Maharashtra Board Class 9 Maths Part 1 Algebra Chapter 4 Ratio and Proportion Set 4.1 Solutions

Question 1. From the following pairs of numbers, find the reduced form of ratio of first number to second number.
(i) 72,60
(ii) 38,57
(iii) 52,78
Answer:
(i) 72, 60
Ratio \( = \frac{72}{60} = \frac{12 \times 6}{12 \times 5} = \frac{6}{5} = 6:5 \)
(ii) 38, 57
Ratio \( = \frac{38}{57} = \frac{19 \times 2}{19 \times 3} = \frac{2}{3} = 2:3 \)
(iii) 52, 78
Ratio \( = \frac{52}{78} = \frac{26 \times 2}{26 \times 3} = \frac{2}{3} = 2:3 \)
In simple words: To find the reduced form of a ratio, divide both numbers by their greatest common divisor until they cannot be simplified further. This expresses the ratio in its simplest integer form.

🎯 Exam Tip: Always look for the greatest common factor (GCF) to simplify ratios to their lowest terms. Ensure both numbers are integers before reduction.

Question 2. Find the reduced form of the ratio of the first quantity to second quantity.
(i) Rs. 700, Rs. 308
(ii) Rs. 14, 12 and 40 paise
(iii) 5 litres, 2500 ml
(iv) 3 years 4 months, 5 years 8 months
(v) 3.8 kg, 1900 gm
(vi) 7 minutes 20 seconds, 5 minutes 6 seconds
Answer:
(i) Rs. 700, Rs. 308
Ratio \( = \frac{700}{308} = \frac{28 \times 25}{28 \times 11} = \frac{25}{11} = 25:11 \)
(ii) Rs. 14, 12 and 40 paise
Rs. 14 = 14 x 100 paise = 1400 paise
12 and 40 paise = 12 x 100 paise + 40 paise
= (1200 + 40) paise
= 1240 paise
Ratio \( = \frac{\text{Rs. }14}{12\text{ and }40\text{ paise}} = \frac{1400\text{ paise}}{1240\text{ paise}} = \frac{1400}{1240} = \frac{40 \times 35}{40 \times 31} = \frac{35}{31} = 35:31 \)
(iii) 5 litres, 2500 ml
5 litres = 5 x 1000 ml = 5000ml
Ratio \( = \frac{5\text{ litres}}{2500\text{ ml}} = \frac{5000\text{ ml}}{2500\text{ ml}} = \frac{5000}{2500} = \frac{2500 \times 2}{2500 \times 1} = \frac{2}{1} = 2:1 \)
(iv) 3 years 4 months, 5 years 8 months
3 years 4 months = \( 3 \times 12 \text{ months} + 4 \text{ months} \)
= (36 + 4) months
= 40 months
5 years 8 months = \( 5 \times 12 \text{ months} + 8 \text{ months} \)
= (60 + 8) months
= 68 months
Ratio \( = \frac{3\text{ years }4\text{ months}}{5\text{ years }8\text{ months}} = \frac{40\text{ months}}{68\text{ months}} = \frac{40}{68} = \frac{4 \times 10}{4 \times 17} = \frac{10}{17} = 10:17 \)
(v) 3.8 kg, 1900 gm
3.8 kg = \( 3.8 \times 1000 \text{ gm} = 3800 \text{ gm} \)
Ratio \( = \frac{3.8\text{kg}}{1900\text{gm}} = \frac{3800\text{ gm}}{1900\text{ gm}} = \frac{3800}{1900} = \frac{1900 \times 2}{1900 \times 1} = \frac{2}{1} = 2:1 \)
(vi) 7 minutes 20 seconds, 5 minutes 6 seconds
7 minutes 20 seconds = \( 7 \times 60 \text{ seconds} + 20 \text{ seconds} \)
= (420 + 20) seconds
= 440 seconds
5 minutes 6 seconds = \( 5 \times 60 \text{ seconds} + 6 \text{ seconds} \)
= (300 + 6) seconds
= 306 seconds
Ratio \( = \frac{7\text{ minutes }20\text{ seconds}}{5\text{ minutes }6\text{ seconds}} = \frac{440\text{ seconds}}{306\text{ seconds}} = \frac{440}{306} = \frac{2 \times 220}{2 \times 153} = \frac{220}{153} = 220:153 \)
In simple words: To find the ratio of two quantities, they must first be converted to the same unit. Once in the same unit, divide the first quantity by the second and simplify the resulting fraction to its lowest terms.

🎯 Exam Tip: Always ensure that both quantities in a ratio are expressed in the same units before performing any calculations. This is a common pitfall that leads to incorrect answers.

Question 3. Express the following percentages as ratios
(i) 75: 100
(ii) 44: 100
(iii) 6.25%
(iv) 52: 100
(v) 0.64%
Answer:
(i) Ratio \( = 75:100 = \frac{75}{100} = \frac{25 \times 3}{25 \times 4} = \frac{3}{4} = 3:4 \)
(ii) Ratio \( = 44:100 = \frac{44}{100} = \frac{4 \times 11}{4 \times 25} = \frac{11}{25} = 11:25 \)
(iii) Ratio \( = 6.25\% = \frac{6.25}{100} = \frac{625}{100 \times 100} = \frac{25 \times 25}{25 \times 4 \times 25 \times 4} = \frac{1}{16} = 1:16 \)
(iv) Ratio \( = 52:100 = \frac{52}{100} = \frac{4 \times 13}{4 \times 25} = \frac{13}{25} = 13:25 \)
(v) Ratio \( = 0.64\% = \frac{0.64}{100} = \frac{64}{100 \times 100} = \frac{4 \times 4 \times 4}{25 \times 4 \times 25 \times 4} = \frac{4}{625} = 4:625 \)
In simple words: To convert a percentage or a ratio in the form X:100 to a reduced ratio, express it as a fraction with a denominator of 100, then simplify the fraction by dividing the numerator and denominator by their greatest common factor.

🎯 Exam Tip: Remember that a percentage is inherently a ratio out of 100. Write it as a fraction with 100 as the denominator and then simplify it to its simplest form for the ratio.

Question 4. Three persons can build a small house in 8 days. To build the same house in 6 days, how many persons are required?
Answer:
Let the persons required to build a house in 6 days be x.
Days required to build a house and number of persons are in inverse proportion.
\( \therefore 6 \times x = 8 \times 3 \)
\( \therefore 6x = 24 \)
\( \therefore x = 4 \)
\( \therefore 4 \text{ persons are required to build the house in } 6 \text{ days.} \)
In simple words: This is an inverse proportion problem; fewer days mean more people are needed. Multiply the initial number of people by the initial days, and set it equal to the unknown number of people multiplied by the new days, then solve for the unknown.

🎯 Exam Tip: Identify whether the problem involves direct or inverse proportion. For inverse proportion, the product of the two quantities remains constant (Person \( \times \) Time = Constant).

Question 5. Convert the following ratios into percentages.
(i) 15:25
(ii) 47: 50
(iii) \( \frac{7}{10} \)
(iv) \( \frac{546}{600} \)
(v) \( \frac{7}{16} \)
Answer:
(i) Let 15:25 = x %
\( \therefore \frac{15}{25} = \frac{x}{100} \)
\( \therefore x = \frac{15}{25} \times 100 = 15 \times 4 = 60\% \)
\( \therefore 15:25 = 60 \% \)
(ii) Let 47:50 = x%
\( \therefore \frac{47}{50} = \frac{x}{100} \)
\( \therefore x = \frac{47}{50} \times 100 = 47 \times 2 = 94\% \)
\( \therefore 47:50 = 94\% \)
(iii) Let \( \frac{7}{10} = x \% \)
\( \therefore \frac{7}{10} = \frac{x}{100} \)
\( \therefore x = \frac{7}{10} \times 100 = 7 \times 10 = 70\% \)
\( \therefore \frac{7}{10} = 70\% \)
(iv) Let \( \frac{546}{600} = x \% \)
\( \therefore \frac{546}{600} = \frac{x}{100} \)
\( \therefore x = \frac{546}{600} \times 100 = \frac{546}{6} = 91\% \)
\( \therefore \frac{546}{600} = 91\% \)
(v) Let \( \frac{7}{16} = x \% \)
\( \therefore \frac{7}{16} = \frac{x}{100} \)
\( \therefore x = \frac{7}{16} \times 100 = \frac{7}{4} \times 25 = \frac{175}{4} = 43.75\% \)
\( \therefore \frac{7}{16} = 43.75\% \)
In simple words: To convert a ratio or a fraction into a percentage, multiply the ratio or fraction by 100. This effectively expresses the proportion as a part of one hundred.

🎯 Exam Tip: Converting fractions to percentages involves multiplying by 100. Ensure to simplify the fraction first if possible, or perform the division to get a decimal before multiplying by 100 for accuracy.

Question 6. The ratio of ages of Abha and her mother is 2 : 5. At the time of Abha's birth her mothers age was 27 years. Find the present ages of Abha and her mother.
Answer:
The ratio of ages of Abha and her mother is 2 : 5.
Let the common multiple be x.
\( \therefore \) Present age of Abha = 2x years and Present age of Abha's mother = 5x years
According to the given condition, the age of Abha's mother at the time of Abha's birth = 27 years
\( \therefore 5x - 2x = 27 \)
\( \therefore 3x = 27 \)
\( \therefore x = 9 \)
\( \therefore \) Present age of Abha = \( 2x = 2 \times 9 = 18 \) years
\( \therefore \) Present age of Abha's mother = \( 5x = 5 \times 9 = 45 \) years
The present ages of Abha and her mother are 18 years and 45 years respectively.
In simple words: Set up the current ages using a common multiple 'x' based on the given ratio. The difference in their ages is constant, so use this information to form an equation and solve for 'x', then calculate their current ages.

🎯 Exam Tip: When dealing with age-related ratio problems, introduce a common multiple 'x' to represent actual ages. Remember that the difference in ages between two people remains constant over time.

Question 7. Present ages of Vatsala and Sara are 14 years and 10 years respectively. After how many years the ratio of their ages will become 5 : 4?
Answer:
Present age of Vatsala = 14 years
Present age of Sara = 10 years
After x years,
Vatsala's age = (14 + x) years
Sara's age = (10 + x) years
According to the given condition,
After x years the ratio of their ages will become 5:4
\( \therefore \frac{14+x}{10+x} = \frac{5}{4} \)
\( \therefore 4(14 + x) = 5(10 + x) \)
\( \therefore 56 + 4x = 50 + 5x \)
\( \therefore 56 - 50 = 5x - 4x \)
\( \therefore 6 = x \)
\( \therefore x = 6 \)
\( \therefore \) After 6 years, the ratio of their ages will become 5 : 4.
In simple words: Represent the future ages by adding 'x' (number of years) to their current ages. Set up a ratio equation with these future ages and the given target ratio, then solve for 'x' to find the number of years.

🎯 Exam Tip: For problems involving future ages, add 'x' to each person's current age. For past ages, subtract 'x'. Set up the ratio of these new ages and cross-multiply to solve the equation.

Question 8. The ratio of present ages of Rehana and her mother is 2 : 7. After 2 years, the ratio of their ages will be 1 : 3. What is Rehana's present age ?
Answer:
The ratio of present ages of Rehana and her mother is 2 : 7
Let the common multiple be x.
\( \therefore \) Present age of Rehana = 2x years and Present age of Rehana's mother = 7x years
After 2 years,
Rehana's age = (2x + 2) years
Age of Rehana's mother = (7x + 2) years
According to the given condition,
After 2 years, the ratio of their ages will be 1:3
\( \therefore \frac{2x+2}{7x+2} = \frac{1}{3} \)
\( \therefore 3(2x + 2) = 1(7x + 2) \)
\( \therefore 6x + 6 = 7x + 2 \)
\( \therefore 6 - 2 = 7x - 6x \)
\( \therefore 4 = x \)
\( \therefore x = 4 \)
\( \therefore \) Rehana's present age = \( 2x = 2 \times 4 = 8 \) years
\( \therefore \) Rehana's present age is 8 years.
In simple words: Use 'x' to define their current ages based on the initial ratio. Then, add 2 years to each age to form expressions for their ages after 2 years. Create a new ratio equation with these expressions and the new ratio (1:3), then solve for 'x' to find Rehana's current age.

🎯 Exam Tip: Clearly define variables for current ages using the given ratio. Accurately modify these age expressions for future or past scenarios before setting up and solving the ratio equation.