Question 1

A profit of Rs. 960 is shared between A and B in the ratio 1/3 : 1/2. What is the difference in rupees between the amounts of profit received by A and B?

Accepted Answer

Introduction / Context: This problem involves dividing a total profit between two people, A and B, according to a ratio expressed in fractional form. The question then asks for the difference between their individual shares. It tests understanding of ratios, handling of fractions, and the ability to translate a profit sharing agreement into actual rupee amounts. Given Data / Assumptions: • Total profit to be shared between A and B is Rs. 960.• The ratio of their shares is 1/3 : 1/2.• The ratio refers to the relative quantities of profit received by A and B.• We must find the difference between B's share and A's share in rupees. Concept / Approach: To work with a ratio like 1/3 : 1/2, it is convenient to clear denominators by multiplying both terms by the least common multiple of 3 and 2, which is 6. This converts the ratio into whole numbers. Once we have a standard ratio in whole numbers, we can represent the shares as multiples of a common variable. Then we use the condition that their sum is equal to Rs. 960 to determine the value of that variable and hence each individual share. The final step is to subtract A's share from B's share to obtain the required difference. Step-by-Step Solution: Step 1: The ratio is 1/3 : 1/2.Step 2: Multiply both terms by 6 to remove fractions: (1/3)*6 = 2 and (1/2)*6 = 3, so the ratio becomes 2 : 3.Step 3: Let A's share be 2k and B's share be 3k, where k is a common multiplier.Step 4: The total profit is 960, so 2k + 3k = 960.Step 5: Combine like terms: 5k = 960.Step 6: Solve for k: k = 960 / 5 = 192.Step 7: Compute A's share: 2k = 2 * 192 = 384.Step 8: Compute B's share: 3k = 3 * 192 = 576.Step 9: Difference between their shares = 576 − 384 = 192 rupees. Verification / Alternative check: We can check by confirming that the ratio of the actual shares equals 1/3 : 1/2. First, form the ratio 384 : 576. Divide both numbers by 192: 384 / 192 = 2 and 576 / 192 = 3, giving the simplified ratio 2 : 3. Now match this with the original fractional ratio. Since 2 : 3 corresponds to (1/3)*6 : (1/2)*6, it is consistent. Also, the total 384 + 576 = 960 equals the given profit. Therefore, the calculation is correct and the difference of Rs. 192 is verified. Why Other Options Are Wrong: • Rs. 120: This would come from using the wrong total or from misinterpreting the ratio.• Rs. 160: This might come from mistakenly taking 1/5 of the total as the difference, but that is not how the distribution works here.• Rs. 240: This value would not preserve both the correct ratio and the correct total profit when back calculated. Common Pitfalls: Some learners forget to convert the fractional ratio into a whole number ratio and try to work directly with fractions in a confusing way. Others may mistakenly add the fractions 1/3 and 1/2 incorrectly or treat them as percentages. Another error is to miscalculate the value of k by incorrectly dividing the total by something other than the sum of the ratio parts. Care must also be taken not to confuse the difference between the ratios (such as 3 − 2) with the difference between the actual rupee amounts. Each step must be done systematically to avoid these mistakes. Final Answer: The difference between the profit shares of A and B is Rs. 192.

Question 2

Find the third proportional to the two numbers 9 and 24, that is, the number which forms a proportion 9 : 24 = 24 : x.

Accepted Answer

Introduction / Context: This question tests the concept of third proportional in ratio and proportion. The idea of third proportional is fundamental in many algebra and aptitude problems, where a proportion of the form a : b = b : c is used. Here, the numbers 9 and 24 are given and we must find the third proportional that completes the proportion with 24 as the middle term. Given Data / Assumptions: • The first number is 9.• The second number is 24.• We need to find the third proportional x such that 9 : 24 = 24 : x.• All numbers are positive real numbers. 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 such a proportion, we have a / b = b / c. Cross multiplication then gives a * c = b * b. From this relation, c can be found as c = (b^2) / a. Applying this directly to 9 and 24 allows us to compute the third proportional quickly and accurately. Step-by-Step Solution: Step 1: Let the third proportional be x so that 9 : 24 = 24 : x.Step 2: Write this proportion in fractional form: 9 / 24 = 24 / x.Step 3: Cross multiply to eliminate the fractions: 9 * x = 24 * 24.Step 4: Compute 24 * 24 = 576.Step 5: So 9 * x = 576.Step 6: Solve for x by dividing both sides by 9: x = 576 / 9.Step 7: Perform the division: 576 / 9 = 64.Step 8: Therefore, the third proportional to 9 and 24 is 64. Verification / Alternative check: To verify, check that 9 : 24 = 24 : 64. First, simplify 9 : 24 by dividing both numbers by 3, giving 3 : 8. Next, simplify 24 : 64 by dividing both numbers by 8, giving 3 : 8. Since both simplified ratios are equal, 9 : 24 = 24 : 64 holds true. This confirms that 64 is indeed the correct third proportional. Why Other Options Are Wrong: • 39: Plugging 39 in place of x would give 9 / 24 ≠ 24 / 39, so the proportion is not satisfied.• 48: If x were 48, then 24 : 48 simplifies to 1 : 2, whereas 9 : 24 simplifies to 3 : 8, which are not equal ratios.• 72: If x were 72, the ratio 24 : 72 simplifies to 1 : 3, again not matching 9 : 24. Common Pitfalls: Students sometimes confuse the concept of mean proportional (or geometric mean) with third proportional or mistakenly use the formula for fourth proportional. Another frequent issue is to reverse the position of numbers in the proportion and write 24 : 9 = 24 : x, which changes the meaning completely. Miscalculating 24 squared or dividing 576 by 9 incorrectly can also lead to wrong results. Being clear about the definition of third proportional and using the formula c = b^2 / a helps avoid such errors. Final Answer: The third proportional to 9 and 24 is 64.

Question 3

In what ratio does the point T(3, 0) divide the line segment joining S(4, 2) and U(1, 4) in the coordinate plane?

Accepted Answer

Introduction / Context: This coordinate geometry question asks for the ratio in which a point divides a line segment joining two other points. Such problems are standard applications of the section formula or distance ratio concepts and are important for understanding how coordinates relate to geometric positions on the plane. Given Data / Assumptions: • Point S has coordinates (4, 2).• Point U has coordinates (1, 4).• Point T has coordinates (3, 0) and lies on the segment joining S and U.• We must find the ratio in which T divides the segment SU, that is, ST : TU. Concept / Approach: There are two common methods: using the section formula or using distances. If a point T divides the segment joining S(x1, y1) and U(x2, y2) in the ratio m : n internally, then its coordinates are given by ((m*x2 + n*x1) / (m + n), (m*y2 + n*y1) / (m + n)). Alternatively, the ratio of the distances ST and TU can be found directly using the distance formula between points. Here, it is simpler to calculate the distances ST and TU and then express their ratio in simplest form. Step-by-Step Solution: Step 1: Coordinates of S are (4, 2) and coordinates of T are (3, 0).Step 2: Use the distance formula to find ST: ST = sqrt((3 − 4)^2 + (0 − 2)^2).Step 3: Compute the differences: 3 − 4 = −1 and 0 − 2 = −2.Step 4: So ST = sqrt((−1)^2 + (−2)^2) = sqrt(1 + 4) = sqrt(5).Step 5: Now find TU using T(3, 0) and U(1, 4): TU = sqrt((1 − 3)^2 + (4 − 0)^2).Step 6: Compute the differences: 1 − 3 = −2 and 4 − 0 = 4.Step 7: So TU = sqrt((−2)^2 + (4)^2) = sqrt(4 + 16) = sqrt(20).Step 8: Simplify sqrt(20) = sqrt(4 * 5) = 2 * sqrt(5).Step 9: Now find the ratio ST : TU = sqrt(5) : 2*sqrt(5).Step 10: Divide both sides by sqrt(5) to simplify the ratio: 1 : 2. Verification / Alternative check: As an alternative check, the section formula can be used. Let T divide SU in the ratio k : 1, where k corresponds to the part near U and 1 is near S, or vice versa. However, once we have already computed ST : TU as 1 : 2, this automatically matches the idea that T is one third of the way from S to U. The fact that ST is exactly half of TU aligns with the distances we computed. The numeric distances and simplified ratio confirm that the ratio 1 : 2 is consistent and correct. Why Other Options Are Wrong: • 2 : 1: This would suggest that ST is twice TU, which is not true because ST = sqrt(5) and TU = 2*sqrt(5).• 2 : 3: This ratio does not match the actual distance comparison and would not place T at (3, 0).• 3 : 2: This would incorrectly suggest ST is 1.5 times TU, which contradicts the explicit distance calculations. Common Pitfalls: One common mistake is to confuse the order of the ratio and write TU : ST instead of ST : TU, which would invert the correct answer. Another error is to forget to take square roots properly or to simplify sqrt(20) incorrectly. Some learners also miscalculate coordinate differences, for example, mixing up x1 − x2 and x2 − x1, which leads to wrong distances. Being careful with arithmetic and the distance formula is crucial in such coordinate geometry problems. Final Answer: The point T divides the segment SU in the ratio 1 : 2 (that is, ST : TU = 1 : 2).

Question 4

In an army selection process, the ratio of selected candidates to unselected candidates is 3 : 1. If 80 fewer candidates had applied and 40 fewer had been selected, the new ratio of selected to unselected would be 4 : 1. How many candidates original…

Accepted Answer

Introduction / Context: This is a word problem involving ratios and algebraic reasoning. It describes an army selection process where candidates are either selected or unselected. We are given an initial ratio of selected to unselected candidates and a hypothetical scenario that changes the total number of applicants and selected candidates, resulting in a new ratio. The goal is to determine how many candidates originally applied. Given Data / Assumptions: • Initially, the ratio of selected candidates to unselected candidates is 3 : 1.• Let the original total number of candidates be N.• From this ratio, selected candidates initially are 3N / 4 and unselected candidates are N / 4.• If 80 fewer candidates had applied, the total would have been N − 80.• If 40 fewer candidates had been selected, the number of selected candidates would be 3N / 4 − 40.• Under these new conditions, the ratio of selected to unselected candidates becomes 4 : 1. Concept / Approach: This problem is solved through algebra. We express the selected and unselected candidates in terms of N using the given ratios. Then, we apply the changes described and set up a new ratio equation. Ratios of the form selected : unselected can be written as fractions, and we equate that fraction to 4 / 1 in the second scenario. Solving the resulting equation for N gives the total number of candidates who originally applied. Careful handling of the selection and unselection counts is crucial. Step-by-Step Solution: Step 1: Initially, selected : unselected = 3 : 1.Step 2: Let total candidates be N. Then selected = 3N / 4 and unselected = N / 4.Step 3: In the hypothetical scenario, 80 fewer apply, so total becomes N − 80.Step 4: Also, 40 fewer are selected, so new selected = 3N / 4 − 40.Step 5: New unselected = total − new selected = (N − 80) − (3N / 4 − 40).Step 6: Simplify new unselected: N − 80 − 3N / 4 + 40 = N / 4 − 40.Step 7: Given that the new ratio of selected to unselected is 4 : 1, we write (3N / 4 − 40) / (N / 4 − 40) = 4.Step 8: Cross multiply: 3N / 4 − 40 = 4 * (N / 4 − 40).Step 9: Expand the right side: 4 * (N / 4) − 4 * 40 = N − 160.Step 10: So the equation is 3N / 4 − 40 = N − 160.Step 11: Bring all terms involving N to one side: 3N / 4 − N = −160 + 40.Step 12: 3N / 4 − N = 3N / 4 − 4N / 4 = −N / 4.Step 13: The right side is −120.Step 14: So −N / 4 = −120, which gives N / 4 = 120.Step 15: Therefore N = 120 * 4 = 480. Verification / Alternative check: With N = 480, initial selected = 3 * 480 / 4 = 360 and initial unselected = 480 / 4 = 120, giving ratio 360 : 120 = 3 : 1, which matches the first condition. In the altered scenario, total candidates = 480 − 80 = 400. New selected = 360 − 40 = 320. New unselected = 400 − 320 = 80. The new ratio is 320 : 80 = 4 : 1, which exactly matches the condition given in the problem. This confirms that N = 480 is correct. Why Other Options Are Wrong: • 960: This would double all the values and would not satisfy the second condition when the hypothetical changes are applied.• 240: This would give non-integer or inconsistent values for selected and unselected counts under the given ratios.• 1,440: This is too large and does not maintain both ratio conditions when checked. Common Pitfalls: A common mistake is to treat 3 : 1 as meaning 3 selected and 1 unselected without relating it to the total N properly. Others incorrectly compute new unselected candidates by subtracting only from the original unselected group rather than from the new total after changing both applied and selected counts. Algebraic manipulation errors, such as misplacing terms when solving the equation, are also frequent. Careful step-by-step reasoning helps avoid such errors. Final Answer: The number of candidates who originally applied for the process is 480.

Question 5

The bus fare between two cities is increased in the ratio 17 : 20. If the original bus fare is Rs. 425, what is the increase in the fare in rupees?

Accepted Answer

Introduction / Context: This is another example of a ratio based price change question, here applied to bus fare. The problem checks understanding of how to compute a new value when an old value is scaled according to a ratio and how to find the absolute difference between the new and old values, which represents the increase in fare. Given Data / Assumptions: • Original bus fare between the two cities is Rs. 425.• The fare is increased in the ratio 17 : 20.• This means the old fare corresponds to 17 parts and the new fare corresponds to 20 parts.• We are required to find the increase in fare, not the final fare. Concept / Approach: If a quantity changes from a ratio of a to b, the new value is (b / a) times the old value. Here, the ratio 17 : 20 indicates that the bus fare has been multiplied by 20 / 17. Once we compute the new fare as original fare multiplied by 20 / 17, the increase is obtained by subtracting the original fare from this new value. This approach directly uses the definition of ratios to avoid confusion with percentage calculations. Step-by-Step Solution: Step 1: Let the old fare correspond to 17 parts and the new fare correspond to 20 parts.Step 2: The scaling factor for the new fare is therefore 20 / 17.Step 3: Original fare is Rs. 425.Step 4: New fare = 425 * 20 / 17.Step 5: First compute 425 / 17. Since 17 * 25 = 425, 425 / 17 = 25.Step 6: Multiply this by 20 to get the new fare: 25 * 20 = 500.Step 7: Increase in fare = new fare − old fare = 500 − 425 = Rs. 75. Verification / Alternative check: We can verify by checking whether old and new fares are in the ratio 17 : 20. Divide both fares by 25. Old fare 425 / 25 = 17 and new fare 500 / 25 = 20. So the ratio is 17 : 20, which matches the given condition. This confirms that the new fare is Rs. 500 and therefore the increase of Rs. 75 is correct. Why Other Options Are Wrong: • Rs. 100: This would make the new fare 525, and 425 : 525 when simplified does not yield 17 : 20.• Rs. 200: This would make the new fare 625, giving a ratio inconsistent with 17 : 20.• Rs. 500: This would imply the old fare was zero, which is impossible in this context. Common Pitfalls: Some students mistakenly think the ratio 17 : 20 refers to a 3 part increase and directly compute an increase as 3 * 25 = 75 without justifying the part value, or they confuse ratio with percent. Others might try to compute the percentage increase from 17 to 20 and then apply that percentage incorrectly. Errors can also occur if they miscalculate 425 / 17 or forget that the question asks for the increase, not the final fare. Keeping the focus on the multiplicative factor 20 / 17 and carefully subtracting the original value helps avoid mistakes. Final Answer: The increase in bus fare is Rs. 75.

Question 6

In a school with 720 students, the ratio of the number of boys to the number of girls is 7 : 5. How many additional girls must be admitted so that the ratio of boys to girls becomes 1 : 1?

Accepted Answer

Introduction / Context: This ratio and proportion question is about adjusting the composition of a group to reach a desired ratio. The school initially has a certain ratio of boys to girls, and we are asked how many more girls must be added to make the numbers of boys and girls equal, which corresponds to a ratio of 1 : 1. Given Data / Assumptions: • Total number of students in the school is 720.• The ratio of boys to girls initially is 7 : 5.• Only girls are added; the number of boys remains the same.• After adding some girls, the ratio of boys to girls must become 1 : 1. Concept / Approach: When a ratio a : b represents two parts of a whole, the actual numbers can be written as 7k and 5k for some integer k. Here, the total number of students is 720, which equals the sum of the numbers of boys and girls. So we first solve for k. Once we know the exact counts of boys and girls, we can determine how many additional girls need to be added so that the number of girls equals the number of boys. This illustrates how ratios translate into real counts and how to adjust one component to reach a target ratio. Step-by-Step Solution: Step 1: Let the number of boys be 7k and the number of girls be 5k.Step 2: Total students are 7k + 5k = 12k.Step 3: We are told that 12k = 720.Step 4: Solve for k: k = 720 / 12 = 60.Step 5: Number of boys = 7k = 7 * 60 = 420.Step 6: Number of girls initially = 5k = 5 * 60 = 300.Step 7: We want the new number of girls to be equal to the number of boys, which is 420.Step 8: Therefore, additional girls needed = desired girls − current girls = 420 − 300 = 120. Verification / Alternative check: After adding 120 girls, the number of girls becomes 300 + 120 = 420. The number of boys remains 420. Thus, boys : girls becomes 420 : 420, which simplifies to 1 : 1. This confirms that adding 120 girls achieves the target ratio. There is no contradiction with the total because the school size would then become 720 + 120 = 840, which is acceptable since the question allows for additional admissions. Why Other Options Are Wrong: • 90: Adding 90 girls would give 390 girls, while boys remain 420, leading to a ratio of 420 : 390, which does not simplify to 1 : 1.• 220: This would result in 520 girls, exceeding the 420 boys and giving a ratio different from 1 : 1.• 240: This would produce 540 girls and 420 boys, again not leading to equality of numbers. Common Pitfalls: Some students mistakenly assume the total number of students must remain 720 and try to adjust both boys and girls, which is not what the problem states. Others might miscalculate k by incorrectly dividing 720 by something other than the sum of the ratio parts. It is also easy to confuse the required final ratio and think that the number of boys should change, when in fact only girls are being admitted. Carefully reading the question and systematically using the ratio representation avoids these errors. Final Answer: The number of additional girls that must be admitted is 120.

Question 7

The reciprocals of the squares of the numbers 1 1/2 and 1 1/3 are taken. In what ratio are these reciprocals related?

Accepted Answer

Introduction / Context: This question combines fractions, squaring, and reciprocals, and then asks for the ratio between the results. It checks the ability to manipulate mixed numbers, convert them to improper fractions, square them correctly, and then take reciprocals, which is common in quantitative aptitude problems involving algebraic fractions. Given Data / Assumptions: • The two given numbers are 1 1/2 and 1 1/3.• We must first find the squares of these numbers.• Then we must take the reciprocals of those squares.• Finally, we must determine the ratio of these two reciprocal values. Concept / Approach: To handle mixed numbers, we first convert them into improper fractions. Then we square each fraction by squaring both numerator and denominator. After squaring, we take the reciprocal by inverting numerator and denominator. Finally, the ratio of two fractions is found by forming a fraction of one over the other and simplifying. Working step by step avoids confusion and ensures accurate handling of fractions and powers. Step-by-Step Solution: Step 1: Convert the mixed numbers to improper fractions.• 1 1/2 = (2 * 1 + 1) / 2 = 3 / 2.• 1 1/3 = (3 * 1 + 1) / 3 = 4 / 3.Step 2: Square each fraction.• Square of 3 / 2 is (3 / 2)^2 = 9 / 4.• Square of 4 / 3 is (4 / 3)^2 = 16 / 9.Step 3: Take the reciprocal of each squared value.• Reciprocal of 9 / 4 is 4 / 9.• Reciprocal of 16 / 9 is 9 / 16.Step 4: Now find the ratio of these reciprocals: (4 / 9) : (9 / 16).Step 5: Express this ratio as a single fraction and simplify: (4 / 9) / (9 / 16) = (4 / 9) * (16 / 9).Step 6: Multiply numerators and denominators: (4 * 16) / (9 * 9) = 64 / 81.Step 7: Therefore, the ratio of the reciprocals is 64 : 81. Verification / Alternative check: We can also think directly in terms of proportionality. Since the squared values are 9 / 4 and 16 / 9, their reciprocals 4 / 9 and 9 / 16 are natural inverses. Writing the ratio as 4 / 9 : 9 / 16 and simplifying through the same multiplication process confirms that no simplification has been overlooked. The final reduced form 64 : 81 is already in simplest form because 64 and 81 have no common factor other than 1. This confirms that the answer is consistent and fully simplified. Why Other Options Are Wrong: • 8 : 9: This might arise from confusing squaring with taking square roots or simplifying incorrectly.• 81 : 64: This is the inverse of the correct ratio and would correspond to reversing the order of the two reciprocals.• 9 : 8: This again reflects mixing up the fraction operations and not handling squares and reciprocals step by step. Common Pitfalls: Typical errors include failing to convert mixed numbers into improper fractions, squaring only the numerator but not the denominator, or misinterpreting the word reciprocal. Some students also directly square 1.5 and 1.333 without expressing them as fractions, which can introduce rounding errors. Another pitfall is to compute the ratio of the squares instead of the ratio of the reciprocals of the squares. Careful reading and systematic handling of fraction operations is essential. Final Answer: The reciprocals of the squares are in the ratio 64 : 81.

Question 8

If 3A = 2B = 4C, what is the simplified ratio A : B : C expressed in whole numbers?

Accepted Answer

Introduction / Context: This algebraic ratio question asks us to determine the relationship between three variables A, B and C given that 3A, 2B and 4C are all equal to the same quantity. It tests the ability to interpret compound equalities and then express a three term ratio in the simplest whole number form. Given Data / Assumptions: • We are given that 3A = 2B = 4C.• All variables A, B and C are assumed to be non zero real numbers.• We must find the ratio A : B : C. Concept / Approach: If 3A, 2B and 4C are all equal to some common value, say k, then we can express A, B and C individually in terms of k. Specifically, A = k / 3, B = k / 2, and C = k / 4. Once we have expressions for A, B and C in terms of the same constant, we can form the ratio A : B : C and clear denominators by multiplying all terms by the least common multiple of the denominators. This produces a clean integer ratio. Step-by-Step Solution: Step 1: Let the common value be k. Then we have:• 3A = k, so A = k / 3.• 2B = k, so B = k / 2.• 4C = k, so C = k / 4.Step 2: Now write the ratio A : B : C in terms of k:A : B : C = (k / 3) : (k / 2) : (k / 4).Step 3: Because k is common and positive, it cancels out from the ratio. So we focus on 1 / 3 : 1 / 2 : 1 / 4.Step 4: To convert this to whole numbers, find the least common multiple of the denominators 3, 2 and 4, which is 12.Step 5: Multiply each term by 12:• (1 / 3) * 12 = 4.• (1 / 2) * 12 = 6.• (1 / 4) * 12 = 3.Step 6: Therefore, the simplified ratio A : B : C is 4 : 6 : 3. Verification / Alternative check: We can verify by assigning any convenient value to k and confirming that 3A, 2B and 4C become equal. For example, let k = 12. Then A = 12 / 3 = 4, B = 12 / 2 = 6, and C = 12 / 4 = 3. Check the equalities: 3A = 3 * 4 = 12, 2B = 2 * 6 = 12, and 4C = 4 * 3 = 12. All are equal to 12, which matches k. This confirms that A : B : C = 4 : 6 : 3 is consistent with the original condition. Why Other Options Are Wrong: • 3 : 2 : 4: This ratio does not satisfy 3A = 2B = 4C when tested with actual values.• 6 : 4 : 3: While similar in shape, interchanging positions leads to inconsistent equalities for 3A, 2B and 4C.• 2 : 3 : 4: This again rearranges the terms incorrectly and fails to make 3A, 2B and 4C equal. Common Pitfalls: Many learners mistakenly take the coefficients 3, 2 and 4 directly as the ratio A : B : C without inverting them. However, 3A = k actually implies A is k / 3, not 3k. Others forget to clear denominators and leave the ratio in fractional form, which makes comparison difficult. It is also easy to confuse the meaning of 3A = 2B = 4C and equate only two at a time without considering the third. Adopting the common value k technique simplifies the reasoning and avoids these mistakes. Final Answer: The correct simplified ratio is A : B : C = 4 : 6 : 3.

Question 9

In what ratio does the Y axis (the line x = 0) divide the line segment joining the points (2, 3) and (−2, 1) in the coordinate plane?

Accepted Answer

Introduction / Context: This coordinate geometry question asks about the ratio in which a vertical line, namely the Y axis (x = 0), divides a segment connecting two given points. It is a standard type of problem that tests understanding of the distance formula and how partition points lie on line segments in the Cartesian plane. Given Data / Assumptions: • The two endpoints of the segment are (2, 3) and (−2, 1).• The Y axis is the vertical line given by the equation x = 0.• The segment joining the points intersects the Y axis at some point.• We must find the ratio in which this intersection point divides the segment, measured from one endpoint to the other. Concept / Approach: The segment intersects the Y axis where x = 0. We can use a parametric representation of the line segment from one endpoint to the other and then solve for the parameter value at which x equals zero. After getting the intersection point, we compute the distances from this point to each endpoint using the distance formula. The ratio of these two distances gives the required partition ratio. Because distances are always positive, the ratio should be simplified to positive integers. Step-by-Step Solution: Step 1: Let A = (2, 3) and B = (−2, 1).Step 2: Parametrize the line segment from A to B as A + t(B − A) for 0 ≤ t ≤ 1.Step 3: Compute B − A = (−2 − 2, 1 − 3) = (−4, −2).Step 4: So a general point on the segment is (2 + t * (−4), 3 + t * (−2)) = (2 − 4t, 3 − 2t).Step 5: The Y axis is x = 0. Set the x coordinate equal to zero: 2 − 4t = 0.Step 6: Solve for t: 4t = 2, so t = 1 / 2.Step 7: Substitute t = 1 / 2 into the coordinates: x = 2 − 4 * 1 / 2 = 0 and y = 3 − 2 * 1 / 2 = 3 − 1 = 2.Step 8: Thus, the Y axis meets the segment at the point P = (0, 2).Step 9: Now compute distances AP and PB using the distance formula.Step 10: AP = sqrt((0 − 2)^2 + (2 − 3)^2) = sqrt((−2)^2 + (−1)^2) = sqrt(4 + 1) = sqrt(5).Step 11: PB = sqrt((−2 − 0)^2 + (1 − 2)^2) = sqrt((−2)^2 + (−1)^2) = sqrt(4 + 1) = sqrt(5).Step 12: Therefore, AP and PB are equal, so the ratio AP : PB is sqrt(5) : sqrt(5) = 1 : 1. Verification / Alternative check: The fact that t = 1 / 2 in the parametric form also indicates that the point P lies exactly halfway between A and B. When a point corresponds to t = 1 / 2 on a linear interpolation between two endpoints, it is the midpoint, and by definition the midpoint divides the segment into two equal parts. Hence the ratio of the distances from A to P and from P to B must be 1 : 1. This gives an algebraic confirmation of the earlier distance calculation. Why Other Options Are Wrong: • 1 : 2 and 2 : 3: These ratios suggest that the point is closer to one endpoint than the other, which contradicts the equal distances AP and PB.• 3 : 1: This would imply a much greater distance on one side, not supported by the numerical distance calculations. Common Pitfalls: Students sometimes do not parametrize correctly or confuse the direction vector B − A with A − B, though this does not affect the final ratio if handled consistently. Another common error is miscalculating the intersection point by solving the equation for the wrong coordinate or misusing the distance formula. Some learners might also forget to simplify the ratio of distances to its simplest integer form. Clearly following the steps of finding the intersection, computing both distances, and then simplifying the ratio prevents these issues. Final Answer: The Y axis divides the segment joining (2, 3) and (−2, 1) in the ratio 1 : 1.

Question 10

The ratio of money with Ram and Gopal is 7 : 17, and the ratio of money with Gopal and Krishan is also 7 : 17. If Ram has Rs. 490, how much money does Krishan have?

Accepted Answer

Introduction / Context: This problem involves chained ratios among three people Ram, Gopal and Krishan. The ratios relate Ram to Gopal and Gopal to Krishan, and the absolute amount with Ram is given. From this information, we are asked to determine the amount with Krishan. It tests the ability to work with linked ratios and to use a common variable approach effectively. Given Data / Assumptions: • The ratio of money with Ram and Gopal is 7 : 17.• The ratio of money with Gopal and Krishan is 7 : 17.• Ram has Rs. 490.• All amounts are positive and measured in rupees.• We are to find the amount of money with Krishan. Concept / Approach: We interpret the given ratios using variables. From Ram and Gopal's ratio, we write Ram's money as 7k and Gopal's as 17k. From Gopal and Krishan's ratio, we write Gopal's money as 7m and Krishan's as 17m. Since Gopal's actual money must be the same in both expressions, we equate 17k and 7m to establish a link between k and m. Once we know k from Ram's actual amount, we can compute m and then Krishan's amount. This method demonstrates how to handle multiple related ratio conditions. Step-by-Step Solution: Step 1: From the ratio Ram : Gopal = 7 : 17, let Ram have 7k and Gopal have 17k.Step 2: We are told that Ram has Rs. 490. So 7k = 490.Step 3: Solve for k: k = 490 / 7 = 70.Step 4: Therefore, Gopal has 17k = 17 * 70 = 1,190 rupees.Step 5: Now use the second ratio Gopal : Krishan = 7 : 17.Step 6: Let Gopal have 7m and Krishan have 17m according to this ratio.Step 7: Gopal's money must be the same in both descriptions, so 7m = 1,190.Step 8: Solve for m: m = 1,190 / 7 = 170.Step 9: Krishan's money = 17m = 17 * 170.Step 10: Multiply to find the product: 17 * 170 = 17 * (17 * 10) = 289 * 10 = 2,890.Step 11: Therefore, Krishan has Rs. 2,890. Verification / Alternative check: We can verify the consistency of ratios and amounts. Using k = 70, Ram has 490 and Gopal has 1,190. The ratio 490 : 1,190 simplifies by dividing both numbers by 70 to 7 : 17, which matches the first given ratio. Using m = 170, Gopal has 1,190 and Krishan has 2,890. The ratio 1,190 : 2,890 simplifies by dividing by 170 to 7 : 17, which matches the second given ratio. Since both ratios hold true with these amounts, our solution is consistent. Why Other Options Are Wrong: • Rs. 2,680: This would give a different ratio than 7 : 17 when compared with Gopal's 1,190.• Rs. 2,330: This would not maintain the ratio 7 : 17 with Gopal's amount and would contradict the given conditions.• Rs. 1,190: This is the amount with Gopal, not with Krishan, and it ignores the second ratio that requires Krishan to have more money. Common Pitfalls: Some students mistakenly assume that the same multiplier applies to all ratios simultaneously and try to use one variable instead of linking two sets of ratios through a shared quantity. Others may incorrectly assume that the ratio 7 : 17 applies directly between Ram and Krishan. Computational mistakes in handling large numbers or in simplifying ratios can also lead to errors. The crucial step is recognizing that Gopal is the common link and equating his two expressions to relate the different ratio multipliers. Final Answer: Krishan has Rs. 2,890.

Question 11

A man spends a part of his monthly income and saves the rest. The ratio of his expenditure to his savings is 61 : 6. If his monthly income is Rs. 8,710, what is the amount of his monthly savings?

Accepted Answer

Introduction / Context: This is a straightforward ratio and proportion question about income, expenditure and savings. The man spends part of his income and saves the rest. The ratio of expenditure to savings is given along with the total income. We must compute the rupee amount that he saves each month, which is a practical example of budgeting and personal finance. Given Data / Assumptions: • Monthly income = Rs. 8,710.• Ratio of expenditure to savings = 61 : 6.• Income = expenditure + savings.• We need to find the amount of monthly savings. Concept / Approach: The ratio 61 : 6 means that for every 61 units of money spent, 6 units are saved. The total of these parts is 61 + 6 = 67. Since income is the sum of expenditure and savings, the income corresponds to 67 equal parts. By dividing the total income by 67, we can find the value of one part. Then, multiplying that single part value by 6 gives the savings amount. This is a standard application of ratios to divide a total into proportional segments. Step-by-Step Solution: Step 1: Let expenditure be 61x and savings be 6x.Step 2: Then income = expenditure + savings = 61x + 6x = 67x.Step 3: We are told that income is Rs. 8,710, so 67x = 8,710.Step 4: Solve for x: x = 8,710 / 67.Step 5: Perform the division: 8,710 / 67 = 130.Step 6: Now compute savings = 6x = 6 * 130 = 780 rupees.Step 7: Therefore, the man saves Rs. 780 each month. Verification / Alternative check: We can verify by also computing the expenditure. Expenditure = 61x = 61 * 130 = 7,930. Now check the total: expenditure + savings = 7,930 + 780 = 8,710, which matches the monthly income. Also, the ratio expenditure : savings = 7,930 : 780. Dividing both by 130 gives 61 : 6, which matches the ratio given in the question. This confirms that the savings amount has been computed correctly. Why Other Options Are Wrong: • Rs. 870: If savings were 870, the implied ratio of expenditure to savings and the total income would not satisfy the given condition.• Rs. 690: This is too low and would yield the wrong ratio when combined with the corresponding expenditure.• Rs. 980: This is too high and would not keep expenditure and savings in the correct ratio while summing to Rs. 8,710. Common Pitfalls: Some students mistakenly interpret the ratio 61 : 6 as expenditure : income instead of expenditure : savings. Others forget to add the ratio parts when equating to total income and incorrectly use either 61 or 6 alone for division. Arithmetic mistakes in dividing 8,710 by 67 can also lead to incorrect answers. Carefully identifying what each part of the ratio represents and keeping track of income as the sum of expenditure and savings is essential. Final Answer: The man saves Rs. 780 every month.

Question 12

Find the third proportional to the expressions (x − y)^2 and (x^2 − y^2)^2 in algebraic form.

Accepted Answer

Introduction / Context: This is an algebraic ratio and proportion problem involving expressions rather than simple numbers. We are asked to find the third proportional to two given expressions. This tests understanding of the definition of third proportional and the ability to manipulate algebraic expressions using identities such as x^2 − y^2 = (x − y)(x + y). Given Data / Assumptions: • The first expression is (x − y)^2.• The second expression is (x^2 − y^2)^2.• We must find an expression Z that is third proportional to these two, so that (x − y)^2 : (x^2 − y^2)^2 = (x^2 − y^2)^2 : Z.• All expressions are defined for values of x and y that make denominators nonzero if any arise. Concept / Approach: For numbers or expressions a and b, a third proportional c is defined so that a : b = b : c. In algebraic terms, this means a / b = b / c, leading to a * c = b^2. Solving for c gives c = b^2 / a. Here a = (x − y)^2 and b = (x^2 − y^2)^2. So Z = ((x^2 − y^2)^2)^2 / (x − y)^2. To simplify this, we use the factorization x^2 − y^2 = (x − y)(x + y). Substituting this into the expression and simplifying powers allows us to express Z in terms of (x − y) and (x + y). Step-by-Step Solution: Step 1: Let a = (x − y)^2 and b = (x^2 − y^2)^2.Step 2: The third proportional Z must satisfy a : b = b : Z.Step 3: Using the relation for third proportional, a * Z = b^2, so Z = b^2 / a.Step 4: Substitute a and b: Z = ((x^2 − y^2)^2)^2 / (x − y)^2.Step 5: Simplify ((x^2 − y^2)^2)^2 as (x^2 − y^2)^4.Step 6: So Z = (x^2 − y^2)^4 / (x − y)^2.Step 7: Use the identity x^2 − y^2 = (x − y)(x + y).Step 8: Then (x^2 − y^2)^4 = ((x − y)(x + y))^4 = (x − y)^4 * (x + y)^4.Step 9: Substitute back: Z = (x − y)^4 * (x + y)^4 / (x − y)^2.Step 10: Simplify by canceling powers of (x − y): (x − y)^4 / (x − y)^2 = (x − y)^2.Step 11: Therefore, Z = (x − y)^2 * (x + y)^4.Step 12: The simplified expression is (x + y)^4 (x − y)^2. Verification / Alternative check: To verify, we check whether the proportion holds: (x − y)^2 : (x^2 − y^2)^2 = (x^2 − y^2)^2 : (x + y)^4 (x − y)^2. Writing this as fractions, we want (x − y)^2 / (x^2 − y^2)^2 = (x^2 − y^2)^2 / ((x + y)^4 (x − y)^2). Using x^2 − y^2 = (x − y)(x + y), the left side becomes (x − y)^2 / ((x − y)^2 (x + y)^2) = 1 / (x + y)^2. The right side becomes ((x − y)^2 (x + y)^2) / ((x + y)^4 (x − y)^2) = 1 / (x + y)^2. Both sides match, confirming the correctness of the third proportional. Why Other Options Are Wrong: • (x + y)^3 (x − y)^2: This has the wrong power of (x + y) and does not satisfy the proportion when checked.• (x + y)^2 (x − y)^2: This corresponds to b, not to the third proportional b^2 / a.• (x + y)^2 (x − y)^3: This has an extra power of (x − y) and does not simplify correctly in the proportion test. Common Pitfalls: Many learners misapply the definition of third proportional and instead compute something like a * b or a + b. Others may mis-handle the algebraic identity x^2 − y^2 = (x − y)(x + y) or incorrectly raise expressions to powers. Canceling powers incorrectly or missing a factor in the numerator or denominator when simplifying is also common. Being systematic with the definition Z = b^2 / a and carefully using standard algebraic identities prevents these mistakes. Final Answer: The third proportional to (x − y)^2 and (x^2 − y^2)^2 is (x + y)^4 (x − y)^2.

Question 13

Two numbers are in the ratio 3 : 5. If 6 is added to both numbers, the new ratio becomes 2 : 3. What are the two original numbers?

Accepted Answer

Introduction / Context: This question deals with two numbers whose ratio changes when a constant is added to both. It tests understanding of ratios, how adding the same quantity to both terms affects the ratio, and how to set up and solve equations based on these conditions. Given Data / Assumptions: • Initially, the ratio of the two numbers is 3 : 5.• When 6 is added to each number, the new ratio becomes 2 : 3.• Both numbers are assumed to be positive integers.• We must find the values of the two original numbers. Concept / Approach: When two numbers are in the ratio 3 : 5, we can write them as 3k and 5k for some positive constant k. After adding 6 to each, they become 3k + 6 and 5k + 6. We are told that their new ratio is 2 : 3, so (3k + 6) / (5k + 6) = 2 / 3. This leads to a linear equation in k that can be solved using cross multiplication. Once k is determined, we substitute back to find the original numbers. Step-by-Step Solution: Step 1: Let the original numbers be 3k and 5k.Step 2: After adding 6 to each, the new numbers are 3k + 6 and 5k + 6.Step 3: The new ratio is given as 2 : 3, so (3k + 6) / (5k + 6) = 2 / 3.Step 4: Cross multiply to eliminate fractions: 3 * (3k + 6) = 2 * (5k + 6).Step 5: Expand both sides: 9k + 18 = 10k + 12.Step 6: Bring all terms involving k to one side: 9k − 10k = 12 − 18.Step 7: This simplifies to −k = −6.Step 8: So k = 6.Step 9: Therefore, the original numbers are 3k = 3 * 6 = 18 and 5k = 5 * 6 = 30. Verification / Alternative check: Check the conditions using the found numbers. Initially, the ratio of 18 to 30 is 18 : 30, which simplifies to 3 : 5 when both numbers are divided by 6. This matches the given initial ratio. After adding 6 to each, the numbers become 24 and 36. The new ratio is 24 : 36, which simplifies to 2 : 3 when both are divided by 12. This matches the given new ratio. Since both conditions are satisfied, the solution is verified as correct. Why Other Options Are Wrong: • 21 and 35: Their initial ratio is 21 : 35 = 3 : 5, but adding 6 gives 27 and 41, which do not have a ratio of 2 : 3.• 30 and 50: The initial ratio 30 : 50 = 3 : 5, but adding 6 yields 36 and 56, not in the ratio 2 : 3.• 24 and 40: The initial ratio is 3 : 5, but adding 6 gives 30 and 46, which again do not simplify to 2 : 3. Common Pitfalls: Some students try to guess the numbers by trial without systematically using algebra, which can be time consuming and error prone. Another common mistake is to misinterpret the new ratio and set up the equation incorrectly, for example by mixing up the order of the numbers. Algebraic errors during cross multiplication or simplification can also lead to wrong values for k. Careful equation setup and simplification step by step ensure accurate results. Final Answer: The two original numbers are 18 and 30.

Question 14

In what ratio does the Y axis (x = 0) divide the line segment joining the points (−1, 3) and (2, −4) in the coordinate plane?

Accepted Answer

Introduction / Context: This coordinate geometry problem asks for the ratio in which the Y axis divides a line segment between two given points. It is another application of the section formula or parametric representation combined with the distance formula, and it tests spatial reasoning and algebraic manipulation in the plane. Given Data / Assumptions: • The endpoints of the segment are A = (−1, 3) and B = (2, −4).• The Y axis is the line x = 0.• The segment AB intersects the Y axis at some point P.• We must determine the ratio AP : PB. Concept / Approach: We can parametrize the segment from A to B, find the parameter value at which x becomes zero, and thus obtain the coordinates of the intersection point P. Once P is known, we use the distance formula to compute distances AP and PB. The ratio of these distances, simplified to whole numbers, gives the required division ratio. This method ensures a clear and systematic solution. Step-by-Step Solution: Step 1: Let A = (−1, 3) and B = (2, −4).Step 2: Find the direction vector B − A = (2 − (−1), −4 − 3) = (3, −7).Step 3: Parametrize the line segment as A + t(B − A) for 0 ≤ t ≤ 1.Step 4: A general point on the segment is (−1 + 3t, 3 − 7t).Step 5: The Y axis has equation x = 0, so set the x coordinate equal to zero: −1 + 3t = 0.Step 6: Solve for t: 3t = 1, so t = 1 / 3.Step 7: Substitute t = 1 / 3 into the coordinates to get P: x = −1 + 3 * (1 / 3) = 0 and y = 3 − 7 * (1 / 3) = 3 − 7 / 3 = (9 − 7) / 3 = 2 / 3.Step 8: So P = (0, 2 / 3).Step 9: Now compute distance AP using A(−1, 3) and P(0, 2 / 3).Step 10: AP = sqrt((0 − (−1))^2 + (2 / 3 − 3)^2) = sqrt(1^2 + (2 / 3 − 9 / 3)^2) = sqrt(1 + (−7 / 3)^2) = sqrt(1 + 49 / 9) = sqrt(58 / 9).Step 11: Next, compute distance PB using P(0, 2 / 3) and B(2, −4).Step 12: PB = sqrt((2 − 0)^2 + (−4 − 2 / 3)^2) = sqrt(2^2 + (−14 / 3)^2) = sqrt(4 + 196 / 9) = sqrt(232 / 9).Step 13: Observe that PB^2 / AP^2 = (232 / 9) / (58 / 9) = 232 / 58 = 4.Step 14: Therefore PB^2 = 4 * AP^2 and PB = 2 * AP (taking positive distances).Step 15: So the ratio AP : PB is 1 : 2. Verification / Alternative check: The parameter t = 1 / 3 also provides a quick verification. In a parametric representation A + t(B − A), t represents the fraction of the distance from A to B. If t = 1 / 3, then AP : PB = 1 : 2 because AP is one third of the entire segment and PB is the remaining two thirds. This simple interpretation confirms the more detailed distance based calculation and assures us that the ratio AP : PB is indeed 1 : 2. Why Other Options Are Wrong: • 2 : 1: This would suggest that AP is longer than PB, which contradicts the finding that PB = 2 * AP.• 1 : 4 and 4 : 1: Both of these imply a much more uneven split of the segment than what the parametric value t = 1 / 3 and distance calculations show. Common Pitfalls: Students sometimes confuse which ratio is being asked (AP : PB versus PB : AP) and may invert the correct ratio. Miscomputing the direction vector or the parameter t can also cause errors. Additionally, some learners skip using a parametric form and try to guess the intersection point, leading to algebraic inconsistencies. A structured approach using the equation for the segment and the line x = 0 avoids these issues. Final Answer: The Y axis divides the segment joining (−1, 3) and (2, −4) in the ratio 1 : 2 (that is, AP : PB = 1 : 2).

Question 15

The present ages of R and S are in the ratio 6 : 7. Twelve years ago, the ratio of their ages was 9 : 11. What is the present age of R (in years)?

Accepted Answer

Introduction / Context: This is a typical ages and ratio problem. It involves the present ratio of ages of two people and their ratio of ages a certain number of years in the past. The problem checks understanding of how ages change over time and how to set up equations based on ratios to determine actual ages. Given Data / Assumptions: • Present age ratio of R and S is 6 : 7.• Twelve years ago, the ratio of their ages was 9 : 11.• Ages increase uniformly over time by the same number of years for each person.• We must find R's present age in years. Concept / Approach: We represent the present ages using a common multiplier. Let R's present age be 6k and S's present age be 7k for some positive integer k. Twelve years ago, their ages would have been 6k − 12 and 7k − 12. At that time, the ratio of their ages was 9 : 11. This gives the equation (6k − 12) / (7k − 12) = 9 / 11. Solving this equation for k allows us to find the actual present ages, particularly R's age. Step-by-Step Solution: Step 1: Let present age of R be 6k and present age of S be 7k.Step 2: Twelve years ago, R's age would have been 6k − 12 and S's age would have been 7k − 12.Step 3: We are told that twelve years ago, their ages were in the ratio 9 : 11, so (6k − 12) / (7k − 12) = 9 / 11.Step 4: Cross multiply to eliminate the fraction: 11 * (6k − 12) = 9 * (7k − 12).Step 5: Expand both sides: 66k − 132 = 63k − 108.Step 6: Bring k terms to one side and constants to the other: 66k − 63k = −108 + 132.Step 7: This simplifies to 3k = 24.Step 8: So k = 24 / 3 = 8.Step 9: Therefore, R's present age = 6k = 6 * 8 = 48 years.Step 10: S's present age could also be found as 7k = 7 * 8 = 56 years if needed. Verification / Alternative check: We can verify by checking the ratio 12 years ago. R's age 12 years ago = 48 − 12 = 36 years. S's age 12 years ago = 56 − 12 = 44 years. The ratio 36 : 44 simplifies by dividing both numbers by 4 to 9 : 11, which matches the condition given in the problem. The present ratio 48 : 56 simplifies by dividing by 8 to 6 : 7, which matches the given present ratio. These checks confirm that the solution is consistent. Why Other Options Are Wrong: • 56: This is the present age of S, not R.• 36: This is R's age twelve years ago, not the present age.• 44: This does not correspond to either the present or past age of R in the correct ratio structure. Common Pitfalls: Some learners confuse which person corresponds to which part of the ratio and may assign 7k to R and 6k to S, which leads to an incorrect equation. Others forget to subtract 12 years from both ages when forming the past ratio, or they set up the ratio in reverse order. Algebraic mistakes, such as expansion errors or sign errors when moving terms across the equation, can also obstruct finding the correct value of k. Careful mapping of the ratios and neat algebraic work help avoid these issues. Final Answer: The present age of R is 48 years.

Question 16

If the wealth of A is 4/9 times the wealth of B and the wealth of C is 7/6 times the wealth of B, what is the ratio of the wealth of C to the wealth of A?

Accepted Answer

Introduction / Context: This ratio problem involves three people A, B and C, whose wealths are related to B by given fractional multipliers. We must find the ratio of the wealth of C to the wealth of A. It tests understanding of proportional relationships when multiple quantities are expressed in terms of a common reference quantity. Given Data / Assumptions: • Wealth of A is 4/9 times the wealth of B.• Wealth of C is 7/6 times the wealth of B.• All wealth values are positive and measured in the same monetary units.• We must determine the ratio C : A. Concept / Approach: When multiple quantities are compared to a common base, it is convenient to represent the base using a variable. Let the wealth of B be some amount, say b. Then A and C can be written in terms of b using the given fractional relationships. The ratio of C to A is then obtained by forming a fraction of C over A, substituting the expressions in terms of b, and simplifying. Because b appears in both numerator and denominator, it cancels out, leaving a ratio in simplest terms. Step-by-Step Solution: Step 1: Let the wealth of B be b (in suitable units).Step 2: A's wealth is given as 4/9 of B's wealth, so A = (4 / 9) * b.Step 3: C's wealth is given as 7/6 of B's wealth, so C = (7 / 6) * b.Step 4: We are required to find the ratio C : A.Step 5: Express C : A as a fraction and simplify: C / A = ((7 / 6) * b) / ((4 / 9) * b).Step 6: Cancel b from numerator and denominator, since b is nonzero: C / A = (7 / 6) / (4 / 9).Step 7: Division of fractions is done by multiplying by the reciprocal, so C / A = (7 / 6) * (9 / 4).Step 8: Multiply numerators and denominators: (7 * 9) / (6 * 4) = 63 / 24.Step 9: Simplify 63 / 24 by dividing numerator and denominator by their greatest common divisor, which is 3. This gives 21 / 8.Step 10: Therefore, the ratio C : A is 21 : 8. Verification / Alternative check: To verify, we can assign a convenient value to b, such as b = 72, which is divisible by both 9 and 6. Then A = (4 / 9) * 72 = 4 * 8 = 32. C = (7 / 6) * 72 = 7 * 12 = 84. The ratio C : A is then 84 : 32. Divide both numbers by 4 to get 21 : 8. This matches the simplified ratio obtained by algebra, confirming that the result is correct and independent of the particular value chosen for b. Why Other Options Are Wrong: • 8 : 21: This is simply the inverse of the correct ratio and would correspond to A : C, not C : A.• 27 : 14 and 14 : 27: These ratios do not arise from the given fractional relationships with B and do not simplify to 21 : 8 when checked with actual values. Common Pitfalls: Some learners misread the question and compute the ratio A : C instead of C : A. Others might incorrectly handle the fractional multipliers and attempt to work with percentages instead, leading to confusion. Mistakes can also occur during the simplification of 63 / 24 if the greatest common divisor is not correctly identified. Using the method of representing B's wealth as a variable, carefully canceling common factors, and methodically simplifying fractions helps avoid these issues. Final Answer: The ratio of the wealth of C to the wealth of A is 21 : 8.

Question 17

What is the fourth proportional to the numbers 24, 120 and 22, that is, the number x such that 24 : 120 = 22 : x?

Accepted Answer

Introduction / Context: This question asks for the fourth proportional to three numbers. The idea of fourth proportional is a classic concept in ratio and proportion, where we want to find a number x such that the proportion a : b = c : x holds. Here, a = 24, b = 120 and c = 22, and we must determine x. Given Data / Assumptions: • The first number is 24.• The second number is 120.• The third number is 22.• The fourth proportional x satisfies 24 : 120 = 22 : x. Concept / Approach: For numbers a, b, c and x, if a : b = c : x, then a / b = c / x. Cross multiplication gives a * x = b * c, and solving for x yields x = (b * c) / a. Applying this formula directly with the given numbers makes the calculation straightforward. We just need to perform a multiplication and a division accurately to obtain the fourth proportional. Step-by-Step Solution: Step 1: Write the proportion: 24 : 120 = 22 : x.Step 2: Convert the ratio into a fraction equation: 24 / 120 = 22 / x.Step 3: Cross multiply: 24 * x = 120 * 22.Step 4: Compute the right side: 120 * 22 = 120 * (20 + 2) = 120 * 20 + 120 * 2 = 2,400 + 240 = 2,640.Step 5: So we have 24 * x = 2,640.Step 6: Solve for x by dividing both sides by 24: x = 2,640 / 24.Step 7: Perform the division. First divide by 8: 2,640 / 8 = 330 and 24 / 8 = 3. So x = 330 / 3 = 110.Step 8: Therefore, the fourth proportional x is 110. Verification / Alternative check: To verify, check that 24 : 120 equals 22 : 110. Simplify 24 : 120 by dividing both terms by 24 to get 1 : 5. Now simplify 22 : 110 by dividing both by 22 to also get 1 : 5. Since both ratios simplify to the same value, the proportion 24 : 120 = 22 : 110 holds, confirming that x = 110 is correct. Why Other Options Are Wrong: • 120: This would give the ratio 22 : 120, which simplifies to 11 : 60 and does not equal 1 : 5.• 100: The ratio 22 : 100 simplifies to 11 : 50, which is different from 1 : 5.• 90: The ratio 22 : 90 simplifies to 11 : 45, again not matching 1 : 5. Common Pitfalls: Some learners reverse the numbers and mistakenly compute x = (a * c) / b instead of x = (b * c) / a. Others may incorrectly simplify the ratios or forget to reduce them to simplest form when checking their work. Arithmetic errors in multiplying 120 by 22 or dividing 2,640 by 24 can also cause trouble. Writing down the proportion carefully and using cross multiplication systematically helps avoid such mistakes. Final Answer: The fourth proportional to 24, 120 and 22 is 110.

Question 18

The flight fare between two cities is increased in the ratio 9 : 11. If the original one-way fare was Rs 18,000, what is the increase in the ticket price (in Rs)?

Accepted Answer

Introduction / Context: This question is a classic application of ratios and percentage increase in the context of travel fares. The fare between two cities has been increased in a specific ratio, and we are given the original ticket price. Our goal is to translate the ratio information into actual rupee values and then compute how much extra a passenger now pays after the hike. This type of question tests understanding of ratios, proportional thinking, and simple arithmetic rather than complicated algebra. Given Data / Assumptions: • Original flight fare between the two cities = Rs 18,000. • Ratio of old fare to new fare = 9 : 11. • We assume the ratio 9 : 11 directly compares old fare : new fare. • We need to find the increase in fare in rupees, that is, new fare minus old fare. Concept / Approach: The key idea is that if the old and new fares are in the ratio 9 : 11, then the new fare can be expressed as (11/9) times the old fare. Once the new fare is found, the increase is simply the difference between the new fare and the original fare. This is a direct use of the concept of proportionality based on ratios. No percentage calculation is strictly required, although we could also interpret the change as a percentage increase if desired. Step-by-Step Solution: Step 1: Let the old fare be represented by the ratio value 9 parts and the new fare by 11 parts. Step 2: We know the actual old fare is Rs 18,000. So 9 parts correspond to Rs 18,000. Step 3: Value of 1 part = 18,000 / 9 = Rs 2,000. Step 4: New fare corresponds to 11 parts = 11 * 2,000 = Rs 22,000. Step 5: Increase in fare = New fare − Old fare = 22,000 − 18,000 = Rs 4,000. Verification / Alternative Check: We can verify by checking the ratio directly. Old fare : New fare = 18,000 : 22,000. Divide both terms by 2,000 to simplify. We get 9 : 11, which matches the given ratio. This confirms that the interpretation and calculations are consistent. Therefore, the increase in fare as Rs 4,000 is correct. Why Other Options Are Wrong: • Rs 22,000: This is the new total fare, not the increase. It confuses the final amount with the increment. • Rs 3,600: This value might arise from a mistaken percentage calculation or incorrect ratio scaling. • Rs 20,000: This looks like another possible total fare guess, but it does not satisfy the 9 : 11 ratio with 18,000. Common Pitfalls: Students often misread the ratio 9 : 11 as meaning an 11 percent increase or directly subtract 9 from 11 and relate this difference incorrectly to the original amount. Another mistake is to compute the new fare correctly but forget that the question asks specifically for the increase, not the final fare. Careful reading of what is being asked is as important as the numerical work. Final Answer: The new fare is Rs 22,000 and the increase over the original fare of Rs 18,000 is Rs 4,000.

Question 19

In coordinate geometry, in what ratio does the point T(x, 0) on the x-axis divide the segment joining the points S(-4, -1) and U(1, 4)?

Accepted Answer

Introduction / Context: This problem uses the concept of internal division of a line segment in coordinate geometry. A point T lies on the x-axis and divides the line segment joining two given points S and U. Because T is on the x-axis, its y-coordinate is zero, which allows us to use the section concept or a simple linear interpolation approach on the y-coordinates to find the ratio in which T divides the segment. Such questions frequently appear in aptitude and entrance exams to test understanding of coordinate geometry basics. Given Data / Assumptions: • Point S has coordinates S(-4, -1). • Point U has coordinates U(1, 4). • Point T lies on the x-axis, so T has coordinates (x, 0). • T divides segment SU internally in some ratio k : (1 − k) or m : n which we must find. Concept / Approach: If a point divides a line segment internally in a given ratio, its coordinates can be obtained using the section formula. However, in this question, a more direct way is to focus on the y-coordinates. The y-coordinate varies linearly between the endpoints of the segment. Because T is on the x-axis, its y-coordinate is zero, so we can write an equation based on the change from -1 to 4. The parameter that takes us from -1 to 0 will give the fraction of the segment from S to T. From that fraction we can deduce the ratio ST : TU. Step-by-Step Solution: Step 1: Let T divide the segment from S to U in the ratio ST : TU = k : (1 − k), where k is the fraction from S. Step 2: Parameterize the y-coordinate along SU. Starting from S, the y-coordinate after fraction k is y = -1 + k * (4 - (-1)). Step 3: Simplify the change: 4 - (-1) = 5. So y = -1 + 5k. Step 4: Since T is on the x-axis, its y-coordinate must be zero. So set -1 + 5k = 0. Step 5: Solve for k: 5k = 1, so k = 1/5. Step 6: This means ST : SU = 1 : 5, so ST : TU = (1/5) : (4/5) = 1 : 4. Verification / Alternative Check: We can find the coordinates of T explicitly. For the x-coordinate, x = -4 + k * (1 - (-4)) = -4 + (1/5) * 5 = -4 + 1 = -3. So T(-3, 0). Now compute distances ST and TU using rough comparisons or actual lengths. T is closer to S than to U, and the ratio of the progress along the segment based on the parameter is indeed 1 : 4. This matches our earlier result. Why Other Options Are Wrong: • 4 : 1 would imply T is much closer to U than to S, which is not consistent with the y-interpolation that gives k = 1/5. • 1 : 2 and 2 : 1 are intermediate ratios and do not satisfy the condition y = 0 when we interpolate between -1 and 4. Common Pitfalls: Students sometimes try to work with the x-coordinates first, which can be more confusing. Another frequent mistake is to assume that equal changes in x and y correspond to equal ratios, which is not correct. The safest approach is to use the parameter on the y-coordinate because the condition y = 0 is very clear and leads directly to the correct ratio. Final Answer: The point T divides the segment joining S(-4, -1) and U(1, 4) in the ratio 1 : 4 (ST : TU).

Question 20

What is the third proportional to the numbers 10 and 25 (that is, 10 : 25 = 25 : x)?

Accepted Answer

Introduction / Context: This question tests the concept of third proportional in ratio and proportion. When two numbers are given, the third proportional is defined using a proportional relationship. It is a standard topic in quantitative aptitude and appears frequently in competitive exams. The numbers 10 and 25 are given, and we are asked to find the value of x such that the proportional relationship is satisfied. Given Data / Assumptions: • First number = 10. • Second number = 25. • Third proportional = x such that 10 : 25 = 25 : x. • All numbers are assumed to be real positive quantities. Concept / Approach: If a, b and c are in continued proportion (or if c is the third proportional to a and b), then a : b = b : c. In terms of fractions, a / b = b / c. From this, we get c = b^2 / a. The question is therefore a direct application of this formula. We simply substitute a = 10 and b = 25 and compute c. This avoids any unnecessary complication and is the fastest way to solve such questions. Step-by-Step Solution: Step 1: Write the proportion according to the definition of third proportional: 10 : 25 = 25 : x. Step 2: Convert the proportion into an equation using fractions: 10 / 25 = 25 / x. Step 3: Cross multiply to remove the denominators: 10 * x = 25 * 25. Step 4: Compute 25 * 25 = 625. Step 5: So 10x = 625. Divide both sides by 10 to get x = 625 / 10. Step 6: Simplify x = 62.5. Verification / Alternative Check: We can check whether the original proportion holds with x = 62.5. Compute 10 : 25 which simplifies to 2 : 5. Now compute 25 : 62.5. Divide both numbers by 12.5 to simplify. We get 25 / 12.5 = 2 and 62.5 / 12.5 = 5, so 25 : 62.5 = 2 : 5. Both ratios match, which means the proportion is correct and x = 62.5 is confirmed as the third proportional. Why Other Options Are Wrong: • 125 and 150 are larger numbers that might come from squaring incorrectly or misplacing terms in the proportion, but they do not satisfy 10 / 25 = 25 / x. • 225 is another distractor that also fails the proportional test when substituted for x. Common Pitfalls: A common mistake is to confuse third proportional with mean proportional (geometric mean). The mean proportional m between a and b satisfies a : m = m : b, which is a different relation. Another pitfall is to invert the proportion and write 25 : 10 = 25 : x, which gives the wrong formula. Remember that the pattern for third proportional is a : b = b : c, and the formula c = b^2 / a follows directly from that. Final Answer: The third proportional to 10 and 25 is 62.5.

Ratio and Proportion Questions