Ratio & Proportion — Exercise 8(A)

Find the ratio of the following in simplest form:

₹7.80 to 65 paise

Answer

₹7.80 to 65 paise

Convert ₹ into paise: ₹7.80 = 7.80 × 100 paise = 780 paise.

Ratio = 780 : 65

H.C.F. of 780 and 65 is 65.

$\dfrac{780 \div 65}{65 \div 65} = \dfrac{12}{1} = 12 : 1$

Hence, the ratio is 12 : 1.

Find the ratio of the following in simplest form:

15 hours to 1 day

Answer

15 hours to 1 day

Convert day into hours: 1 day = 24 hours.

Ratio = 15 : 24

H.C.F. of 15 and 24 is 3.

$\dfrac{15 \div 3}{24 \div 3} = \dfrac{5}{8} = 5 : 8$

Hence, the ratio is 5 : 8.

Find the ratio of the following in simplest form:

750 gm to 2 kg

Answer

750 gm to 2 kg

Convert kg into gm: 2 kg = 2 × 1000 gm = 2000 gm.

Ratio = 750 : 2000

H.C.F. of 750 and 2000 is 250.

$\dfrac{750 \div 250}{2000 \div 250} = \dfrac{3}{8} = 3 : 8$

Hence, the ratio is 3 : 8.

Find the ratio of the following in simplest form:

64 ml to 1 litre

Answer

64 ml to 1 litre

Convert litre into ml: 1 litre = 1000 ml.

Ratio = 64 : 1000

H.C.F. of 64 and 1000 is 8.

$\dfrac{64 \div 8}{1000 \div 8} = \dfrac{8}{125} = 8 : 125$

Hence, the ratio is 8 : 125.

Find the ratio of the following in simplest form:

18 mm to 3 cm

Answer

18 mm to 3 cm

Convert cm into mm: 3 cm = 3 × 10 mm = 30 mm.

Ratio = 18 : 30

H.C.F. of 18 and 30 is 6.

$\dfrac{18 \div 6}{30 \div 6} = \dfrac{3}{5} = 3 : 5$

Hence, the ratio is 3 : 5.

Find the ratio of the following in simplest form:

80 kg to 2 quintals

Answer

80 kg to 2 quintals

Convert quintals into kg: 1 quintal = 100 kg, so 2 quintals = 200 kg.

Ratio = 80 : 200

H.C.F. of 80 and 200 is 40.

$\dfrac{80 \div 40}{200 \div 40} = \dfrac{2}{5} = 2 : 5$

Hence, the ratio is 2 : 5.

Express each of the following ratios in simplest form:

15 : 25

Answer

15 : 25

H.C.F. of 15 and 25 is 5.

$\dfrac{15 \div 5}{25 \div 5} = \dfrac{3}{5} = 3 : 5$

Hence, the answer is 3 : 5.

Express each of the following ratios in simplest form:

49 : 35

Answer

49 : 35

H.C.F. of 49 and 35 is 7.

$\dfrac{49 \div 7}{35 \div 7} = \dfrac{7}{5} = 7 : 5$

Hence, the answer is 7 : 5.

Express each of the following ratios in simplest form:

90 : 72

Answer

90 : 72

H.C.F. of 90 and 72 is 18.

$\dfrac{90 \div 18}{72 \div 18} = \dfrac{5}{4} = 5 : 4$

Hence, the answer is 5 : 4.

Express each of the following ratios in simplest form:

146 : 365

Answer

146 : 365

H.C.F. of 146 and 365 is 73.

$\dfrac{146 \div 73}{365 \div 73} = \dfrac{2}{5} = 2 : 5$

Hence, the answer is 2 : 5.

Express each of the following ratios in simplest form:

111 : 259

Answer

111 : 259

H.C.F. of 111 and 259 is 37.

$\dfrac{111 \div 37}{259 \div 37} = \dfrac{3}{7} = 3 : 7$

Hence, the answer is 3 : 7.

Express each of the following ratios in simplest form:

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

Answer

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

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

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

L.C.M. = 2 × 2 × 2 × 3 = 24.

Multiplying each term by 24:

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

= 4 : 3

Hence, the answer is 4 : 3.

Express each of the following ratios in simplest form:

$\dfrac{3}{4} : \dfrac{4}{5} : \dfrac{5}{6}$

Answer

$\dfrac{3}{4} : \dfrac{4}{5} : \dfrac{5}{6}$

Let us find the L.C.M. of 4, 5 and 6:

$\begin{array}{l|rrr} 2 & 4, & 5, & 6 \\ \hline 2 & 2, & 5, & 3 \\ \hline 3 & 1, & 5, & 3 \\ \hline 5 & 1, & 5, & 1 \\ \hline & 1, & 1, & 1 \end{array}$

L.C.M. = 2 × 2 × 3 × 5 = 60.

Multiplying each term by 60:

$\dfrac{3}{4} : \dfrac{4}{5} : \dfrac{5}{6} = \left(\dfrac{3}{4} \times 60\right) : \left(\dfrac{4}{5} \times 60\right) : \left(\dfrac{5}{6} \times 60\right) = 45 : 48 : 50$

Since the H.C.F. of 45, 48 and 50 is 1, the ratio is already in simplest form.

Hence, the answer is 45 : 48 : 50.

Express each of the following ratios in simplest form:

$2\dfrac{1}{3} : 3\dfrac{1}{4} : 5\dfrac{1}{6}$

Answer

$2\dfrac{1}{3} : 3\dfrac{1}{4} : 5\dfrac{1}{6}$

Convert the mixed fractions into improper fractions:

$2\dfrac{1}{3} = \dfrac{7}{3}, \quad 3\dfrac{1}{4} = \dfrac{13}{4}, \quad 5\dfrac{1}{6} = \dfrac{31}{6}$

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

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

L.C.M. = 2 × 2 × 3 = 12.

Multiplying each term by 12:

$\dfrac{7}{3} : \dfrac{13}{4} : \dfrac{31}{6} = \left(\dfrac{7}{3} \times 12\right) : \left(\dfrac{13}{4} \times 12\right) : \left(\dfrac{31}{6} \times 12\right)$

= 28 : 39 : 62

Since the H.C.F. of 28, 39 and 62 is 1, the ratio is already in simplest form.

Hence, the answer is 28 : 39 : 62.

In a school, there are 420 boys and 180 girls. Find the ratio of:

(i) girls to boys

(ii) girls to total number of students

(iii) boys to total number of students

Answer

Given:

Number of boys = 420

Number of girls = 180

Total number of students = 420 + 180 = 600

(i) Ratio of girls to boys = 180 : 420

H.C.F. of 180 and 420 is 60.

$\dfrac{180 \div 60}{420 \div 60} = \dfrac{3}{7} = 3 : 7$

Hence, the ratio of girls to boys is 3 : 7.

(ii) Ratio of girls to total number of students = 180 : 600

H.C.F. of 180 and 600 is 60.

$\dfrac{180 \div 60}{600 \div 60} = \dfrac{3}{10} = 3 : 10$

Hence, the ratio of girls to total number of students is 3 : 10.

(iii) Ratio of boys to total number of students = 420 : 600

H.C.F. of 420 and 600 is 60.

$\dfrac{420 \div 60}{600 \div 60} = \dfrac{7}{10} = 7 : 10$

Hence, the ratio of boys to total number of students is 7 : 10.

The total population of a village is 3540, out of which 2065 are males. Find the ratio of males to females.

Answer

Given:

Total population = 3540

Number of males = 2065

Number of females = Total population − Number of males

Number of females = 3540 − 2065 = 1475

Ratio of males to females = 2065 : 1475

H.C.F. of 2065 and 1475 is 295.

$\dfrac{2065 \div 295}{1475 \div 295} = \dfrac{7}{5} = 7 : 5$

Hence, the ratio of males to females is 7 : 5.

Find the ratio of the price of coffee to that of tea, if coffee costs ₹48 per 100 gm and tea costs ₹280 per kg.

Answer

Given:

Cost of coffee = ₹48 per 100 gm

Cost of tea = ₹280 per kg

To compare the two prices, express the cost of coffee per kg (1 kg = 1000 gm).

Cost of coffee per kg = ₹48 × 10 = ₹480 per kg

Ratio of price of coffee to that of tea = 480 : 280

H.C.F. of 480 and 280 is 40.

$\dfrac{480 \div 40}{280 \div 40} = \dfrac{12}{7} = 12 : 7$

Hence, the ratio of the price of coffee to that of tea is 12 : 7.

The ratio of tin and zinc in an alloy is 3 : 4. In 21 gm of alloy, find the quantity of (i) tin and (ii) zinc.

Answer

Given:

Total weight of alloy = 21 gm

Ratio (Tin : Zinc) = 3 : 4

Sum of ratio terms = 3 + 4 = 7.

(i) Quantity of tin = $21 \times \dfrac{3}{7}$ = 3 × 3 gm = 9 gms.

Hence, the quantity of tin is 9 gms.

(ii) Quantity of zinc = $21 \times \dfrac{4}{7}$ = 4 × 3 gm = 12 gms.

Hence, the quantity of zinc is 12 gms.

Divide ₹1,155 between Amit and Sanya in the ratio 7 : 4.

Answer

Given:

Total money = ₹1,155

Ratio (Amit : Sanya) = 7 : 4

Sum of ratio terms = 7 + 4 = 11.

Amit's share = $1{,}155 \times \dfrac{7}{11}$ = 7 × ₹105 = ₹735.

Sanya's share = $1{,}155 \times \dfrac{4}{11}$ = 4 × ₹105 = ₹420.

Hence, Amit gets ₹735 and Sanya gets ₹420.

Divide ₹3,500 among A, B and C in the ratio 3 : 4 : 7.

Answer

Given:

Total money = ₹3,500

Ratio (A : B : C) = 3 : 4 : 7

Sum of ratio terms = 3 + 4 + 7 = 14.

A's share = $3{,}500 \times \dfrac{3}{14}$ = 3 × ₹250 = ₹750.

B's share = $3{,}500 \times \dfrac{4}{14}$ = 4 × ₹250 = ₹1,000.

C's share = $3{,}500 \times \dfrac{7}{14}$ = 7 × ₹250 = ₹1,750.

Hence, A gets ₹750, B gets ₹1,000 and C gets ₹1,750.

Divide ₹3,010 among A, B and C in such a way that A gets double of what B gets and B gets double of what C gets.

Answer

Given:

Total money = ₹3,010

Suppose C gets ₹1. Then B gets double of C = ₹2, and A gets double of B = ₹4.

So, Ratio (A : B : C) = 4 : 2 : 1

Sum of ratio terms = 4 + 2 + 1 = 7.

A's share = $3{,}010 \times \dfrac{4}{7}$ = 4 × ₹430 = ₹1,720.

B's share = $3{,}010 \times \dfrac{2}{7}$ = 2 × ₹430 = ₹860.

C's share = $3{,}010 \times \dfrac{1}{7}$ = 1 × ₹430 = ₹430.

Hence, A gets ₹1,720, B gets ₹860 and C gets ₹430.

Divide 182 in three parts in the ratio $\dfrac{1}{10} : \dfrac{1}{15} : \dfrac{1}{20}$.

Answer

Given:

Total = 182

Ratio = $\dfrac{1}{10} : \dfrac{1}{15} : \dfrac{1}{20}$

Let us find the L.C.M. of 10, 15 and 20:

$\begin{array}{l|rrr} 2 & 10, & 15, & 20 \\ \hline 2 & 5, & 15, & 10 \\ \hline 3 & 5, & 15, & 5 \\ \hline 5 & 5, & 5, & 5 \\ \hline & 1, & 1, & 1 \end{array}$

L.C.M. = 2 × 2 × 3 × 5 = 60.

Multiplying each term by 60:

$\Rightarrow \left(\dfrac{1}{10} \times 60\right) : \left(\dfrac{1}{15} \times 60\right) : \left(\dfrac{1}{20} \times 60\right) \\[1em] \Rightarrow 6 : 4 : 3$

Sum of ratio terms = 6 + 4 + 3 = 13.

First part = $182 \times \dfrac{6}{13}$ = 6 × 14 = 84.

Second part = $182 \times \dfrac{4}{13}$ = 4 × 14 = 56.

Third part = $182 \times \dfrac{3}{13}$ = 3 × 14 = 42.

Hence, the three parts are 84, 56 and 42.

The sides of a triangle are in the ratio 2 : 3 : 4. If its perimeter is 54 cm, find the lengths of the sides of the triangle.

Answer

Given:

Perimeter of the triangle = 54 cm

Ratio of sides = 2 : 3 : 4

Sum of ratio terms = 2 + 3 + 4 = 9.

First side = $54 \times \dfrac{2}{9}$ = 2 × 6 cm = 12 cm.

Second side = $54 \times \dfrac{3}{9}$ = 3 × 6 cm = 18 cm.

Third side = $54 \times \dfrac{4}{9}$ = 4 × 6 cm = 24 cm.

Hence, the lengths of the sides of the triangle are 12 cm, 18 cm and 24 cm.

The angles of a triangle are in the ratio 3 : 5 : 7. Find the measure of each angle of the triangle.

Answer

Given:

Ratio of angles = 3 : 5 : 7

We know that the sum of the angles of a triangle is 180°.

Sum of ratio terms = 3 + 5 + 7 = 15.

First angle = $180 \times \dfrac{3}{15}$ = 3 × 12° = 36°.

Second angle = $180 \times \dfrac{5}{15}$ = 5 × 12° = 60°.

Third angle = $180 \times \dfrac{7}{15}$ = 7 × 12° = 84°.

Hence, the angles of the triangle are 36°, 60° and 84°.

A ratio in simplest form is 11 : 15. If its antecedent is 121, find its consequent.

Answer

Given:

Ratio in simplest form = 11 : 15

Antecedent = 121

It is given that $\dfrac{11}{15} = \dfrac{121}{?}$.

We know that 121 = 11 × 11.

So, we multiply the numerator and denominator of $\dfrac{11}{15}$ by 11.

$\dfrac{11}{15} = \dfrac{11 \times 11}{15 \times 11} = \dfrac{121}{165}$

Hence, the required consequent is 165.

A ratio in simplest form is 7 : 12. If its consequent is 144, find its antecedent.

Answer

Given:

Ratio in simplest form = 7 : 12

Consequent = 144

It is given that $\dfrac{7}{12} = \dfrac{?}{144}$.

We know that 144 = 12 × 12.

So, we multiply the numerator and denominator of $\dfrac{7}{12}$ by 12.

$\dfrac{7}{12} = \dfrac{7 \times 12}{12 \times 12} = \dfrac{84}{144}$

Hence, the required antecedent is 84.