Question 1

The monthly salaries of Ravi and Sumit are in the ratio 2 : 3. If the salary of each is increased by Rs 4000 and the new ratio becomes 40 : 57, what is the present salary of Sumit in rupees?

Accepted Answer

Introduction / Context: This question checks understanding of ratios combined with linear equations. Salaries that increase by a fixed amount but still keep a new ratio are a standard pattern in aptitude tests and help to build algebraic thinking about proportional relationships. Given Data / Assumptions: Let the original salaries of Ravi and Sumit be in the ratio 2 : 3. Each salary is increased by Rs 4000. The new ratio of their salaries becomes 40 : 57. We have to find the present salary of Sumit in rupees. Concept / Approach: If two quantities are in the ratio m : n, we can represent them as m*k and n*k for some common factor k. When the same fixed increment is added to both, a new ratio is formed. We can use this information to form an equation in k and then solve for the actual values. This is a typical application of ratio and proportion with algebra. Step-by-Step Solution: Step 1: Let the original salaries be 2x for Ravi and 3x for Sumit. Step 2: After an increase of Rs 4000 each, the new salaries are 2x + 4000 and 3x + 4000. Step 3: We are told that (2x + 4000) : (3x + 4000) = 40 : 57. Step 4: Form the equation (2x + 4000) / (3x + 4000) = 40 / 57. Step 5: Cross multiply: 57(2x + 4000) = 40(3x + 4000). Step 6: Expand: 114x + 2,28,000 = 120x + 1,60,000. Step 7: Rearrange: 2,28,000 - 1,60,000 = 120x - 114x, so 68,000 = 6x. Step 8: Therefore x = 68,000 / 6 = 34,000 / 3. Step 9: Sumit salary = 3x = 3 * (34,000 / 3) = 34,000. Verification / Alternative check: Original salaries are 2x = 22,666.67 and 3x = 34,000 approximately. After adding Rs 4000, Ravi salary is about 26,666.67 and Sumit salary is 38,000.67 but in exact algebraic form they fit the 40 : 57 ratio perfectly. Computing 2x + 4000 and 3x + 4000 with x = 34,000 / 3 confirms that their ratio simplifies to 40 : 57, which validates our result. Why Other Options Are Wrong: Values like 38,000, 46,800, 50,000 and 36,700 do not satisfy the new ratio when you solve backward to get Ravi salary and then check the 40 : 57 proportion after adding Rs 4000. Only 34,000 is consistent with both the original ratio and the new ratio information. Common Pitfalls: Many learners wrongly treat the ratio 2 : 3 as a fixed difference 1x instead of a multiplicative relation. Another frequent error is to form the equation as (2x + 4000) : (3x + 4000) = 2 : 3, which ignores the new ratio given in the statement. Forgetting to cross multiply carefully or making arithmetic mistakes in subtraction of large numbers can also lead to wrong answers. Final Answer: The present salary of Sumit is Rs 34,000.

Question 2

A sum of Rs 312 is divided among 100 children consisting of boys and girls so that each boy receives Rs 3.60 and each girl receives Rs 2.40. How many girls are there in the group?

Accepted Answer

Introduction / Context: This problem combines simple linear equations with an everyday context of distributing money among boys and girls. It tests the ability to translate a word problem into algebraic equations and to work with two variables under given constraints. Given Data / Assumptions: Total number of children (boys plus girls) = 100 Total amount distributed = Rs 312 Each boy gets Rs 3.60 Each girl gets Rs 2.40 We must determine the number of girls. Concept / Approach: Let the number of boys and girls be variables. Use the information about total number of children to form one equation, and the information about total money distributed to form another equation. Solving these two simultaneous equations will give us the required numbers. This is a classic two variable linear equation setup frequently asked in aptitude exams. Step-by-Step Solution: Step 1: Let the number of boys be b and the number of girls be g. Step 2: From the total headcount, we have b + g = 100. Step 3: Each boy gets Rs 3.60 and each girl gets Rs 2.40, so total amount is 3.60b + 2.40g = 312. Step 4: Divide the second equation by 1.2 to simplify: 3b + 2g = 260. Step 5: Now we have the system: b + g = 100 and 3b + 2g = 260. Step 6: Multiply the first equation by 2: 2b + 2g = 200. Step 7: Subtract this from 3b + 2g = 260 to get b = 60. Step 8: Substitute b = 60 into b + g = 100 to get g = 40. Verification / Alternative check: With 60 boys and 40 girls, total amount distributed is 60 * 3.60 + 40 * 2.40. That equals 216 + 96 = 312, which matches the given total. Also, the headcount 60 + 40 is 100. Therefore, the calculated numbers satisfy both conditions and confirm that the answer is correct. Why Other Options Are Wrong: If we assume 35, 45, 50 or 30 girls, the resulting number of boys will not satisfy both the total amount and total children conditions simultaneously. Either the total money will not sum to Rs 312 or the headcount will not be exactly 100. Hence those options are inconsistent with at least one of the given equations. Common Pitfalls: A common mistake is to forget to convert the rupee values properly when removing decimals, or to mix up the coefficients when forming the total amount equation. Some learners try to solve by guessing combinations instead of setting up equations; this can be slow and error prone. It is always safer to write the equations clearly and solve them systematically. Final Answer: The number of girls in the group is 40.

Question 3

What is the compounded ratio obtained by combining the ratios (2 : 3), (6 : 11) and (11 : 2)?

Accepted Answer

Introduction / Context: This question checks the understanding of compounded ratio, which is formed by multiplying corresponding terms of two or more ratios. It is a straightforward but important concept in ratio and proportion used in many competitive exams. Given Data / Assumptions: First ratio = 2 : 3 Second ratio = 6 : 11 Third ratio = 11 : 2 We have to find the compounded ratio of these three ratios. Concept / Approach: The compounded ratio of several ratios is obtained by multiplying together all the antecedents (first terms) to get the new antecedent and multiplying together all the consequents (second terms) to get the new consequent. After that, we simplify the resulting ratio to its lowest terms by dividing both sides by their greatest common divisor. Step-by-Step Solution: Step 1: Identify the antecedent and consequent of each ratio. Step 2: Antecedents are 2, 6 and 11. Consequents are 3, 11 and 2. Step 3: Multiply the antecedents: 2 * 6 * 11 = 132. Step 4: Multiply the consequents: 3 * 11 * 2 = 66. Step 5: So the compounded ratio before simplification is 132 : 66. Step 6: Divide both sides by their greatest common divisor 66 to simplify. Step 7: 132 ÷ 66 = 2 and 66 ÷ 66 = 1, so the simplified compounded ratio is 2 : 1. Verification / Alternative check: We can quickly check by seeing that 132 : 66 clearly has a factor of 66 common to both terms. Dividing both by 66 is exact and yields 2 : 1. No further simplification is possible, so 2 : 1 is the final simplest form of the compounded ratio. Why Other Options Are Wrong: Ratios like 1 : 2, 11 : 24 or 36 : 121 come from incorrect multiplication or from reducing with a wrong common factor. For example, 11 : 24 would suggest that 132 and 66 were not multiplied correctly, while 36 : 121 would correspond to a completely different pair of products. These do not match the actual arithmetic of 2 * 6 * 11 and 3 * 11 * 2. Common Pitfalls: A common error is to add ratios instead of multiplying them or to mix antecedents with consequents incorrectly. Some students also forget to simplify the final ratio to lowest terms. Remember that compounded ratio must be formed by multiplying corresponding terms and then always simplified by dividing by the greatest common divisor. Final Answer: The compounded ratio is 2 : 1.

Question 4

Jo has a collection of US, Indian and British stamps. The ratio of US to Indian stamps is 5 : 2 and the ratio of Indian to British stamps is 5 : 1. What is the ratio of US stamps to British stamps in the collection?

Accepted Answer

Introduction / Context: This question involves chaining together two given ratios to find a third related ratio. It is a standard type of ratio and proportion problem where we must express all quantities using a common middle term to avoid confusion. Given Data / Assumptions: Ratio of US to Indian stamps = 5 : 2 Ratio of Indian to British stamps = 5 : 1 We need the ratio of US to British stamps. Concept / Approach: When two ratios share a common term, we can convert them to equivalent ratios where the common term has the same value in both ratios. Then, we can link them to find the required ratio. Here, Indian stamps are the common middle term, so we will make the Indian part equal in both ratios by appropriate multiplication and then read off the US to British relationship. Step-by-Step Solution: Step 1: Let US : Indian = 5 : 2. Step 2: Let Indian : British = 5 : 1. Step 3: To connect these, we want the Indian term in both ratios to be equal. Step 4: The first ratio has Indian = 2 parts, the second has Indian = 5 parts. Step 5: Make them equal by a common multiple. Multiply the first ratio by 5 to get US : Indian = 25 : 10. Step 6: Multiply the second ratio by 2 to get Indian : British = 10 : 2. Step 7: Now Indian = 10 in both, so US : British = 25 : 2. Verification / Alternative check: We can assume a concrete number of Indian stamps, say 10. Then US stamps would be (5 / 2) * 10 = 25 and British stamps would be (1 / 5) * 10 * 5? A simpler way: from the combined ratios we already have US = 25, Indian = 10 and British = 2. The ratio US : British is 25 : 2, which matches the derived expression. Why Other Options Are Wrong: Ratios like 10 : 5, 15 : 2 or 20 : 2 come from multiplying only one of the given ratios or from not aligning the Indian term correctly in both ratios. They do not preserve the given relationships simultaneously. Only 25 : 2 is consistent with both original ratios when the Indian term is matched. Common Pitfalls: Students sometimes add or subtract ratios or try to combine them without equalizing the common term. Another mistake is to equate 2 directly with 5 without using a common multiple, which leads to incorrect scaling. Always bring the shared middle term to a common value in both ratios before reading off the desired ratio. Final Answer: The ratio of US to British stamps is 25 : 2.

Question 5

If 0.75 : x :: 5 : 8 (that is, 0.75 is to x as 5 is to 8), what is the value of x?

Accepted Answer

Introduction / Context: This is a direct proportion question using the notation a : b :: c : d, which means a is to b as c is to d. The problem checks the ability to convert a verbal proportion into an equation and solve for an unknown variable using simple cross multiplication. Given Data / Assumptions: The proportion is 0.75 : x :: 5 : 8. This means 0.75 / x = 5 / 8. We must find the numerical value of x. Concept / Approach: In any proportion a : b :: c : d, we have a / b = c / d. Therefore, we equate the two fractions and solve for the unknown using cross multiplication. This is a fundamental technique for proportional reasoning and ratio problems. Step-by-Step Solution: Step 1: Translate the proportion into an equation: 0.75 / x = 5 / 8. Step 2: Cross multiply to remove the denominators: 0.75 * 8 = 5 * x. Step 3: Compute 0.75 * 8 = 6.0. Step 4: So 6.0 = 5x. Step 5: Divide both sides by 5 to get x = 6.0 / 5. Step 6: Therefore, x = 1.2 which can be written as 1.20. Verification / Alternative check: Check the proportion with x = 1.2. We have 0.75 / 1.2 and 5 / 8. Calculate 0.75 / 1.2 = 0.625 and 5 / 8 = 0.625. Since both fractions are equal, the proportion holds exactly. Hence x = 1.20 is correct. Why Other Options Are Wrong: Values like 1.12, 1.16, 1.30 or 1.05 do not satisfy the proportion 0.75 : x :: 5 : 8. If you plug any of them into 0.75 / x, the result will not be equal to 5 / 8, so the fundamental proportional relationship fails. They are merely distractors based on likely arithmetic slips. Common Pitfalls: A frequent error is to invert the proportion incorrectly, for example by writing x / 0.75 = 5 / 8 and then solving, which gives a different value. Another mistake is to approximate intermediate values too early instead of keeping them exact. It is also easy to confuse the colon notation and write 0.75 : x = 5 : 8 without understanding that it really means 0.75 / x = 5 / 8. Final Answer: The value of x is 1.20.

Question 6

A sum of Rs 427 is to be divided among A, B and C so that three times the share of A, four times the share of B and seven times the share of C are all equal. What is the share of C in rupees?

Accepted Answer

Introduction / Context: This question tests proportional thinking in a slightly indirect way. Instead of giving a direct ratio, it expresses relationships using equal multiples of the unknown shares. Such problems help train the ability to convert verbal relationships into algebraic expressions involving a common constant. Given Data / Assumptions: Total amount to be divided = Rs 427. Three times the share of A, four times the share of B and seven times the share of C are all equal. We must find the value of C share. Concept / Approach: If three times A share, four times B share and seven times C share are equal, then there is a common constant k such that 3A = 4B = 7C = k. From this, we can express A, B and C in terms of k, then use the total sum to solve for k and hence find C. This is equivalent to a ratio setup but expressed in terms of equal products. Step-by-Step Solution: Step 1: Let 3A = 4B = 7C = k. Step 2: Then A = k / 3, B = k / 4 and C = k / 7. Step 3: Total amount is A + B + C = 427. Step 4: Substitute the expressions: k / 3 + k / 4 + k / 7 = 427. Step 5: Find the common denominator 84 and rewrite: (28k + 21k + 12k) / 84 = 427. Step 6: This simplifies to 61k / 84 = 427. Step 7: Therefore, k = 427 * 84 / 61 = 427 * (84 / 61) = 427 * (84 / 61). Step 8: Since 61 * 7 = 427, we get k = 7 * 84 = 588. Step 9: Now C = k / 7 = 588 / 7 = 84. Verification / Alternative check: Compute A and B to verify. A = k / 3 = 588 / 3 = 196, B = k / 4 = 588 / 4 = 147 and C = 84. Total amount is 196 + 147 + 84 = 427, which matches the given sum. Also, 3A = 588, 4B = 588 and 7C = 588, so the condition about equal multiples holds exactly. Why Other Options Are Wrong: Values such as 140, 196, 240 or 112 do not satisfy both the total sum and the equal multiples condition when substituted into the equations. For example, if C were 196, then 7C would be much larger than 3A or 4B for any division totalling 427, so the core constraint would break. Common Pitfalls: Learners sometimes assume the ratio of A : B : C is 3 : 4 : 7 directly, which is incorrect because the given statement is about 3A, 4B and 7C being equal, not A, B and C themselves. Forgetting to use a common constant k or mismanaging the fraction arithmetic when summing k / 3, k / 4 and k / 7 can also cause errors. Final Answer: The share of C is Rs 84.

Question 7

In a zoo there are rabbits and pigeons. The total number of heads is 340 and the total number of legs is 1060. How many pigeons are there?

Accepted Answer

Introduction / Context: This is a classic heads and legs puzzle used to test linear equation modelling. Rabbits and pigeons have different numbers of legs, and the problem asks you to deduce how many pigeons are present using total head and leg counts. Given Data / Assumptions: There are only two types of animals: rabbits and pigeons. Total number of heads = 340. Total number of legs = 1060. Each rabbit has 4 legs. Each pigeon has 2 legs. We must find the number of pigeons. Concept / Approach: Let the number of rabbits and pigeons be variables. Each animal contributes one head, but different numbers of legs. We can write one equation for heads and one for legs, and then solve the resulting system of two linear equations. This is a direct application of algebra to a word problem. Step-by-Step Solution: Step 1: Let r be the number of rabbits and p be the number of pigeons. Step 2: From heads, we have r + p = 340. Step 3: From legs, rabbits contribute 4r legs and pigeons contribute 2p legs, so 4r + 2p = 1060. Step 4: Divide the leg equation by 2 to simplify: 2r + p = 530. Step 5: Now subtract the head equation (r + p = 340) from 2r + p = 530. Step 6: This gives (2r + p) - (r + p) = 530 - 340, so r = 190. Step 7: Substitute r = 190 into r + p = 340 to get 190 + p = 340, hence p = 150. Verification / Alternative check: Check the leg count with r = 190 and p = 150. Total legs = 190 * 4 + 150 * 2 = 760 + 300 = 1060, which matches the given number of legs. The head count is 190 + 150 = 340, also matching the given heads. Both conditions are satisfied, confirming that p = 150 is correct. Why Other Options Are Wrong: If pigeons were 120, 170, 180 or 190, the corresponding rabbit count from r + p = 340 would lead to a different total leg count that does not equal 1060. For example, with 170 pigeons, rabbits would be 170 and legs would sum to 170 * 4 + 170 * 2 = 1020, which is incorrect. Common Pitfalls: Some learners try to guess combinations without writing equations, which can be time consuming and error prone. Others mistakenly assign 2 legs to rabbits and 4 to pigeons. Another frequent error is forgetting to divide the leg equation by 2 to simplify, which makes the algebra unnecessarily complicated. Final Answer: The number of pigeons in the zoo is 150.

Question 8

Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. If the seats for these three subjects are increased by 40%, 50% and 75% respectively, what will be the new ratio of seats for Mathematics, Physics and Biology?

Accepted Answer

Introduction / Context: This question checks understanding of how percentage increases affect an existing ratio. Instead of working with actual numbers of seats, we can treat the ratio terms as basic units and apply the percentage changes directly to them. This technique is useful whenever proportional scaling is involved. Given Data / Assumptions: Initial ratio of seats for Mathematics, Physics, Biology = 5 : 7 : 8. Mathematics seats are increased by 40%. Physics seats are increased by 50%. Biology seats are increased by 75%. We must find the new ratio of seats. Concept / Approach: We can assume the original number of seats as 5k, 7k and 8k respectively. A percentage increase changes each term by a multiplicative factor. For example, a 40% increase corresponds to multiplying by 1.40. After updating each term, we obtain new numbers that can be simplified to their smallest integer ratio by dividing by a common factor. Step-by-Step Solution: Step 1: Let original seats be 5k, 7k and 8k. Step 2: Increase Mathematics seats by 40%. New Mathematics seats = 5k * 1.40 = 7k. Step 3: Increase Physics seats by 50%. New Physics seats = 7k * 1.50 = 10.5k. Step 4: Increase Biology seats by 75%. New Biology seats = 8k * 1.75 = 14k. Step 5: The new ratio before simplification is 7k : 10.5k : 14k. Step 6: Remove the common factor k to get 7 : 10.5 : 14. Step 7: To avoid decimals, multiply all terms by 2 to get 14 : 21 : 28. Step 8: Divide all terms by 7 to simplify: 14 ÷ 7 = 2, 21 ÷ 7 = 3 and 28 ÷ 7 = 4. Step 9: So the simplified new ratio is 2 : 3 : 4. Verification / Alternative check: We can select a convenient value for k, for example k = 10. Then original seats are 50, 70 and 80. After the respective increases, Mathematics has 70 seats, Physics has 105 seats and Biology has 140 seats. The ratio 70 : 105 : 140 simplifies by dividing all terms by 35 to 2 : 3 : 4, which confirms the earlier result. Why Other Options Are Wrong: Ratios such as 1 : 2 : 3 or 3 : 4 : 5 and 4 : 5 : 6 are not equivalent to 2 : 3 : 4 when you compare the relative increases implied. They do not match the actual scaled seat counts after applying 40%, 50% and 75% increases to the original ratio 5 : 7 : 8. Common Pitfalls: Learners sometimes add the percentage values directly to the ratio numbers rather than treating the percentages as multipliers. For example, adding 40 to 5 instead of doing 5 * 1.40 is incorrect. Another mistake is to forget to simplify the final numbers to their lowest integer ratio, leaving answers with decimals instead of neat whole numbers. Final Answer: The new ratio of seats for Mathematics, Physics and Biology is 2 : 3 : 4.

Question 9

What is the third proportional to the numbers 9 and 12?

Accepted Answer

Introduction / Context: The concept of third proportional is a classic topic in ratio and proportion. It helps in understanding proportional relationships where three numbers satisfy a specific ratio condition and is often tested in quantitative aptitude sections of competitive exams. Given Data / Assumptions: We are given two numbers: 9 and 12. We must find the third proportional to 9 and 12. Concept / Approach: If a, b and c are such that a : b = b : c, then c is called the third proportional to a and b. In equation form, a / b = b / c, which implies a * c = b * b. So c = b^2 / a. We will substitute a = 9 and b = 12 into this formula to find c. Step-by-Step Solution: Step 1: Let the required third proportional be x. Step 2: By definition, 9 : 12 = 12 : x. Step 3: Convert this ratio statement into a fraction equation: 9 / 12 = 12 / x. Step 4: Cross multiply to get 9x = 12 * 12. Step 5: Compute 12 * 12 = 144. Step 6: So 9x = 144. Therefore x = 144 / 9. Step 7: Simplify 144 / 9 = 16. Step 8: Hence the third proportional is 16. Verification / Alternative check: Check whether the ratio 9 : 12 equals 12 : 16. Compute 9 / 12 = 0.75 and 12 / 16 = 0.75. Since they are equal, the condition a : b = b : c is satisfied. This confirms that 16 is indeed the correct third proportional to 9 and 12. Why Other Options Are Wrong: Values like 14, 18, 20 or 24 do not satisfy the proportional relationship 9 : 12 = 12 : x. For each of these, 12 / x gives a fraction that is not equal to 9 / 12, so they cannot be the third proportional. For example, 12 / 18 = 2 / 3 which is different from 9 / 12 = 3 / 4. Common Pitfalls: Students sometimes confuse third proportional with mean proportional (geometric mean). For a third proportional, the equation is 9 : 12 = 12 : x, not 9 : x = x : 12. Another error is to attempt to subtract numbers or take an average instead of using the correct proportion formula. Always remember that third proportional uses the pattern a : b = b : c which translates into b^2 = a * c. Final Answer: The third proportional to 9 and 12 is 16.

Question 10

The salaries of A, B and C are in the ratio 1 : 2 : 3. The combined salary of B and C is Rs 6000. By what percentage is the salary of C more than the salary of A?

Accepted Answer

Introduction / Context: This question combines simple ratio with percentage comparison. It is a typical aptitude problem where we first convert a ratio into actual values using a given sum, and then compare two of the resulting quantities as a percentage difference. Given Data / Assumptions: Ratio of salaries A : B : C = 1 : 2 : 3. Combined salary of B and C is Rs 6000. We must find how many percent the salary of C is more than the salary of A. Concept / Approach: Let the common ratio factor be k, so A = k, B = 2k and C = 3k. Then B + C = 5k. Using the given total for B and C, we can find k and then compute A and C. The percentage by which C is more than A is calculated as (C - A) / A * 100. Step-by-Step Solution: Step 1: Let A = k, B = 2k and C = 3k. Step 2: Then B + C = 2k + 3k = 5k. Step 3: It is given that B + C = Rs 6000, so 5k = 6000. Step 4: Therefore k = 6000 / 5 = 1200. Step 5: So A = 1200, B = 2400 and C = 3600. Step 6: We need the percentage by which C is more than A. Step 7: Difference C - A = 3600 - 1200 = 2400. Step 8: Percentage increase from A to C = (2400 / 1200) * 100 = 2 * 100 = 200%. Verification / Alternative check: Notice that in the ratio 1 : 2 : 3, C is three times A. Being three times larger means an increase of two times A over the base A, which is a 200% increase. This reasoning independently confirms the earlier calculation based on actual salaries. Why Other Options Are Wrong: Options like 100%, 300% or 600% correspond to misinterpreting the ratio or the calculation formula. For instance, 100% would mean C is only double of A, which is not correct since 3k is three times k. A value of 300% would imply C is four times A, which also does not match the given ratio. Common Pitfalls: Some learners mistakenly compute the percentage as C / A * 100 instead of (C - A) / A * 100, which omits the idea of increase over the base. Others misread the question and compute how much A is less than C rather than how much C is more than A, although in this case the numerical result for percentage is the same, the interpretation can still cause confusion. Final Answer: The salary of C is 200% more than the salary of A.

Question 11

A dog takes 3 leaps for every 5 leaps of a hare. If one leap of the dog is equal in length to 3 leaps of the hare, what is the ratio of the speed of the dog to the speed of the hare?

Accepted Answer

Introduction / Context: This question involves relative speed expressed through leap counts and leap lengths. It checks understanding that speed is distance travelled per unit time, and both frequency of leaps and the length of each leap contribute to the effective speed of an animal. Given Data / Assumptions: The dog takes 3 leaps while the hare takes 5 leaps in the same time. One leap of the dog covers the same distance as 3 leaps of the hare. We must find the ratio of the speed of the dog to the speed of the hare. Concept / Approach: Speed is proportional to (number of leaps per unit time) multiplied by (length of each leap). If we assume a unit of time (for example one second), we can calculate distances covered by each animal in that time and compare them. Using symbolic leap length for the hare makes the comparison easy. Step-by-Step Solution: Step 1: Let length of one hare leap be L units. Step 2: Then length of one dog leap is 3L, because it is equal to 3 hare leaps. Step 3: In a fixed time interval, the dog takes 3 leaps. Step 4: In the same time, the hare takes 5 leaps. Step 5: Distance covered by dog in that time = number of leaps * length per leap = 3 * 3L = 9L. Step 6: Distance covered by hare in that time = 5 * L = 5L. Step 7: Since time is the same for both, speed is directly proportional to distance covered. Step 8: Therefore, speed of dog : speed of hare = 9L : 5L = 9 : 5. Verification / Alternative check: We can choose a concrete value for L, for example L = 1 metre. Then in one time unit, the dog covers 9 metres and the hare covers 5 metres. Their speed ratio is 9 : 5. Any other positive value of L would cancel out in the ratio, so 9 : 5 is robust and independent of actual leap length. Why Other Options Are Wrong: Ratios like 2 : 3, 4 : 7 or 5 : 6 ignore either the difference in leap frequency or the difference in leap length. For instance, 3 leaps versus 5 leaps alone would give 3 : 5, while leap length alone of 3L versus L would give 3 : 1. Correct speed comparison must combine both effects, leading to 9 : 5 rather than any of the distractor ratios. Common Pitfalls: A common mistake is to take ratio of leaps only (3 : 5) and ignore leap length, or to use only leap length and forget that the hare makes more leaps in the same time. Another error is to multiply the ratios incorrectly instead of multiplying the actual distance components. Always remember that speed depends on both frequency and distance per action. Final Answer: The ratio of the speed of the dog to the speed of the hare is 9 : 5.

Question 12

In a college, the ratio of boys to girls is 8 : 5. If there are 160 girls in the college, what is the total number of students?

Accepted Answer

Introduction / Context: This problem tests the ability to apply a simple ratio to find total numbers when one category count is known. It is a direct and common use of ratios in everyday contexts such as class composition and population data. Given Data / Assumptions: Ratio of boys to girls = 8 : 5. Number of girls = 160. We must find the total number of students in the college. Concept / Approach: If boys : girls = 8 : 5, we can represent the numbers as 8k and 5k for some positive factor k. The actual count of girls is 5k, which is given as 160. Solving for k allows us to compute the total number of students as 8k + 5k. Step-by-Step Solution: Step 1: Let number of boys = 8k and number of girls = 5k. Step 2: It is given that the number of girls is 160, so 5k = 160. Step 3: Solve for k by dividing 160 by 5: k = 160 / 5 = 32. Step 4: Now find the number of boys: boys = 8k = 8 * 32 = 256. Step 5: Total number of students = boys + girls = 256 + 160. Step 6: Total = 416. Verification / Alternative check: We can verify by checking the ratio of 256 to 160. Compute 256 / 160 = 1.6 which is the same as 8 / 5 (since 8 / 5 = 1.6). This confirms that the boy to girl ratio remains 8 : 5 and the total 416 is consistent with the given data. Why Other Options Are Wrong: Totals like 100, 250, 260 or 520 do not yield 160 girls when the ratio 8 : 5 is applied. For example, if total is 260, the girls count would be 5 / 13 * 260 = 100, which contradicts the condition of 160 girls. Thus these options are inconsistent with the given ratio and the provided count of girls. Common Pitfalls: Learners sometimes treat 8 and 5 as actual counts rather than as parts of a ratio. Another mistake is to divide 160 by 8 instead of 5, or to add 8 and 5 incorrectly. Remember that the known count corresponds to its ratio part (5k in this case), and you must solve for k before calculating the total. Final Answer: The total number of students in the college is 416.

Question 13

The concentration of petrol in three different mixtures of petrol and kerosene is 1/2, 3/5 and 4/5 respectively. If 2 litres, 3 litres and 1 litre are taken from these three mixtures and combined, what is the ratio of petrol to kerosene in the resul…

Accepted Answer

Introduction / Context: This question involves mixing solutions with different concentrations and then finding the composition of the resulting mixture. It tests comfort with fractions, weighted averages and the idea that total amount of petrol and kerosene is additive across the mixtures. Given Data / Assumptions: Mixture 1: petrol fraction = 1/2, kerosene fraction = 1/2, volume taken = 2 litres. Mixture 2: petrol fraction = 3/5, kerosene fraction = 2/5, volume taken = 3 litres. Mixture 3: petrol fraction = 4/5, kerosene fraction = 1/5, volume taken = 1 litre. We must find petrol : kerosene in the combined mixture. Concept / Approach: We calculate the quantity of petrol and kerosene from each mixture separately using the relation quantity = fraction * total volume. Then we add petrol amounts together and kerosene amounts together. The ratio of these totals gives the composition in the final mixture. This is a straightforward application of mixture and alligation concepts. Step-by-Step Solution: Step 1: From mixture 1 (2 litres, petrol 1/2), petrol quantity = 2 * 1/2 = 1 litre and kerosene quantity = 2 * 1/2 = 1 litre. Step 2: From mixture 2 (3 litres, petrol 3/5), petrol quantity = 3 * 3/5 = 9/5 litres = 1.8 litres and kerosene quantity = 3 * 2/5 = 6/5 litres = 1.2 litres. Step 3: From mixture 3 (1 litre, petrol 4/5), petrol quantity = 1 * 4/5 = 0.8 litres and kerosene quantity = 1 * 1/5 = 0.2 litres. Step 4: Total petrol = 1 + 1.8 + 0.8 = 3.6 litres. Step 5: Total kerosene = 1 + 1.2 + 0.2 = 2.4 litres. Step 6: Petrol : kerosene in final mixture = 3.6 : 2.4. Step 7: Divide both sides by 1.2 to simplify: 3.6 ÷ 1.2 = 3 and 2.4 ÷ 1.2 = 2. Step 8: So the ratio becomes 3 : 2. Verification / Alternative check: Total mixture volume is 2 + 3 + 1 = 6 litres. Petrol fraction in final mixture is 3.6 / 6 = 0.6, kerosene fraction is 2.4 / 6 = 0.4. The ratio 0.6 : 0.4 simplifies to 6 : 4 which further simplifies to 3 : 2. This confirms that the ratio 3 : 2 is consistent. Why Other Options Are Wrong: Ratios such as 4 : 5, 3 : 5 or 2 : 3 represent either an overall lower or higher petrol fraction, which does not match the computed 60% petrol and 40% kerosene. For example, 2 : 3 would correspond to 40% petrol and 60% kerosene, which contradicts our calculations. Common Pitfalls: One common mistake is to average the fractions directly as (1/2 + 3/5 + 4/5) / 3 without weighting by volumes, which is wrong. Another error is to add the ratios instead of quantities. Always remember that ratios describe relative amounts but actual computations must be done on absolute quantities obtained by multiplying fractions with volumes. Final Answer: The ratio of petrol to kerosene in the resulting mixture is 3 : 2.

Question 14

Alloy A contains 40% gold and 60% silver. Alloy B contains 35% gold, 40% silver and 25% copper. If alloys A and B are mixed in the ratio 1 : 4 by weight, what are the respective percentages of gold and silver in the newly formed alloy?

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Introduction / Context: This question is about mixing two alloys with known percentage compositions to form a new alloy. It tests understanding of weighted averages and percentage calculations when different components like gold, silver and copper are involved. Given Data / Assumptions: Alloy A: 40% gold and 60% silver. Alloy B: 35% gold, 40% silver and 25% copper. Alloys A and B are mixed in the ratio 1 : 4 by weight. We must find the percentage of gold and the percentage of silver in the new alloy. Concept / Approach: Assume convenient total weights for the alloys according to the given mixing ratio, for example 1 kg of alloy A and 4 kg of alloy B. Compute the mass of gold and silver from each alloy using the given percentages. Add the masses and then express them as percentages of the total mass of the new alloy. Copper is present but only gold and silver percentages are required. Step-by-Step Solution: Step 1: Assume we mix 1 kg of alloy A and 4 kg of alloy B (total 5 kg). Step 2: In 1 kg of alloy A, gold = 40% of 1 kg = 0.40 kg and silver = 60% of 1 kg = 0.60 kg. Step 3: In 4 kg of alloy B, gold = 35% of 4 kg = 0.35 * 4 = 1.40 kg. Step 4: In 4 kg of alloy B, silver = 40% of 4 kg = 0.40 * 4 = 1.60 kg. Step 5: Total gold in new alloy = 0.40 + 1.40 = 1.80 kg. Step 6: Total silver in new alloy = 0.60 + 1.60 = 2.20 kg. Step 7: Total mass of new alloy = 1 + 4 = 5 kg. Step 8: Percentage of gold = (1.80 / 5) * 100 = 36%. Step 9: Percentage of silver = (2.20 / 5) * 100 = 44%. Verification / Alternative check: Gold percentage 36 and silver percentage 44 sum to 80%. The remaining 20% of the alloy must be copper, which is consistent with the fact that only alloy B originally contained copper and that it occupied 25% of B but only four fifths of the total mixture is B. Thus the distribution is internally consistent and the calculations are correct. Why Other Options Are Wrong: Options such as 20% and 30% or 25% and 35% would imply total gold and silver percentages far below the 75 to 80 percent range implied by the original compositions, which is not possible given that alloy A has 100% gold and silver and alloy B has 75% gold and silver combined. Only 36% and 44% align with the weighted contribution from each alloy. Common Pitfalls: A common mistake is to average the percentages directly as (40 + 35) / 2 without weighting by the mixing ratio 1 : 4. Another error is to forget that the total alloy mass changes when mixing, so gold and silver percentages must be taken over the total 5 parts, not individually. Careless handling of percentage to mass conversion also causes confusion, so always express quantities explicitly in kilograms or grams before computing final percentages. Final Answer: In the new alloy, gold is 36% and silver is 44% by weight.

Question 15

The numbers of oranges in three baskets are in the ratio 3 : 4 : 5. By what ratio must the numbers of oranges in the first two baskets be increased (keeping the third basket unchanged) so that the new ratio of the three baskets becomes 5 : 4 : 3?

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Introduction / Context: This question deals with adjusting quantities in two of three groups so that the overall ratio changes to a new specified ratio while one group remains unchanged. It tests the ability to set up algebraic equations from ratio conditions and to think carefully about which quantities are changed. Given Data / Assumptions: Original ratio of oranges in three baskets (first : second : third) = 3 : 4 : 5. The number of oranges in the third basket remains unchanged. After increasing the numbers in the first two baskets, the new ratio becomes 5 : 4 : 3. We must find the ratio in which the numbers of oranges in the first and second baskets are increased. Concept / Approach: Let the original numbers be 3k, 4k and 5k. After changes, let the new numbers be 5t, 4t and 3t according to the new ratio. Since the third basket is unchanged, 5k must equal 3t, which links k and t. The increments in the first and second baskets can then be expressed in terms of k, and their ratio gives the required answer. Step-by-Step Solution: Step 1: Let original numbers be 3k, 4k and 5k. Step 2: New ratio is 5 : 4 : 3, so new numbers can be written as 5t, 4t and 3t. Step 3: The third basket is unchanged, so 5k = 3t. Step 4: From 5k = 3t, we get t = 5k / 3. Step 5: New number in first basket = 5t = 5 * (5k / 3) = 25k / 3. Step 6: New number in second basket = 4t = 4 * (5k / 3) = 20k / 3. Step 7: Increase in first basket = (25k / 3) - 3k = (25k - 9k) / 3 = 16k / 3. Step 8: Increase in second basket = (20k / 3) - 4k = (20k - 12k) / 3 = 8k / 3. Step 9: Ratio of increases in first and second baskets = (16k / 3) : (8k / 3) = 16 : 8 = 2 : 1. Verification / Alternative check: Pick a convenient value for k, for example k = 3. Then original numbers are 9, 12 and 15. The new numbers, according to ratio 5 : 4 : 3, must be in the form 5t, 4t and 3t with 3t = 15, so t = 5. Thus new numbers are 25, 20 and 15. Increases in first and second baskets are 25 - 9 = 16 and 20 - 12 = 8, which are in the ratio 16 : 8 = 2 : 1, matching our earlier result. Why Other Options Are Wrong: Ratios such as 1 : 3, 3 : 4 or 2 : 3 do not satisfy the constraints when tested with actual numbers. For these, the adjustments to the first and second baskets would not lead to the new overall ratio 5 : 4 : 3 while keeping the third basket fixed at 5k. Common Pitfalls: A typical mistake is to change all three baskets, which violates the condition that the third basket remains unchanged. Another error is to assume that the new ratio must share the same base multiple k instead of introducing a new variable t. Misinterpreting ratios as absolute numbers rather than relative quantities can also cause confusion. Final Answer: The numbers of oranges in the first and second baskets must be increased in the ratio 2 : 1.

Question 16

What is the third proportional to the numbers 16 and 36?

Accepted Answer

Introduction / Context: This problem again uses the concept of third proportional, similar to earlier questions but with different numbers. It reinforces the formula and the understanding that third proportional is defined through a specific proportional relationship between three numbers. Given Data / Assumptions: First number = 16. Second number = 36. We must find the third proportional. Concept / Approach: If a, b and c are in proportion such that a : b = b : c, then c is called the third proportional to a and b. In equation form this is a / b = b / c leading to a * c = b^2 and therefore c = b^2 / a. We will substitute a = 16 and b = 36 into this formula. Step-by-Step Solution: Step 1: Let the third proportional be x. Step 2: By definition, 16 : 36 = 36 : x. Step 3: Convert this to a fraction equation: 16 / 36 = 36 / x. Step 4: Cross multiply: 16x = 36 * 36. Step 5: Compute 36 * 36 = 1296. Step 6: So 16x = 1296, which gives x = 1296 / 16. Step 7: Simplify 1296 / 16 by dividing numerator and denominator by 16: 1296 ÷ 16 = 81. Step 8: Hence the third proportional is 81. Verification / Alternative check: Check if 16 : 36 equals 36 : 81. Compute 16 / 36 = 4 / 9, and 36 / 81 = 4 / 9 after simplification. Since both ratios are equal, the proportional relation holds and confirms that 81 is the correct third proportional to 16 and 36. Why Other Options Are Wrong: Options such as 79, 84 or 88 do not satisfy the equality 16 : 36 = 36 : x. For these values, the fraction 36 / x will not simplify to 4 / 9. For instance, 36 / 84 simplifies to 3 / 7, which is not equal to 4 / 9, so 84 cannot be the third proportional. Common Pitfalls: A common mistake is to mix up third proportional with mean proportional and wrongly set 16 : x = x : 36. Another error is to perform the square on the wrong number or to miscalculate 36 squared. Remember that the formula c = b^2 / a comes from the proportionality a / b = b / c, which is the defining property of third proportional. Final Answer: The third proportional to 16 and 36 is 81.

Question 17

Rs 6500 is to be divided among A, B and C so that after A spends 90% of his share, B spends 75% of her share and C spends 60% of his share, their remaining savings are in the ratio 3 : 5 : 6. What are the respective shares of A, B and C?

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Introduction / Context: This problem involves reverse working from final savings back to original shares when different spending percentages are applied. It combines ratio, percentage and linear equations in a practical money distribution context, which is common in aptitude exams. Given Data / Assumptions: Total amount to be divided = Rs 6500. A spends 90% of his share, so he saves 10%. B spends 75% of her share, so she saves 25%. C spends 60% of his share, so he saves 40%. Their savings are in the ratio 3 : 5 : 6. We must find the original shares of A, B and C. Concept / Approach: Let the savings of A, B and C be 3k, 5k and 6k for some k. Relate these savings back to their original shares by using the percentage of saving for each person. Then use the fact that the sum of the original shares equals 6500 to determine k and hence each share. This is a standard method for such reverse percentage problems. Step-by-Step Solution: Step 1: Let savings of A, B and C be 3k, 5k and 6k. Step 2: A saves 10% of his share. So 0.10 * share of A = 3k, giving share of A = 3k / 0.10 = 30k. Step 3: B saves 25% of her share. So 0.25 * share of B = 5k, giving share of B = 5k / 0.25 = 20k. Step 4: C saves 40% of his share. So 0.40 * share of C = 6k, giving share of C = 6k / 0.40 = 15k. Step 5: Total of all shares = share of A + share of B + share of C = 30k + 20k + 15k = 65k. Step 6: It is given that the total is Rs 6500, so 65k = 6500. Step 7: Therefore k = 6500 / 65 = 100. Step 8: Share of A = 30k = 30 * 100 = Rs 3000. Step 9: Share of B = 20k = 20 * 100 = Rs 2000. Step 10: Share of C = 15k = 15 * 100 = Rs 1500. Verification / Alternative check: Compute savings from these shares. A saves 10% of 3000 = 300. B saves 25% of 2000 = 500. C saves 40% of 1500 = 600. Savings ratio is 300 : 500 : 600. Dividing by 100 gives 3 : 5 : 6, exactly as required. Sum of shares 3000 + 2000 + 1500 is 6500, matching the total amount. Thisdouble check confirms the shares are correct. Why Other Options Are Wrong: Other combinations such as A = 1000, B = 2000, C = 1500 do not satisfy both the total Rs 6500 and the required savings ratio when the spending percentages are applied. At least one of the constraints will fail for each incorrect option, especially the precise 3 : 5 : 6 savings ratio. Common Pitfalls: Some learners confuse spending percentage with saving percentage, using 90% as the saving for A instead of 10%. Another common mistake is to allocate the 3 : 5 : 6 ratio directly to the shares instead of to the savings. It is important to read carefully that the ratio applies to savings after different spending percentages, not to the original amounts themselves. Final Answer: The shares are A = Rs 3000, B = Rs 2000 and C = Rs 1500.

Question 18

The ratio of males to females in a city is 7 : 8. Among males, 25% are children and among females, 20% are children. If the number of adult females is 1,56,800, what is the total population of the city?

Accepted Answer

Introduction / Context: This question tests combined use of ratios and percentages in a population problem. It requires careful distinction between adults and children among both male and female populations and then reconstruction of the total population from partial information. Given Data / Assumptions: Ratio of males to females = 7 : 8. Among males, 25% are children and 75% are adults. Among females, 20% are children and 80% are adults. Number of adult females = 1,56,800. We must find the total population (males plus females, adults plus children). Concept / Approach: Let the numbers of males and females be 7x and 8x respectively. Since 80% of females are adults, adult females count is 0.80 * 8x which is given. Solving for x provides actual male and female populations. The total population is then the sum of 7x and 8x. The internal adult and child counts can also be verified using the given percentages. Step-by-Step Solution: Step 1: Let number of males = 7x and number of females = 8x. Step 2: Among females, 80% are adults, so adult females = 0.80 * 8x = 6.4x. Step 3: Adult female population is given as 1,56,800, so 6.4x = 1,56,800. Step 4: Solve for x: x = 1,56,800 / 6.4. Step 5: Calculate x = 24,500. Step 6: Total males = 7x = 7 * 24,500 = 1,71,500. Step 7: Total females = 8x = 8 * 24,500 = 1,96,000. Step 8: Total population = 1,71,500 + 1,96,000 = 3,67,500. Verification / Alternative check: Check adult females: 80% of 1,96,000 = 0.80 * 1,96,000 = 1,56,800, which matches the given data. Adult males would be 75% of 1,71,500 = 1,28,625 and male children 42,875. Female children are 20% of 1,96,000 = 39,200. Sum of all four categories (adult males, adult females, male children, female children) is 1,28,625 + 1,56,800 + 42,875 + 39,200 = 3,67,500, which agrees with the total population computed earlier. Why Other Options Are Wrong: Other totals such as 4,12,480 or 5,44,700 would not yield an adult female population exactly equal to 1,56,800 when the ratio 7 : 8 and the given percentages are applied. They fail to maintain consistency between overall ratio, adult percentages and the specific adult female count. Common Pitfalls: A common mistake is to assume that 1,56,800 is total females instead of adult females, which leads to wrong calculations. Another error is miscomputing the adult percentage, for example using 20% instead of 80% for adult females. Always pay close attention to whether a percentage relates to adults or children, and to which subgroup it applies. Final Answer: The total population of the city is 3,67,500.

Question 19

What is the third proportional to x² − y² and x − y?

Accepted Answer

Introduction / Context: This algebraic question generalises the idea of third proportional to symbolic expressions involving x and y. It requires comfort with both the definition of proportionality and algebraic factorisation, particularly the difference of squares identity. Given Data / Assumptions: First term a = x² − y². Second term b = x − y. We must find the third proportional c such that a : b = b : c. Concept / Approach: For numbers a, b and c in proportion a : b = b : c, the relation a / b = b / c holds. This implies a * c = b² and c = b² / a. Here, a is x² − y² and b is x − y. We use the identity x² − y² = (x − y)(x + y) to simplify b² / a and to obtain a simplified expression for the third proportional. Step-by-Step Solution: Step 1: Let the third proportional be T. Step 2: By definition, (x² − y²) : (x − y) = (x − y) : T. Step 3: This means (x² − y²) / (x − y) = (x − y) / T. Step 4: Rearranging, (x² − y²) * T = (x − y)². Step 5: Therefore T = (x − y)² / (x² − y²). Step 6: Factorise x² − y² using the difference of squares: x² − y² = (x − y)(x + y). Step 7: Substitute this in the expression for T: T = (x − y)² / [(x − y)(x + y)]. Step 8: Cancel one common factor (x − y) from numerator and denominator, assuming x ≠ y. Step 9: This gives T = (x − y) / (x + y). Verification / Alternative check: Check that the proportion holds: (x² − y²) : (x − y) = (x − y) : T. Since x² − y² = (x − y)(x + y), the first ratio becomes [(x − y)(x + y)] : (x − y) which simplifies to (x + y) : 1. The second ratio is (x − y) : [(x − y) / (x + y)] which simplifies, by dividing numerator and denominator by (x − y), to (x + y) : 1 as well. Thus both ratios are equal, verifying that T is correct. Why Other Options Are Wrong: Expressions such as x + y or x − y alone do not satisfy the proportionality a : b = b : c for a = x² − y² and b = x − y. Likewise, (x + y) / (x − y) inverts the required expression. Only (x − y) / (x + y) maintains the equality of ratios when the algebraic simplifications are carried out properly. Common Pitfalls: Some learners forget the identity x² − y² = (x − y)(x + y) or cancel terms incorrectly. Others mistakenly treat the relation as a three-term arithmetic progression rather than a proportion. It is also easy to invert the final fraction and select (x + y) / (x − y) if the cross multiplication step is rushed. Careful factorisation and cancellation are essential. Final Answer: The third proportional to x² − y² and x − y is (x − y) / (x + y).

Question 20

If the integer 76 is divided into four parts that are proportional to 7 : 5 : 3 : 4, what is the numerical value of the smallest part obtained from this division?

Accepted Answer

Introduction / Context: In many aptitude exams, questions involving division of a number into parts in a given ratio are very common. Such questions test a candidate's understanding of ratios, proportional division and basic arithmetic manipulation. In this problem, the total number 76 is split into four parts that are proportional to 7, 5, 3 and 4 respectively. We are asked to determine the smallest of these four parts, which is a straightforward application of the concept of ratios and proportional allocation of a total. Given Data / Assumptions: • Total value to be divided = 76. • The four parts are in the ratio 7 : 5 : 3 : 4. • Parts are directly proportional to these ratio values. • All parts are positive real numbers. Concept / Approach: The key idea is that if a quantity S is divided into parts in the ratio a : b : c : d, then the actual parts are S * a / (a + b + c + d), S * b / (a + b + c + d), and so on. Therefore, we first compute the sum of the ratio terms, then find the common multiplying factor, and finally compute each individual part. The smallest ratio term will correspond to the smallest part, provided the total is positive. Step-by-Step Solution: Step 1: Sum of ratio terms = 7 + 5 + 3 + 4 = 19. Step 2: Common multiplying factor = total / sum of ratios = 76 / 19. Step 3: Compute 76 / 19. Since 19 * 4 = 76, the factor is 4. Step 4: Actual parts = 7*4, 5*4, 3*4, 4*4 = 28, 20, 12, 16. Step 5: Among these, the smallest part is 12. Verification / Alternative check: We can quickly verify by adding the four parts: 28 + 20 + 12 + 16 = 76, which matches the original total, so the proportional division is correct. Also, the ratio 28 : 20 : 12 : 16 simplifies to 7 : 5 : 3 : 4 if each term is divided by 4. This confirms both the correctness of the factor and the parts. Why Other Options Are Wrong: • 16 is one of the parts but it is not the smallest; 12 is smaller. • 19 does not appear as any part; it is only close to the ratio sum. • 15 is not part of the computed distribution and does not satisfy the ratio. Common Pitfalls: A frequent mistake is to divide 76 by 4, assuming there are four parts of equal size, ignoring the given ratio. Another common error is to use the smallest ratio term (3) directly as the answer without scaling it by the common factor. Missing the step of summing the ratio terms can also lead to incorrect factors and wrong final parts. Final Answer: Hence, the smallest part obtained when 76 is divided in the ratio 7 : 5 : 3 : 4 is 12.

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