text{Answer:}
\displaystyle \text{(i) Let the third proportional to }12\text{ and }16\text{ be }x.
\displaystyle 12:16=16:x.
\displaystyle \frac{12}{16}=\frac{16}{x}.
\displaystyle 12x=16\times16.
\displaystyle \therefore x=\frac{256}{12}=\frac{64}{3}.

\displaystyle \text{(ii) Let the third proportional to }4.5\text{ and }6\text{ be }x.
\displaystyle 4.5:6=6:x.
\displaystyle \frac{4.5}{6}=\frac{6}{x}.
\displaystyle 4.5x=36.
\displaystyle \therefore x=8.

\displaystyle \text{(iii) Let the third proportional to }5\frac{1}{2} \text{ and }16\frac{1}{2}\text{ be }x.
\displaystyle 5\frac{1}{2}:16\frac{1}{2}=16\frac{1}{2}:x.
\displaystyle \frac{11}{2}:\frac{33}{2}=\frac{33}{2}:x.
\displaystyle \frac{1}{3}=\frac{33}{2x}.
\displaystyle 2x=99.
\displaystyle \therefore x=\frac{99}{2}.

\displaystyle \text{(iv) Let the third proportional to }3\frac{1}{2} \text{ and }8\frac{3}{4}\text{ be }x.
\displaystyle 3\frac{1}{2}:8\frac{3}{4}=8\frac{3}{4}:x.
\displaystyle \frac{7}{2}:\frac{35}{4}=\frac{35}{4}:x.
\displaystyle \frac{2}{5}=\frac{35}{4x}.
\displaystyle 8x=175.
\displaystyle \therefore x=\frac{175}{8}=21\frac{7}{8}.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Find the mean proportional between:}
\displaystyle \text{(i) }8\text{ and }18
\displaystyle \text{(ii) }0.3\text{ and }2.7
\displaystyle \text{(iii) }66\frac{2}{3}\text{ and }6
\displaystyle \text{(iv) }1.25\text{ and }0.45
\displaystyle \text{(v) }\frac{1}{7}\text{ and }\frac{4}{63}
\displaystyle \text{Answer:}
\displaystyle \text{(i) Mean proportional}=\sqrt{8\times18} =\sqrt{144}=12.

\displaystyle \text{(ii) Mean proportional}=\sqrt{0.3\times2.7} =\sqrt{0.81}=0.9.

\displaystyle \text{(iii) Mean proportional} =\sqrt{66\frac{2}{3}\times6}
\displaystyle =\sqrt{\frac{200}{3}\times6} =\sqrt{400}=20.

\displaystyle \text{(iv) Mean proportional} =\sqrt{1.25\times0.45}
\displaystyle =\sqrt{0.5625}=0.75.

\displaystyle \text{(v) Mean proportional} =\sqrt{\frac{1}{7}\times\frac{4}{63}}
\displaystyle =\sqrt{\frac{4}{441}} =\frac{2}{21}.
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{If }28\text{ is the third proportional to }7\text{ and }x, \text{ find the value of }x.
\displaystyle \text{Answer:}
\displaystyle 7:x=x:28.
\displaystyle x^2=7\times28=196.
\displaystyle \therefore x=\sqrt{196}=14.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{If }18,\ x,\ 50\text{ are in continued proportion, find the value of }x.
\displaystyle \text{Answer:}
\displaystyle 18:x=x:50.
\displaystyle x^2=18\times50=900.
\displaystyle \therefore x=\sqrt{900}=30.
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{A rod was cut into two pieces in the ratio }7:5. \text{ If the length of the}
\displaystyle \text{smaller piece was }45.5\text{ cm, find the length of the longer piece.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the length of the longer piece be }x\text{ cm}.
\displaystyle 7:5=x:45.5.
\displaystyle \frac{7}{5}=\frac{x}{45.5}.
\displaystyle x=\frac{7\times45.5}{5}=63.7.
\displaystyle \therefore \text{the length of the longer piece is }63.7\text{ cm}.
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{The areas of two rectangular fields are in the ratio }5:9. \text{ Find the area of}
\displaystyle \text{the smaller field if that of the larger field is }2331\text{ square metres}.
\displaystyle \text{Answer:}
\displaystyle \text{Let the area of the smaller field be }x\text{ square metres}.
\displaystyle 5:9=x:2331.
\displaystyle \frac{5}{9}=\frac{x}{2331}.
\displaystyle x=\frac{5\times2331}{9}=1295.
\displaystyle \therefore \text{the area of the smaller field is }1295\text{ square metres}.
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{What number must be subtracted from each of the numbers}
\displaystyle 41,\ 55,\ 36,\ 48\text{ so that the differences are proportional?}
\displaystyle \text{Answer:}
\displaystyle \text{Let the number to be subtracted be }x.
\displaystyle \frac{41-x}{55-x}=\frac{36-x}{48-x}.
\displaystyle (41-x)(48-x)=(36-x)(55-x).
\displaystyle 1968-89x+x^2=1980-91x+x^2.
\displaystyle 1968-89x=1980-91x.
\displaystyle 2x=12.
\displaystyle \therefore x=6.
\displaystyle \therefore \text{the required number is }6.
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{An alloy is to contain copper and zinc in the ratio }9:4.
\displaystyle \text{Find the quantity of zinc to be melted with }2\frac{2}{5}\text{ kg of copper}
\displaystyle \text{to get the desired alloy.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the quantity of zinc required be }x\text{ kg}.
\displaystyle 9:4=2\frac{2}{5}:x.
\displaystyle \frac{9}{4}=\frac{\frac{12}{5}}{x}.
\displaystyle 9x=4\times\frac{12}{5}.
\displaystyle x=\frac{48}{45}=\frac{16}{15}=1\frac{1}{15}.
\displaystyle \therefore 1\frac{1}{15}\text{ kg of zinc is required}.
\displaystyle \\

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