Ratio & Proportion - Exercise 7(A)

Express each of the following ratios in simplest form :

(i) 20 : 35

(ii) 95 : 57

(iii) 2.1 : 1.2

(iv) $3\dfrac{1}{4} : 6\dfrac{1}{2}$

(v) 6a : 9b

(vi) 8ab² : 6a²b

Answer

(i) 20 : 35

The given ratio is 20 : 35.

Let us find H.C.F. of 20 and 35:

$\begin{array}{r} 1 \\ 20 \overline{) 35} \\ \underline{-20} \\ 15 \overline{) 20} ( 1 \\ \underline{-15} \\ 5 \overline{) 15} ( 3 \\ \underline{-15} \\ 0 \end{array}$

H.C.F. = 5

$∴ 20 : 35 = \dfrac{20 \div 5}{35 \div 5} = \dfrac{4}{7} = 4 : 7$

Hence, the answer is 4 : 7

(ii) 95 : 57

The given ratio is 95 : 57.

Let us find H.C.F. of 95 and 57:

$\begin{array}{r} 1 \\ 57 \overline{) 95} \\ \underline{-57} \\ 38 \overline{) 57} ( 1 \\ \underline{-38} \\ 19 \overline{) 38} ( 2 \\ \underline{-38} \\ 0 \end{array}$

H.C.F. = 19

$∴ 95 : 57 = \dfrac{95 \div 19}{57 \div 19} = \dfrac{5}{3} = 5 : 3$

Hence, the answer is 5 : 3

(iii) 2.1 : 1.2

First, multiply both by 10 to remove decimals:

(2.1 x 10 : 1.2 x 10) = 21 : 12

Let us find H.C.F. of 21 and 12:

$\begin{array}{r} 1 \\ 12 \overline{) 21} \\ \underline{-12} \\ 9 \overline{) 12} ( 1 \\ \underline{-9} \\ 3 \overline{) 9} ( 3 \\ \underline{-9} \\ 0 \end{array}$

H.C.F. = 3

$∴ 21 : 12 = \dfrac{21 \div 3}{12 \div 3} = \dfrac{7}{4} = 7 : 4$

Hence, the answer is 7 : 4

(iv) $3\dfrac{1}{4} : 6\dfrac{1}{2}$

Convert mixed fractions to improper fractions: $\dfrac{13}{4} : \dfrac{13}{2}$

Let us find L.C.M. of 4 and 2:

$\begin{array}{l|rr} 2 & 4, & 2 \\ \hline 2 & 2, & 1 \\ \hline & 1, & 1 \end{array}$

L.C.M. = 2 x 2 = 4

Multiply both terms by 4 to convert fractions to whole number:

$\Big(\dfrac{13}{4} \times 4\Big) : \Big(\dfrac{13}{2} \times 4\Big) = 13 : 26$

Let us find H.C.F of 13 and 26:

$\begin{array}{r} 2 \\ 13 \overline{) 26} \\ \underline{-26} \\ 0 \end{array}$

H.C.F = 13

$\dfrac{13 \div 13}{26 \div 13}$ = 1 : 2

Hence, the answer is 1 : 2

(v) 6a : 9b

Given ratio is 6a : 9b

Let us find H.C.F. of 6 and 9:

$\begin{array}{r} 1 \\ 6 \overline{) 9} \\ \underline{-6} \\ 3 \overline{) 6} ( 2 \\ \underline{-6} \\ 0 \end{array}$

H.C.F. = 3

$\dfrac{6a}{9b} = \dfrac{6a \div 3}{9b \div 3} = \dfrac{2a}{3b} = 2a : 3b$

Hence, the answer is 2a : 3b

(vi) 8ab² : 6a²b

Find the common factors in the coefficients and variables:

Let us find H.C.F. of 8 and 6:

$\begin{array}{r} 1 \\ 6 \overline{) 8} \\ \underline{-6} \\ 2 \overline{) 6} ( 3 \\ \underline{-6} \\ 0 \end{array}$

H.C.F. = 2

Common variables: ab.

Divide both terms by 2ab:

$\dfrac{8ab^2}{2ab} : \dfrac{6a^2b}{2ab} = 4b : 3a$

Hence, the answer is 4b : 3a

Express each of the following ratios in simplest form :

(i) $\dfrac{1}{8} : \dfrac{1}{12}$

(ii) $\dfrac{1}{9} : \dfrac{1}{6}$

(iii) $\dfrac{2}{3} : \dfrac{5}{6}$

(iv) $\dfrac{1}{4} : \dfrac{1}{6} : \dfrac{1}{8}$

(v) $1\dfrac{2}{3} : 2\dfrac{1}{2} : 1\dfrac{3}{4}$

(vi) $2\dfrac{1}{3} : 3\dfrac{1}{4} : 1\dfrac{1}{6}$

Answer

(i) $\dfrac{1}{8} : \dfrac{1}{12}$

Let us find L.C.M of 8 and 12:

$\begin{array}{l|l} 2 & 8, 12 \\ \hline 2 & 4, 6 \\ \hline 2 & 2, 3 \\ \hline 3 & 1, 3 \\ \hline & 1, 1 \end{array}$

L.C.M. = 2 x 2 x 2 x 3 = 24.

Multiplying both terms by 24 to convert fractions to whole number:

$\left(\dfrac{1}{8} \times 24\right) : \left(\dfrac{1}{12} \times 24\right) = 3 : 2$

Hence, the answer is 3 : 2

(ii) $\dfrac{1}{9} : \dfrac{1}{6}$

Let us find L.C.M of 9 and 6:

$\begin{array}{l|l} 3 & 9, 6 \\ \hline 3 & 3, 2 \\ \hline 2 & 1, 2 \\ \hline & 1, 1 \end{array}$

L.C.M. = 3 x 3 x 2 = 18.

Multiplying both terms by 18 to convert fractions to whole number:

$\left(\dfrac{1}{9} \times 18\right) : \left(\dfrac{1}{6} \times 18\right) = 2 : 3$

Hence, the answer is 2 : 3

(iii) $\dfrac{2}{3} : \dfrac{5}{6}$

Let us find L.C.M of 3 and 6:

$\begin{array}{l|l} 3 & 3, 6 \\ \hline 2 & 1, 2 \\ \hline & 1, 1 \end{array}$

L.C.M. = 3 x 2 = 6.

Multiplying both terms by 6 to convert fractions to whole number:

$\left(\dfrac{2}{3} \times 6\right) : \left(\dfrac{5}{6} \times 6\right) = 4 : 5$

Hence, the answer is 4 : 5

(iv) $\dfrac{1}{4} : \dfrac{1}{6} : \dfrac{1}{8}$

Let us find L.C.M of 4, 6 and 8:

$\begin{array}{l|l} 2 & 4, 6, 8 \\ \hline 2 & 2, 3, 4 \\ \hline 2 & 1, 3, 2 \\ \hline 3 & 1, 3, 1 \\ \hline & 1, 1, 1 \end{array}$

L.C.M. = 2 x 2 x 2 x 3 = 24.

Multiplying each term by 24 to convert fractions to whole number:

$\left(\dfrac{1}{4} \times 24\right) : \left(\dfrac{1}{6} \times 24\right) : \left(\dfrac{1}{8} \times 24\right) = 6 : 4 : 3$

Hence, the answer is 6 : 4 : 3

(v) $1\dfrac{2}{3} : 2\dfrac{1}{2} : 1\dfrac{3}{4}$

Convert to improper fractions: $\dfrac{5}{3} : \dfrac{5}{2} : \dfrac{7}{4}$.

Let us find L.C.M of 3, 2 and 4:

$\begin{array}{l|l} 2 & 3, 2, 4 \\ \hline 2 & 3, 1, 2 \\ \hline 3 & 3, 1, 1 \\ \hline & 1, 1, 1 \end{array}$

L.C.M. = 2 x 2 x 3 = 12.

Multiplying each term by 12 to convert fractions to whole number:

$\left(\dfrac{5}{3} \times 12\right) : \left(\dfrac{5}{2} \times 12\right) : \left(\dfrac{7}{4} \times 12\right) = 20 : 30 : 21$

Hence, the answer is 20 : 30 : 21

(vi) $2\dfrac{1}{3} : 3\dfrac{1}{4} : 1\dfrac{1}{6}$

Convert to improper fractions: $\dfrac{7}{3} : \dfrac{13}{4} : \dfrac{7}{6}$.

Let us find L.C.M of 3, 4 and 6:

$\begin{array}{l|l} 2 & 3, 4, 6 \\ \hline 2 & 3, 2, 3 \\ \hline 3 & 3, 1, 3 \\ \hline & 1, 1, 1 \end{array}$

L.C.M. = 2 x 2 x 3 = 12.

Multiplying each term by 12 to convert fractions to whole number:

$\left(\dfrac{7}{3} \times 12\right) : \left(\dfrac{13}{4} \times 12\right) : \left(\dfrac{7}{6} \times 12\right) = 28 : 39 : 14$

Hence, the answer is 28 : 39 : 14

Express each of the following ratios in simplest form :

(i) 35 paise : ₹ 1

(ii) 1 kg : 375 g

(iii) 60 cm : 1 m

(iv) 400 m : 1 km

(v) 8 months : $1\dfrac{2}{3}$ years

(vi) $6\dfrac{2}{3}$ hours : 1 day

Answer

(i) 35 paise : ₹ 1

Convert rupee into paise: ₹ 1 = 100 paise.

The ratio is 35 : 100.

Let us find H.C.F. of 35 and 100:

$\begin{array}{r} 2 \\ 35 \overline{) 100} \\ \underline{-70} \\ 30 \overline{) 35} ( 1 \\ \underline{-30} \\ 5 \overline{) 30} ( 6 \\ \underline{-30} \\ 0 \end{array}$

H.C.F. = 5

$\dfrac{35 \div 5}{100 \div 5} = \dfrac{7}{20} = 7 : 20$

Hence, the answer is 7 : 20

(ii) 1 kg : 375 g

Convert kg into g: 1 kg = 1000 g.

The ratio is 1000 : 375.

Let us find H.C.F. of 1000 and 375:

$\begin{array}{r} 2 \\ 375 \overline{) 1000} \\ \underline{-750} \\ 250 \overline{) 375} ( 1 \\ \underline{-250} \\ 125 \overline{) 250} ( 2 \\ \underline{-250} \\ 0 \end{array}$

H.C.F. = 125

$\dfrac{1000 \div 125}{375 \div 125} = \dfrac{8}{3} = 8 : 3$

Hence, the answer is 8 : 3

(iii) 60 cm : 1 m

Convert m into cm: 1 m = 100 cm.

The ratio is 60 : 100.

Let us find H.C.F. of 60 and 100:

$\begin{array}{r} 1 \\ 60 \overline{) 100} \\ \underline{-60} \\ 40 \overline{) 60} ( 1 \\ \underline{-40} \\ 20 \overline{) 40} ( 2 \\ \underline{-40} \\ 0 \end{array}$

H.C.F. = 20

$\dfrac{60 \div 20}{100 \div 20} = \dfrac{3}{5} = 3 : 5$

Hence, the answer is 3 : 5

(iv) 400 m : 1 km

Convert km into m: 1 km = 1000 m.

The ratio is 400 : 1000.

Let us find H.C.F. of 400 and 1000:

$\begin{array}{r} 2 \\ 400 \overline{) 1000} \\ \underline{-800} \\ 200 \overline{) 400} ( 2 \\ \underline{-400} \\ 0 \end{array}$

H.C.F. = 200

$\dfrac{400 \div 200}{1000 \div 200} = \dfrac{2}{5} = 2 : 5$

Hence, the answer is 2 : 5

(v) 8 months : $1\dfrac{2}{3}$ years

Convert $1\dfrac{2}{3}$ years to improper fraction: $\dfrac{5}{3}$ years.

Convert years into months: $\dfrac{5}{3} \times 12 = 20$ months.

The ratio is 8 : 20.

Let us find H.C.F. of 8 and 20:

$\begin{array}{r} 2 \\ 8 \overline{) 20} \\ \underline{-16} \\ 4 \overline{) 8} ( 2 \\ \underline{-8} \\ 0 \end{array}$

H.C.F. = 4

$\dfrac{8 \div 4}{20 \div 4} = \dfrac{2}{5} = 2 : 5$

Hence, the answer is 2 : 5

(vi) $6\dfrac{2}{3}$ hours : 1 day

Convert $6\dfrac{2}{3}$ hours to improper fraction: $\dfrac{20}{3}$ hours.

Convert 1 day into hours: 24 hours.

The ratio is $\dfrac{20}{3}$ : 24.

Multiply both sides by 3 to remove the denominator:

$\Big(\dfrac{20}{3} \times 3\Big) : 24 \times 3 = 20 : 72$

Let us find H.C.F. of 20 and 72:

$\begin{array}{r} 3 \\ 20 \overline{) 72} \\ \underline{-60} \\ 12 \overline{) 20} ( 1 \\ \underline{-12} \\ 8 \overline{) 12} ( 1 \\ \underline{-8} \\ 4 \overline{) 8} ( 2 \\ \underline{-8} \\ 0 \end{array}$

H.C.F. = 4

$\dfrac{20 \div 4}{72 \div 4} = \dfrac{5}{18} = 5 : 18$

Hence, the answer is 5 : 18

The angles of a triangle are in the ratio 5 : 6 : 7. Find each angle.

Answer

Given:

Ratio = 5 : 6 : 7.

The sum of all angles in any triangle is always 180°.

Find total parts: 5 + 6 + 7 = 18 parts.

Find the value of 1 part: 180° ÷ 18 = 10°

Now, calculate each angle by multiplying each part with 10°:

First angle: 5 x 10° = 50°

Second angle: 6 x 10° = 60°

Third angle: 7 x 10° = 70°

Angles are 50°, 60°, and 70°.

A line segment of length 84 cm has been divided into two parts in the ratio 4 : 3. Find the length of each part.

Answer

Given:

Length = 84 cm

Ratio = 4 : 3

Find total parts: 4 + 3 = 7 parts.

Find the value of 1 part: 84 ÷ 7 = 12 cm.

Now, calculate each part by multiplying with 12 cm:

First part = 4 x 12 cm = 48 cm

Second part = 3 x 12 cm = 36 cm

Lengths are 48 cm and 36 cm.

Divide ₹ 3900 between Kunal and Kirti in the ratio 15 : 11.

Answer

Given:

Total money = ₹ 3900

Ratio = 15 : 11

Find total parts: 15 + 11 = 26 parts.

Find the value of 1 part: Divide ₹ 3900 by 26.

3900 ÷ 26 = 150. So, 1 part = ₹ 150.

Now calculate shares by multiplying each part with ₹ 150:

Kunal's share: 15 x ₹ 150 = ₹ 2250

Kirti's share: 11 x ₹ 150 = ₹ 1650

Kunal's share = ₹2250 and Kirti's share = ₹1650.

Divide ₹ 16250 among A, B, C in the ratio 5 : 7 : 13.

Answer

Given:

Total Money: ₹ 16250

Ratio: 5 : 7 : 13

Find total parts: 5 + 7 + 13 = 25 parts.

Find the value of 1 part: 16250 ÷ 25 = 650. So, 1 part = ₹ 650.

Now calculate shares by multiplying each part with ₹ 650:

A gets: 5 x ₹ 650 = ₹ 3250

B gets: 7 x ₹ 650 = ₹ 4550

C gets: 13 x ₹ 650 = ₹ 8450

A's share = ₹3250, B's share = ₹4550, C's share = ₹8450

Divide 35 sweets between Sudha and Tanvy in the ratio $\dfrac{1}{2} : \dfrac{1}{3}$.

Answer

Given:

Total Sweets: 35

Ratio = $\dfrac{1}{2} : \dfrac{1}{3}$

Let us find L.C.M. of 2 and 3.

$\begin{array}{l|rr} 2 & 2, & 3 \\ \hline 3 & 1, & 3 \\ \hline & 1, & 1 \end{array}$

L.C.M. = 2 x 3 = 6

Multiply both by 6 to convert fractions to whole numbers:

$\Big(\dfrac{1}{2} \times 6\Big) : \Big(\dfrac{1}{3} \times 6\Big) = 3 : 2$

Total parts: 3 + 2 = 5 parts.

Value of 1 part: 35 ÷ 5 = 7 sweets.

Calculate sweets by multiplying each part with 7 sweets:

Sudha gets: 3 x 7 sweets = 21 sweets

Tanvy gets: 2 x 7 sweets = 14 sweets

Sudha gets 21 sweets and Tanvy gets 14 sweets.

Divide 468 g of rice into three heaps containing the quantities in the ratio $\dfrac{1}{6} : \dfrac{1}{8} : \dfrac{1}{12}$.

Answer

Given:

Total Rice: 468 g

Ratio: $\dfrac{1}{6} : \dfrac{1}{8} : \dfrac{1}{12}$

Let us find L.C.M. of 6, 8, and 12:

$\begin{array}{l|rrr} 2 & 6, & 8, & 12 \\ \hline 2 & 3, & 4, & 6 \\ \hline 2 & 3, & 2, & 3 \\ \hline 3 & 3, & 1, & 3 \\ \hline & 1, & 1, & 1 \end{array}$

L.C.M. = 2 x 2 x 2 x 3 = 24

Multiply all three terms with 24 to convert fractions to whole numbers:

$\Big(\dfrac{1}{6} \times 24\Big) : \Big(\dfrac{1}{8} \times 24\Big) : \Big(\dfrac{1}{12} \times 24\Big)$

= 4 : 3 : 2

Total parts: 4 + 3 + 2 = 9 parts.

Value of 1 part: 468 g ÷ 9 = 52 g.

Calculate heaps by multiplying each part with 52 g:

First heap: 4 x 52 g = 208 g

Second heap: 3 x 52 g = 156 g

Third heap: 2 x 52 g = 104 g

The three heaps are 208 g, 156 g and 104 g.

Divide ₹ 715 among A, B, C in such a way that B gets three times as much as A gets and C gets half of as much as B gets.

Answer

Given:

Total Money: ₹ 715

B gets 3 times A

C gets half of B.

Let A's share = 1 part.

Then B's share = 3 times A = 3 x 1 = 3 parts.

Then C's share = Half of B = 3 ÷ 2 = $\dfrac{3}{2}$ parts.

Ratio (A : B : C) = 1 : 3 : $\dfrac{3}{2}$

Multiply by 2, to convert fraction into whole number = 2 : 6 : 3.

Total parts: 2 + 6 + 3 = 11 parts.

Value of 1 part: 715 ÷ 11 = ₹ 65.

Calculate shares by multiplying each part with ₹ 65:

A gets: 2 x ₹ 65 = ₹ 130

B gets: 6 x ₹ 65 = ₹ 390

C gets: 3 x ₹ 65 = ₹ 195

A's share = ₹130, B's share = ₹390, C's share = ₹195.

Divide ₹ 760 among A, B, C such that A gets $\dfrac{5}{6}$ of what B gets and the ratio between the shares of B and C is 3 : 4.

Answer

Given:

Total Money: ₹ 760

A = $\dfrac{5}{6}$ of B

B : C = 3 : 4.

Since B : C is 3 : 4, let's look at A.

A = $\dfrac{5}{6} \times 3 = \dfrac{5}{2}$ parts.

Ratio (A : B : C) = $\dfrac{5}{2}$ : 3 : 4.

Multiply by 2 to convert fraction into whole number = 5 : 6 : 8.

Total parts: 5 + 6 + 8 = 19 parts.

Value of 1 part: 760 ÷ 19 = ₹ 40.

Calculate shares by multiplying each part with ₹ 40:

A gets: 5 x ₹ 40 = ₹ 200

B gets: 6 x ₹ 40 = ₹ 240

C gets: 8 x ₹ 40 = ₹ 320

A's share = ₹200, B's share = ₹240, C's share = ₹320

The boys and girls in a school are in the ratio 8 : 3. If the number of girls is 405, how many boys are there in the school?

Answer

Given:

Ratio (Boys : Girls) = 8 : 3

Total number of Girls = 405

Since 3 parts represent the girls, 3 parts = 405.

1 part = 405 ÷ 3 = 135.

Boys = 8 parts = 8 x 135 = 1080.

There are 1080 boys.

An alloy contains copper and zinc in the ratio 7 : 3. If it contains 12.6 g of copper, how much does this alloy weigh?

Answer

Given:

Ratio (Copper : Zinc) = 7 : 3

Copper = 12.6 g

Since 7 parts represent copper, 7 parts = 12.6 g

1 part = 12.6 ÷ 7 = 1.8 g

Zinc = 3 x 1.8 g = 5.4 g

Total weight = Copper + Zinc

Substituting the values in above:

Total weight = 12.6 g + 5.4 g = 18 g

The weight of the alloy is 18 g.

Reena weighed 63 kg. She reduced her weight in the ratio 9 : 8. Find her new weight.

Answer

Given:

Original Weight = 63 kg

Reduction Ratio = Old weight : New weight = 9 : 8

9 parts = 63 kg

1 part = 63 ÷ 9 = 7 kg.

The new weight is represented by 8 parts.

New weight 8 x 7 kg = 56 kg.

Her new weight is 56 kg.

Two numbers are in the ratio 9 : 13 and their sum is 176. Find the numbers.

Answer

Given:

Ratio = 9 : 13

Total Sum = 176

Total parts = 9 + 13 = 22 parts.

Value of 1 part: 176 ÷ 22 = 8. (So, 1 part = 8).

Calculate the numbers by multiplying each part with 8:

First number = 9 x 8 = 72

Second number = 13 x 8 = 104

The numbers are 72 and 104.

Two numbers are in the ratio 5 : 8 and their difference is 12. Find the numbers.

Answer

Given:

Ratio = 5 : 8

Difference = 12

The larger part is 8 and the smaller is 5.

Difference in parts = 8 - 5 = 3 parts.

Difference given = 12

3 parts = 12

1 part = 12 ÷ 3 = 4

Calculate the numbers by multiplying each part with 4:

First number = 5 x 4 = 20

Second number = 8 x 4 = 32

The numbers are 20 and 32.

If a : b = 3 : 2 and b : c = 4 : 5, find a : c.

Answer

Given:

a : b = 3 : 2

b : c = 4 : 5

$a : b = \dfrac{a}{b} = \dfrac{3}{2} \\[1em] b : c = \dfrac{b}{c} = \dfrac{4}{5}$

Now,

$a : c = \dfrac{a}{c} = \Big( \dfrac{a}{b} \times \dfrac{b}{c} \Big) \\[1em] = \Big( \dfrac{3}{2} \times \dfrac{4}{5} \Big) \\[1em] = \Big( \dfrac{3}{1} \times \dfrac{2}{5} \Big) \\[1em] = \dfrac{6}{5} \\[1em] = 6 : 5$

∴ a : c = 6 : 5

If a : b = 5 : 6 and b : c = 2.8 : 3.5, find a : c.

Answer

Given:

a : b = 5 : 6

b : c = 2.8 : 3.5

Multiply by 10 to remove decimals in the second ratio.

b : c = (2.8 x 10) : (3.5 x 10) = 28 : 35

b : c = 4 : 5 $\quad$ [Dividing by 7]

$a : b = \dfrac{a}{b} = \dfrac{5}{6} \\[1em] b : c = \dfrac{b}{c} = \dfrac{4}{5}$

Now,

$a : c = \dfrac{a}{c} = \left( \dfrac{a}{b} \times \frac{b}{c} \right) \\[1em] = \left( \dfrac{5}{6} \times \dfrac{4}{5} \right) \\[1em] = \left( \dfrac{1}{6} \times \dfrac{4}{1} \right) \\[1em] = \dfrac{4}{6} \\[1em] = \dfrac{2}{3} \\[1em] = 2 : 3$

∴ a : c = 2 : 3

If p : q = $1\dfrac{1}{3} : 1\dfrac{1}{2}$ and q : r = $\dfrac{1}{2} : \dfrac{1}{3}$, find p : r.

Answer

Given:

p : q = $1\dfrac{1}{3} : 1\dfrac{1}{2}$

q : r = $\dfrac{1}{2} : \dfrac{1}{3}$

Convert p : q to improper fractions:

p : q = $\dfrac{4}{3} : \dfrac{3}{2}$

L.C.M. of 3 and 2 is 6.

Multiply p : q and q : r with 6 to convert fractions to whole numbers:

$p : q = (\dfrac{4}{3} \times 6) : (\dfrac{3}{2} \times 6) = 8 : 9$

And

$q : r = (\dfrac{1}{2} \times 6) : (\dfrac{1}{3} \times 6) = 3 : 2$

$\dfrac{p}{q} = \dfrac{8}{9} \\[1em] \dfrac{q}{r} = \dfrac{3}{2}$

Now,

$p : r = \dfrac{p}{r} = \left( \dfrac{p}{q} \times \dfrac{q}{r} \right) \\[1em] = \left( \dfrac{8}{9} \times \dfrac{3}{2} \right) \\[1em] = \left( \dfrac{8}{3} \times \dfrac{1}{2} \right) \\[1em] = \dfrac{8}{6} \\[1em] = \dfrac{4}{3} \\[1em] = 4 : 3$

∴ p : r = 4 : 3

If a : b = 3 : 4 and b : c = 8 : 9, find a : b : c.

Answer

Given:

a : b = 3 : 4

b : c = 8 : 9

We make b, the same in both the ratios.

Let us find L.C.M. of 4 and 8:

$\begin{array}{l|rr} 2 & 4, & 8 \\ \hline 2 & 2, & 4 \\ \hline 2 & 1, & 2 \\ \hline & 1, & 1 \end{array}$

L.C.M. = 2 x 2 x 2 = 8

So, we make b equal to 8 in each case.

Now, multiply the first ratio by 2:

$a : b = 3 : 4 \\[1em] = \dfrac{3 \times 2}{4 \times 2} \\[1em] = \dfrac{6}{8} \\[1em] = 6 : 8 \\[1em]$

In second ratio b has 8 in denominator already, so keep it as it is.

b : c = 8 : 9

∴ a : b : c = 6 : 8 : 9

If l : m = $2\dfrac{1}{2} : 1\dfrac{2}{3}$ and m : n = $1\dfrac{1}{4} : 3\dfrac{1}{2}$, find l : m : n.

Answer

Given:

l : m = $2\dfrac{1}{2} : 1\dfrac{2}{3}$

m : n = $1\dfrac{1}{4} : 3\dfrac{1}{2}$

Convert mixed to improper fractions:

l : m = $\dfrac{5}{2} : \dfrac{5}{3}$

m : n = $\dfrac{5}{4} : \dfrac{7}{2}$

Let us simplify the ratios into whole numbers.

l : m = $\dfrac{5}{2} : \dfrac{5}{3}$

Let us find L.C.M. of 2 and 3:

$\begin{array}{l|rr} 2 & 2, & 3 \\ \hline 3 & 1, & 3 \\ \hline & 1, & 1 \end{array}$

L.C.M. = 2 x 3 = 6

l : m = $\Big(\dfrac{5}{2} \times 6\Big) : \Big(\dfrac{5}{3} \times 6\Big)$ = 15 : 10

l : m = 3 : 2 $\quad$[Dividing by 5]

m : n = $\dfrac{5}{4} : \dfrac{7}{2}$

Let us find L.C.M. of 4 and 2:

$\begin{array}{l|rr} 2 & 4, & 2 \\ \hline 2 & 2, & 1 \\ \hline & 1, & 1 \end{array}$

L.C.M. = 2 x 2 = 4

m : n = $\Big(\dfrac{5}{4} \times 4\Big) : \Big(\dfrac{7}{2} \times 4\Big)$

m : n = 5 : 14

We make m, the same in both the ratios.

Let us find L.C.M. of 2 and 5:

$\begin{array}{l|rr} 2 & 2, & 5 \\ \hline 5 & 1, & 5 \\ \hline & 1, & 1 \end{array}$

L.C.M. = 2 x 5 = 10

So, we make m equal to 10 in each case.

Now, multiply the first ratio by 5:

$l : m = 3 : 2 \\[1em] = \dfrac{3 \times 5}{2 \times 5} \\[1em] = \dfrac{15}{10} \\[1em] = 15 : 10$

Multiply the second ratio by 2:

$m : n = 5 : 14 \\[1em] = \dfrac{5 \times 2}{14 \times 2} \\[1em] = \dfrac{10}{28} \\[1em] = 10 : 28$

Now we have,

l : m = 15 : 10 and m : n = 10 : 28

∴ l : m : n = 15 : 10 : 28

Which ratio is greater?

(i) (3 : 4) or (5 : 7)

(ii) (11 : 21) or (19 : 28)

Answer

(i)

The given ratios are 3 : 4 and 5 : 7, which are equivalent to the fractions $\dfrac{3}{4}$ and $\dfrac{5}{7}$ respectively.

Let us find the L.C.M. of 4 and 7:

$\begin{array}{l|rr} 2 & 4, & 7 \\ \hline 2 & 2, & 7 \\ \hline 7 & 1, & 7 \\ \hline & 1, & 1 \end{array}$

L.C.M. = 2 x 2 x 7 = 28.

$∴ 3 : 4 = \dfrac{3}{4} = \dfrac{3 \times 7}{4 \times 7} = \dfrac{21}{28}$

$5 : 7 = \dfrac{5}{7} = \dfrac{5 \times 4}{7 \times 4} = \dfrac{20}{28}$

Clearly, $\dfrac{21}{28} \gt \dfrac{20}{28}$.

Hence, (3 : 4) > (5 : 7).

(ii)

The given ratios are 11 : 21 and 19 : 28, which are equivalent to the fractions $\dfrac{11}{21}$ and $\dfrac{19}{28}$ respectively.

Let us find L.C.M. of 21 and 28:

$\begin{array}{l|l} 7 & 21, 28 \\ \hline 3 & 3, 4 \\ \hline 4 & 1, 4 \\ \hline & 1, 1 \end{array}$

L.C.M. = 7 x 3 x 4 = 84.

$\therefore 11 : 21 = \dfrac{11}{21} = \dfrac{11 \times 4}{21 \times 4} = \dfrac{44}{84}$

$19 : 28 = \dfrac{19}{28} = \dfrac{19 \times 3}{28 \times 3} = \dfrac{57}{84}$

Clearly, $\dfrac{57}{84} \gt \dfrac{44}{84}$.

Hence, (19 : 28) > (11 : 21).