Question 1: Which of the following statements are true?

$\displaystyle \text{ i) } 27:36=4.5:6$      $\displaystyle \text{ ii) } \frac{3}{4} : \frac{15}{16}= \frac{2}{3} : \frac{5}{6}$

$\displaystyle \text{ iii) } \text{ Rs. 14 : Rs 21=2 pens :3 pens}$      $\displaystyle \text{ iv) 6.5 km :2.6 km=Rs. 60 :Rs 24}$

$\displaystyle \text{ i) } 27:36=4.5:6$

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

Hence True.

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

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

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

Hence True

$\displaystyle \text{ iii) } Rs. 14 : \ Rs 21=2 \ pens :3 \ pens$

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

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

Hence True

$\displaystyle \text{ iv) } 6.5 \ km :2.6 \ km=Rs. \ 60 :Rs \ 24$

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

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

Hence True.

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

$\displaystyle \text{ i) } 8, 12, 18, 24$      $\displaystyle \text{ ii) } 6.4, 3.6, 4.8, 2. 7$      $\displaystyle \text{ iii) } 11 \frac{1}{3} , 9 \frac{1}{3} , 8 \frac{1}{2} , 7$

$\displaystyle \text{ iv) } 0.36, 1.8, 6.4, 32$      $\displaystyle \text{ v) } \frac{3}{4} , \frac{5}{6} , \frac{7}{8} , \frac{9}{10}$

$\displaystyle \text{ i) } 8, 12, 18, 24$

We have:

$\displaystyle 8 : 12 = 2 : 3$

$\displaystyle 18 : 24 = 3: 4$

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

$\displaystyle \text{ Hence } 8, 12, 18, 24 \text{ are not in proportion }$

$\displaystyle \text{ ii) } 6.4, 3.6, 4.8, 2. 7$

We have

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

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

Therefore $\displaystyle 6.4 :3.6=4.8 :27$

$\displaystyle \text{ Hence } 6.4, 3.6, 4.8, 2.7 \text{ are in proportion }$.

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

We have

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

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

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

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

$\displaystyle \text{ iv) } 0.36, 1.8, 6.4, 32$

We have:

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

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

Therefore $\displaystyle 0.36 : 1.8 = 6.4 : 32$

$\displaystyle \text{ Hence } 0.36, 1.8, 6.4, 32 \text{ are in proportion }$

$\displaystyle \text{ v) } \frac{3}{4} , \frac{5}{6} , \frac{7}{8} , \frac{9}{10}$

We have:

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

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

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

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

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

$\displaystyle \text{ i) } 8 :x :: 6:27$      $\displaystyle \text{ ii) } 5.6 : 3.5 : :x : 1.25$      $\displaystyle \text{ iii) } 1 \frac{4}{5} : 2 \frac{4}{5} :: x :3 \frac{1}{2}$      $\displaystyle \text{ iv) } \frac{2}{3} : \frac{4}{7} ::1 \frac{5}{6} :x$

$\displaystyle \text{ i) } 8 :x :: 6:27$

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

$\displaystyle \text{ ii) } 5.6 : 3.5 : :x : 1.25$

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

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

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

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

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

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

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Q.4. Find the fourth proportional to:

$\displaystyle \text{ i) } 2.8, 14 \text{ and }3.5$      $\displaystyle \text{ ii) } 3 \frac{1}{3} , 1 \frac{2}{3} , 2 \frac{1}{2}$      $\displaystyle \text{ iii) } 1 \frac{5}{7} , 2 \frac{3}{14} , 3 \frac{3}{5}$      $\displaystyle \text{ iv) } 1 \frac{1}{5} , 1 \frac{3}{5} , 2.1$

$\displaystyle \text{ i) } 2.8, 14 \text{ and }3.5$

Let the fourth proportional term be $\displaystyle x$

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

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

Let the fourth proportional term be $\displaystyle x$

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

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

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

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

Let the fourth proportional term be $\displaystyle x$

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

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

$\displaystyle \text{ iv) } 1 \frac{1}{5} , 1 \frac{3}{5} , 2.1$

Let the fourth proportional term be $\displaystyle x$

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

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

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Q.5. Find the third proportional to:

$\displaystyle \text{ i) } 12, 16$      $\displaystyle \text{ ii) } 4.5, 6$      $\displaystyle \text{ iii) } 5 \frac{1}{2} , 16 \frac{1}{2}$      $\displaystyle \text{ iv) } 3 \frac{1}{2} ,8 \frac{3}{4}$

$\displaystyle \text{ i) } 12, 16$

$\displaystyle \text{ Let the third proportional to } 12 \text{ and }16 \text{ be }x$

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

$\displaystyle \text{ ii) } 4.5, 6$

$\displaystyle \text{ Let the third proportional to } 4.5 \text{ and }6 \text{ be } x$

$\displaystyle \text{ Then } 4.5 :6 ::6 :x \text{ or }x=8$

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

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

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

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

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

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

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Q.6. Find the mean proportional between:

$\displaystyle \text{ i) } 8 \text{ and }18$      $\displaystyle \text{ ii) } 0.3 \text{ and }2. 7$      $\displaystyle \text{ iii) } 66 \frac{2}{3} \text{ and }6$      $\displaystyle \text{ iv) } 1.25 \text{ and } 0.45$      $\displaystyle \text{ v) } \frac{1}{7} \text{ and } \frac{4}{63}$

$\displaystyle \text{ i) } 8 \text{ and }18$

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

$\displaystyle \text{ ii) } 0.3 \text{ and }2. 7$

$\displaystyle \text{ Mean proportional between } 0.3 \text{ and }2.7 = \frac{\sqrt{3\times 27}}{10} \frac{9}{10} =0.9$

$\displaystyle \text{ iii) } 66 \frac{2}{3} \text{ and }6$

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

$\displaystyle \text{ iv) } 1.25 \text{ and }0.45$

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

$\displaystyle \text{ v) } \frac{1}{7} \ and \ \frac{4}{63}$

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

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

$\displaystyle 7:x ::x :28 \text{ or }x^2=7 \times 28 \text{ or }x=14$

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

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

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

$\displaystyle \text{ We have } 7 :5 ::x :45.5 \text{ or }x=63.7$

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

$\displaystyle \text{ We have } 5 :9 ::x :2331 \text{ or }x=1295$

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

Let the number to be subtracted $\displaystyle = x$

Therefore

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

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

solving for $\displaystyle x \text{ we get } x=2$

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

Let the quantity of zinc be $\displaystyle x$
$\displaystyle \text{ We have } 9 :4=2 \frac{2}{5} : x \text{ or } x = \frac{12\times 4}{5\times 9} = \frac{16}{15} \text{ or } \frac{1 1}{15}$