Question 1:  Which of the following statements are true?

i) $27:36=4.5:6$

$4.5:6 =$ $\frac{4.5\times 6}{6\times 6}=\frac{27}{36}$

Hence True.

$\\$

ii) $\frac{3}{4} : \frac{15}{16}= \frac{2}{3} : \frac{5}{6}$

$\frac{3}{4} : \frac{15}{16}= \frac{3}{4}\times \frac{16}{15}= \frac{4}{5}$

$\frac{2}{3} : \frac{5}{6}= \frac{2}{3}\times \frac{6}{5}= \frac{4}{5}$

Hence True

$\\$

iii) $Rs. 14 : \ Rs 21=2 \ pens :3 \ pens$

$\frac{Rs. \ 14}{Rs. \ 21}= \frac{2}{3}$

$\frac{2 \ pens}{3 \ pens}= \frac{2}{3}$

Hence True

$\\$

iv) $6.5 \ km :2.6 \ km=Rs. \ 60 :Rs \ 24$

$\frac{6.5 \ km}{2.6 \ km}= \frac{5}{2}$

$\frac{Rs. \ 60}{Rs. \ 24}= \frac{5}{2}$

Hence True

$\\$

Q.2. Check whether the following numbers are in proportion or not:

i) $8, 12, 18, 24$

We have:

$8 : 12 = 2 : 3$

$18 : 24 = 3: 4$

$Therefore \ 8 : 12 \neq 18 : 24$

Hence $8, 12, 18, 24$ are not in proportion

$\\$

ii) $6.4, 3.6, 4.8, 2. 7$

We have

$6.4:3.6=$ $\frac{64}{36}= \frac{32}{18}= \frac{16}{9}$

$4.8 :2.7=$ $\frac{48}{27}= \frac{16}{9}$

Therefore $6.4 :3.6=4.8 :27$

Hence $6.4, 3.6, 4.8, 2.7$ are in proportion.

$\\$

iii) $11 \frac{1}{3}, 9 \frac{1}{3}, 8 \frac{1}{2}, 7$

We have

$11 \frac{1}{3}:9 \frac{1}{3}= \frac{34}{3}:\frac{28}{3}= \frac{34}{28}= \frac{17}{14}$

$8 \frac{1}{2}: 7= \frac{17}{2} :7= \frac{17}{14}$

Therefore $11 \frac{1}{3}:9 \frac{1}{3}= 8 \frac{1}{2}: 7$

Hence $11 \frac{1}{3}, 9 \frac{1}{3}, 8 \frac{1}{2}, 7$ are in proportional

$\\$

iv) $0.36, 1.8, 6.4, 32$

We have:

$0.36 :1.8=$ $\frac{36}{180} = \frac{1}{5}$

$6.4 :32=$ $\frac{64}{320}= \frac{1}{5}$

Therefore  $0.36 : 1.8 = 6.4 : 32$

Hence $0.36, 1.8, 6.4, 32$ are in proportion

$\\$

vi) $\frac{3}{4}, \frac{5}{6}, \frac{7}{8}, \frac{9}{10}$

We have:

$\frac{3}{4} :\frac{5}{6}= \frac{3}{4}\times \frac{6}{5}= \frac{9}{10}$

$\frac{7}{8}:\frac{9}{10}= \frac{7}{8}\times \frac{10}{9}= \frac{70}{72}$

Therefore  $\frac{3}{4}:\frac{5}{6} \neq \frac{7}{8}:\frac{9}{10}$

Hence  $\frac{3}{4}, \frac{5}{6}, \frac{7}{8}, \frac{9}{10}$ are not in proportion

$\\$

Q.3. Find the value of x in each of the following:

i) $8 :x :: 6:27$

We have $\frac{8}{x}= \frac{6}{27}$ $\ or\ x=$ $\frac{ 8 \times 27}{6}$ $=36$

$\\$

ii) $5.6 : 3.5 : :x : 1.25$

We have $\frac{56}{35}= \frac{x}{1.25}$ $\ or\ x=$ $\frac{56\times 1.25}{35}$ $= 2$

$\\$

iii) $1 \frac{4}{5} : 2 \frac{4}{5} :: x :3 \frac{1}{2}$

We have $\frac{9}{5} : \frac{14}{5}= x : \frac{7}{2}$

$\frac{9}{14}= \frac{2x}{7}$ $\ or\ x=$ $\frac{9\times 7}{14\times 2}=\frac{9}{4}$

$\\$

iv) $\frac{2}{3} : \frac{4}{7} ::1 \frac{5}{6} :x$

We have $\frac{2}{3} \times \frac{7}{4}=\frac{11}{6x}$ $\ or\ x=$ $\frac{11}{7}$

$\\$

Q.4. Find the fourth proportional to:

i) $2.8, 14 \ and\ 3.5$

Let the fourth proportional term be $x$

We have   $\frac{2.8}{14}= \frac{3.5}{x} \ or\ x= \frac{35}{2}$

$\\$

ii) $3 \frac{1}{3}, 1 \frac{2}{3}, 2 \frac{1}{2}$

Let the fourth proportional term be $x$

$3 \frac{1}{3} :1 \frac{2}{3}= 2 \frac{1}{2} :x$

$\frac{10}{3} :\frac{5}{3}= \frac{5}{2} :x$

$\frac{10}{3} \times \frac{3}{5}= \frac{5}{2x} \ or\ x= \frac{5}{4}$

$\\$

iii) $1 \frac{5}{7}, 2 \frac{3}{14}, 3 \frac{3}{5}$

Let the fourth proportional term be $x$

$1 \frac{5}{7}:2 \frac{3}{14}= 3 \frac{3}{5} :x$

$\frac{12}{7}:\frac{31}{14}= \frac{18}{5} :x \ or\ x=\frac{93}{20}$

$\\$

$1 \frac{1}{5}, 1 \frac{3}{5}, 2.1$

Let the fourth proportional term be $x$

$1 \frac{1}{5} :1 \frac{3}{5} :: 2.1 :x$

$\frac{6}{5} :\frac{8}{5} :: 2.1 :\ or\ x=2.8$

$\\$

Q.5. Find the third proportional to:

i) $12, 16$

Let the third proportional to $12 \ and\ 16 \ be \ x$

Then, $12 :16 :: 16 :x \ or\ x= \frac{64}{3}$

$\\$

ii) $4.5, 6$

Let the third proportional to $4.5 \ and\ 6 \ be\ x$

Then $4.5 :6 ::6 :x \ or\ x=8$

$\\$

iii) $5 \frac{1}{2}, 16 \frac{1}{2}$

Let the third proportional to $5 \frac{1}{2}, 16 \frac{1}{2} \ be\ x$

Then $5 \frac{1}{2} : 16 \frac{1}{2} :: 16 \frac{1}{2} :x \ or\ x= \frac{99}{2}$

$\\$

iv) $3 \frac{1}{2},8 \frac{3}{4}$

Let the third proportional to $3 \frac{1}{2}, 8 \frac{3}{4} \ be\ x$

Then $3 \frac{1}{2} : 8 \frac{3}{4} :: 8 \frac{3}{4} :x \ or\ x= 21 \frac{7}{8}$

$\\$

Q.6. Find the mean proportional between:

$8 \ and\ 18$

Mean proportional between $8 \ and\ 18 = \sqrt{8\times 18=144}=12$

$\\$

$0.3 \ and\ 2. 7$

Mean proportional between $0.3 \ and\ 2.7 = \frac{\sqrt{3\times 27=81}}{10}=0.9$

$\\$

$66 \frac{2}{3} \ and\ 6$

Mean proportional between $66 \frac{2}{3} \ and\ 6 = \sqrt{\frac{200}{3}\times 6=400}=20$

$\\$

$1.25 \ and\ 0.45$

Mean proportional between $1.25 \ and \ 0.45 = \sqrt{1.25 \times 0.45}=0.75$

$\\$

$\frac{1}{7} \ and \ \frac{4}{63}$

Mean proportional between  $\frac{1}{7} \ and\ \frac{4}{63} = \frac{\sqrt{1\times 4}}{7\times 63}= \frac{2}{21}$

$\\$

Q.7. If $28$ is the third proportional to $7 \ and\ x$, find the value of

$7:x ::x :28 \ or\ x^2=7 \times 28 \ or\ x=14$

$\\$

Q.8. If $18, x, 50$ are in continued proportion, find the value of

$x= \sqrt{18\times 50}=30$

$\\$

Q.9. A rod was cut into two pieces in the ratio $7: 5$. If the length of the smaller piece was $45.5 \ cm$, then find the length of the longer piece.

We have $7 :5 ::x :45.5 \ or\ x=63.7$

$\\$

Q.10. The areas of two rectangular fields are in the ratio $5: 9$. Find the area of the smaller field if that of the larger field is $2331$ sq. meters.

We have $5 :9 ::x :2331 \ or\ x=1295$

$\\$

Q.11. What number must be subtracted from each of the numbers $41, 55, 36, 48$ so that the differences are proportional?

Let the number to be subtracted $= x$

Therefore

$\frac{41-x}{55-x}= \frac{36-x}{48-x}$

$41 \times 48-48 x-41 x+ x^2=36\times 55-55x-36x+ x^2$

solving for $x \ we \ get \ x=2$

$\\$

Q.12. An alloy is to contain copper and zinc in the ratio $9 : 4$. Find the quantity of zinc to be melted with $2 \frac{2}{5} \ kg$ of copper, to get the desired alloy.

Let the quantity of zinc be $x$

We have $9 :4=2 \frac{2}{5} :x \ or\ x= \frac{12\times 4}{5\times 9}= \frac{16}{15} \ or\ \frac{1 1}{15}$