\begin{array}{l}
\text {Q1. Express each of the following in the language of ratios: } \\
\text {(i) In a class, the number of girls in the merit list of the board examination } \\
\text { is two times that of boys. } \\
\text {(ii) The number of students passing mathematics test is } \frac{2}{3} \text { of the number that appeared. } \\ \\\text {Sol. (i) Ratio of the number of girls to that of boys in the merit list is 2: 1 } \\ \\\text {(ii) Ratio of the number of students passing a mathematics test to that of } \\
\text { total students appearing in the test is 2: 3} \\ \\\text {Q2. Express the following ratios in language of daily life: } \\
\text {(i) The ratio of the number of bad pencils to that of good pencils produced in a factory is 1: 9 } \\
\text {(ii) In India, the ratio of the number of villages to that of cities is about 2000: 1 } \\ \\\text {Sol. (i) The number of bad pencils produced in a factory is } \frac {1}{9} \\
\text { of the number of good pencils produced in the factory.} \\ \\\text {(ii) The number of villages is 2000 times that of cities in India.} \\ \\\text {Q3. Express each of the following ratios in its simplest form: } \\
\text {(i) } \quad 60: 72 \quad
\text {(ii) } \quad 324: 144 \\
\text {(iii) } \quad 85: 391 \\
\text {(iv) } \quad 186: 403 \\ \\\text {Sol. (i) The given ration is } 60 : 72 = \frac {60}{72} \\
\text {To express the given ratio in simplest form, first we have to find HCF of 60 and 72} \\
\text {HCF of 60 and 72 is 12} \\
\text {By dividing each term by of the ratio by HCF of its term i.e 12, we get } \\
\frac {60}{72} = \frac {60 \div 12}{72 \div 12} = \frac {5}{6} = 5:6 \\
\text {Hence, the simplest form of ratio 60 : 72 is 5 : 6 } \\ \\\text {(ii) The given ration is } 324 : 144= \frac {60}{72} \\
\text {To express the given ratio in simplest form, first we have to find HCF of 324 and 144} \\
\text {HCF of 324 and 144 is 36} \\
\text {By dividing each term by of the ratio by HCF of its term i.e 36, we get } \\
\frac {324}{144} = \frac {324 \div 36}{144 \div 36} \frac {9}{4} = 9 : 4 \\
\text {Hence, the simplest form of ratio 324 : 144 is 9 : 4 } \\ \\\text {(iii) The given ration is } 85: 391 = \frac {60}{72} \\
\text {To express the given ratio in simplest form, first we have to find HCF of 85 and 391} \\
\text {HCF of 85 and 391 is 17} \\
\text {By dividing each term by of the ratio by HCF of its term i.e 17, we get } \\
\frac {85}{391} = \frac {85 \div 17}{391 \div 17} = \frac {5}{23} = 5 : 23 \\
\text {Hence, the simplest form of ratio 85: 391 is 5 : 23 } \\ \\\text {(iv) The given ration is } 186: 403= \frac {60}{72} \\
\text {To express the given ratio in simplest form, first we have to find HCF of 186 and 403} \\
\text {HCF of 186 and 403 is 31} \\
\text {By dividing each term by of the ratio by HCF of its term i.e 31, we get } \\
\frac {186}{403} = \frac {186 \div 31}{403 \div 31} = \frac {6}{3} = 6 : 3 \\
\text {Hence, the simplest form of ratio 186: 403 is 6 : 3 } \\ \\\text {Q4. Find the ratio of the following in the simplest form: } \\
\text {(i) 75 paise to Rs 3 } \\
\text {(ii) 35 minutes to 45 minutes } \\
\text {(iii) 8 kg to 400 gm } \\
\text {(iv) 48 minutes to 1 hour } \\
\text {(v) 2 metres to 35 cm} \\
\text {(vi) 35 minutes to 45 seconds } \\
\text {(vii) 2 dozen to 3 scores } \\
\text {(viii) 3 weeks to 3 days } \\
\text {(ix) 48 min to 2 hours 40 min } \\
\text {(x) 3 m 5 cm to 35 cm} \\ \\\text {Sol. (i) 75 paise : Rs. 3 } \\
= \text {75 paise : 300 paise } \quad \quad [\because \text {Rs. } 1 = 100 \text { paisa}] \\
\text {Dividing the two terms by HCF of 75 and 300 i.e 75, we get } \\
= 75 \div 75 : 300 \div 75 \\ \\
= \text {1 : 4 } \\ \\\text {(ii) 35 minutes : 45 minutes} \\
\text {Dividing the two terms by HCF of 35 and 45 i.e 5, we get } \\
= 35 \div 5 : 45 \div 5 \\
= \text {7 : 9 } \\\text {(iii) 8 kg : 400 gm } \\
= \text {8000 gm : 400 gm } \quad \quad [\because 1 \text { kg } = 1000 \text { gm}] \\
\text {Dividing the two terms by HCF of 8000 and 400 i.e 400, we get } \\
= 8000 \div 400 : 400 \div 400 \\ \\
= \text {20 : 1 } \\ \\\text {(iv) 48 minutes : 1 hr } \\
= \text {48 minutes : 60 minutes } \quad \quad [\because 1 \text { hr} = 60 \text { minutes}] \\
\text {Dividing the two terms by HCF of 48 and 60 i.e 12, we get } \\
= 48 \div 12 : 60 \div 12 \\
= \text {4 : 5 } \\ \\\text {(v) 2 metres : 35 cm } \\
= \text {200 cm : 35 cm } \quad \quad [\because 1 \text { m} = 100 \text { cm}] \\
\text {Dividing the two terms by HCF of 200 and 35 i.e 5, we get } \\
= 200 \div 5 : 35 \div 5 \\
= \text {40 : 7 } \\ \\\text {(vi) 35 minutes : 45 seconds } \\
= \text {35 \times 60 seconds : 45 seconds } \quad \quad [\because 1 \text { minute} = 60 \text { seconds}] \\
= \text {2100 seconds : 45 seconds } \\
\text {Dividing the two terms by HCF of 2100 and 45 i.e 15, we get } \\
= 2100 \div 15 : 45 \div 15 \\
= \text {140 : 3 } \\ \\\text {(vii) 2 dozen : 3 scores } \\
\text {We know that } 1 \text { dozen} = 12 \text { units}] \\
\text {And } 1 \text { score} = 20 \text { units}] \\
\therefore \text { 2 dozen : 3 scores }
= 2 \times 12 \text { units} : 3 \times 20 \text { units } \\= \text {24 units : 60 units} \\
\text {Dividing the two terms by HCF of 24 and 60 i.e 12, we get } \\
= 24 \div 12 : 60 \div 12 \\
= \text {2 : 5 } \\ \\\text {(viii) 3 week : 3 days } \\
= \text {3 \times 7 days : 3 days } \quad \quad [\because 1 \text { week} = 7 \text { days}] \\
= \text {21 days : 3 days } \\
\text {Dividing the two terms by HCF of 21 and 3 i.e 3, we get } \\
= 21 \div 3 : 3 \div 3 \\
= \text {7 : 1 } \\ \\\text {(ix) 48 min : 2 hours 40 min } \\
= \text {48 min : 2 \times 60 min + 40 min } \quad \quad [\because 1 \text { hr} = 60 \text { min}] \\
= \text {48 min : 160 min } \\
\text {Dividing the two terms by HCF of 48 and 160 i.e 16, we get } \\
= \text {48 \div 16 : 160 \div 16 } \\
= \text {3 : 10 } \\ \\\text {(x) 3 m 5 cm : 35 cm } \\
= \text {3 \times 100 cm + 5 cm : 35 cm } \quad \quad [\because 1 \text { m} = 100 \text { cm}] \\
= \text {305 cm : 35 cm } \\
\text {Dividing the two terms by HCF of 305 and 35 i.e 5, we get } \\
= 305 \div 5 : 35 \div 5 \\
= \text {61 : 7 } \\ \\\text {Q5. Find the ratio of } \\
\text {(i) 3.2 metres to 56 metres } \\
\text {(ii) 10 metres to 25 cm } \\
\text {(iii) 25 paise to Rs. 60 } \\
\text {(iv) 10 litres to 0.25 litre. } \\ \\\text {Sol. (i) 3.2 metres : 56 metres } = \frac {3.2}{56} \\
= \frac {32}{560} \\
\text {Dividing the two terms by HCF of 32 and 560 i.e 16, we get } \\
= 32 \div 16 : 560 \div 16 \\
= \text {2 : 35 } \\
\therefore \text {(i) 3.2 metres : 56 metres } = = \text {2 : 35 } \\ \\\text {(ii) 10 metres : 25 cm } \\
= \text {10 \times 100 cm : 25 cm } \quad \quad [\because 1 \text { m} = 100 \text { cm}] \\
= \text {1000 cm : 25 cm } \\
\text {Dividing the two terms by HCF of 1000 and 25 i.e 25, we get } \\
= 1000 \div 25 : 25 \div 25 \\
= \text {40 : 1 } \\ \\\text {(iii) 25 paise to Rs. 60} \\
= \text {25 paise : 60 \times 100 paise } \quad \quad [\because \text {Rs. } 1 = 100 \text { paisa}] \\
= \text {25 paise : 6000 paise } \\
\text {Dividing the two terms by HCF of 25 and 6000 i.e 25, we get } \\
= 25 \div 25 : 6000 \div 25 \\
= \text {1 : 240 } \\ \\\text {(iv) 10 litres : 0.25 litre } \\
\text {Dividing the two terms by HCF of 10 and 0.25 i.e 0.25, we get } \\
= 10 \div 0.25 : 0.25 \div 0.25 \\
= \text {40 : 1 } \\ \\\text {Q6. The number of boys and girls in a school are 1168 and 1095 respectively. } \\
\text { Express the ratio of the number of boys to that of the girls in the simplest form.} \\ \\\text {Sol. No. of boys } =1168 \\
\text {No. of girls }=1095 \\
\text {The ratio of the number of boys to that of the girls } =1168 : 1095 \\
\text {Dividing the two terms by HCF of 1168 and 1095 i.e 73, we get } \\
\text {Ratio of number of boys to that of the girls in simplest form}=1168 \div 73 : 1095 \div 73 \\
= 16: 15 \\ \\\text {Q7. Avinash works as a lecturer and earns Rs 12000 per month. His wife who } \\
\text { is a doctor earns Rs. 15000 per month. Find the following ratios: } \\
\text {(i) Avinash’s income to the income of his wife.} \\
\text {(ii) Avinash’s income to their total income. } \\ \\\text {Sol. Avinash monthly income } = \text {Rs. } 12000 \\
\text {Avinash wife monthly income } = \text {Rs. } 15000 \\
\text {Total monthly income } = \text {Rs. } (12000 + 15000 )= \text {Rs. } 27000 \\\text {(i) Ratio of Avinash’s income to the income of his wife }=12000 : 15000=4: 5 \\
\text {(ii) Ratio of Avinash’s income to their total income } \\
=12000 : 27000 =4: 9 \\ \\\text {Q8. Of the 72 persons working in an office, 28 are men and the remaining are women. } \\
\text {Find the ratio of the number of:} \\
\text {(i) men to that of women, } \\
\text {(ii) men to the total number of persons } \\
\text {(iii) persons to that of women. } \\ \\\text {Sol. Total no. of persons working in an office } =72 \\
\text {No. of men } =28 \\
\therefore \text {The number of women }=72-28=44 \\\text {(i) Ratio of men to that of women } =28: 44 \\
\text {Dividing the two terms by HCF of 28 and 44 i.e 4, we get } \\
=28 \div 4 : 44 \div 4 \\
=7: 11 \\ \\\text {(ii) Ratio of the total number of persons } =28: 72 \\
\text {Dividing the two terms by HCF of 28 and 72 i.e 4, we get } \\
=28 \div 4 : 72 \div 4 \\
=7: 18 \\ \\\text {(iii) Ratio of total persons to that of women } =72: 44 \\
\text {Dividing the two terms by HCF of 72 and 44 i.e 4, we get } \\
=72 \div 4 : 44 \div 4 \\
=18: 11 \\ \\\text {Q9. The length of a steel tape for measurements of buildings is 10 m and } \\
\text { its width is 2.4 cm. What is the ratio of its length to width?} \\ \\\text {Sol. Given that Length of a steel tape } =10 \text { m} \\
= 10 \times 100 \text { cm} \quad \quad [\because 1 \text { m} = 100 \text { cm}] \\
= 1000 \text { cm} \\
\text {Width of steel tape } =2.4 \text { cm} \\
\therefore \text {The ratio of its length to width }= \frac {1000 \text { cm} }{2.4 \text { cm}} \\
= \frac {10000 }{24} \\
\text {Dividing the two terms by HCF of 10000 and 24 i.e 8, we get } \\
=10000 \div 8 : 24 \div 8 \\
=1250: 3 \\
\text {Hence, the ratio of its length to width is 1250: 3} \\ \\\text {Q10. An office opens at 9 a.m. and closes at 5 p.m. with a lunch interval of 30 minutes.} \\
\text {What is the ratio of lunch interval to the total period in office? } \\ \\\text {Sol. Total duration of office = 9 am to 5 pm } \\
= \text {9 am to 12 noon and 12 pm to 5 pm } \\
= 3 \text { hours} + 5 \text { hours} \\
= 8 \text { hours } \\
= 8 \times 60 \text { minutes} \quad \quad [\because 1 \text { hour} = 60 \text { minutes}] \\
= 480 \text { minutes} \\\text {Ratio of lunch interval to the total period in office } =30 : 480 \\
\text {Dividing the two terms by HCF of 30 and 480 i.e 30, we get } \\
=30 \div 30 : 480 \div 30 \\
=1: 16 \\
\text {Hence, the ratio of lunch interval to the total period in office is } 1: 16 \\ \\\text {Q11. A bullock-cart travels 24 km in 3 hours and a train } \\
\text {travels 120 km in 2 hours.Find the ratio of their speeds.} \\ \\\text {Sol. We know that Speed } = \frac {\text {Distnace}}{\text {Time Taken}} \\
\text {Distance covered by bullock-cart }=24 \text { km} \\
\text {Time taken } = 3 \text { hours} \\
\text {Speed of bullock-cart} = \frac {24}{3} = 8 \text { km}/\text {hr} \\\text {Distance covered by train }=120 \text { km} \\
\text {Time taken } = 2 \text { hours} \\
\text {Speed of train} = \frac {120}{2} = 60 \text { km}/\text {hr} \\\text {Ratio of the speed of bullock-cart to train} =8 : 60 \\
\text {Dividing the two terms by HCF of 8 and 60 i.e 4, we get } \\
=8 \div 4 : 15 \div 4 \\
=2 : 15 \\
\text {Hence, the ratio of their speeds is } 2 : 15 \\ \\\text {Q12. Margarette works in a factory and earns Rs. 955 per month.} \\
\text {She saves Rs. 185 per month from her earnings. Find the ratio of:} \\
\text {(i) her savings to her income } \\
\text {(ii) her income to her expenditure} \\
\text {(iii) her savings to her expenditure. } \\ \\\text {Sol. Margarette monthly income }= \text {Rs. } 955 \\ \text {Margarette monthly savings }=\text {Rs. } 185 \\
\text {Margarette expenditure }=955 – 185 =\text {Rs.} 770 \\
\text {(i) Margarette savings to her income } =185 : 955 \\
\text {Dividing the two terms by HCF of 185 and 955 i.e. 5, we get } \\
= 185 \div 5 : 955 \div 5 \\
=37: 191 \\ \\\text {(ii) Margarette income to her expenditure } =955 : 770 \\
\text {Dividing the two terms by HCF of 955 and 770 i.e. 5, we get } \\
= 955 \div 5 : 770 \div 5 \\
= 191 : 154 \\ \\\text {(iii) Margarette savings to her expenditure } = 185 : 770 \\
\text {Dividing the two terms by HCF of 185 and 770 i.e. 5, we get } \\
= 185 \div 5 : 770 \div 5 \\
=37: 154 \\ \\\end{array}
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