Q: Direction: Answer the questions based on the information given below. The given line graph shows the revenue generated from debit card transactions and cash transactions for five different stores, A, B, C, D and E. <img src="/api/images/revenue-…

Concept & Logic: The problem requires finding the unknown variables p and q using the given conditions, and then calculating specific revenue values to find the required ratio. Step-by-Step Solution: • Find p: According to Note (d), the sum of revenue from debit card transactions for stores 'A' and 'E' is 420 million. • From the graph data, revenue from debit cards for store A is 6p and for store E is 8p. • Therefore, 6p + 8p = 420 • 14p = 420 \Rightarrow p = 30 • Find q: According to Note (d), the difference between the revenue from cash transactions for stores 'C' and 'D' is 48 million. • From the graph data, revenue from cash transactions for store C is 10q and for store D is 8q. • Therefore, 10q - 8q = 48 • 2q = 48 \Rightarrow q = 24 • Calculate Required Values: • Debit card revenue for store A = 6p = 6 × 30 = 180 • Debit card revenue for store B = 9p = 9 × 30 = 270 • Total debit card revenue for stores A and B together = 180 + 270 = 450 • Total revenue for store B worldwide (debit + cash): • Debit card revenue for B = 270 • Cash transaction revenue for B = 7.5q = 7.5 × 24 = 180 • Total revenue for B = 270 + 180 = 450 • Find the Ratio: • Ratio = (Total debit revenue of A and B) : (Total revenue of B) • Ratio = 450 : 450 = 1 : 1 Exam Strategy & Shortcut: Focus calculations only on the variables and specific data points requested by the question. Avoid calculating the revenue for all stores or determining their ratings, as it wastes valuable time. Common Pitfall: A common mistake is misreading the graph values or mixing up the debit card equation with the cash transaction variables. Always double-check the legend mapping. Final Answer: Therefore, the correct answer is 1 : 1.

Q: Direction: Study the following information carefully and answer the questions. The following pie chart shows the distribution of the number of laptops sold by five stores. <img src="/api/images/laptop-sales-distribution-pie-chart.png" alt="P…

Concept & Ratio Applications: By finding the exact number of laptops sold by Store D and E from the pie chart variables, we can use the given percentages and ratios to find unknown values for an external Store F. Step-by-Step Solution: • Given: z = 14 and y = 56 (solved from base conditions). Store D = (2z - 5)% = 23%. Store E = (8.5z + 1)^\circ = 120^\circ. • Calculation: Total laptops = 7200. • Store D laptops = 0.23 × 7200 = 1656. • Store E laptops = (120/360) × 7200 = 2400. • Deduction: Difference between D and E = 2400 - 1656 = 744. • This difference is 75% less than laptops sold by store F (S_F). Thus, it represents 25% (or (1/4)) of S_F. • 744 = 0.25 × S_F \implies S_F = 744 × 4 = 2976. • The ratio of sold to unsold (U_F) by store F is 62:75. • (2976/U_F) = (62/75) \implies U_F = (2976 × 75/62). • U_F = 48 × 75 = 3600. Exam Strategy & Shortcut: Recognize that "75% less than" means exactly 25% or (1/4) of the value. Multiplying the difference 744 by 4 quickly gives the sold units. For the ratio, directly scaling up 62 to 2976 (a multiplier of 48) allows you to just multiply 75 by 48 (which is 75 × 50 - 75 × 2 = 3750 - 150 = 3600) to find the unsold units mentally. Common Pitfall: A common pitfall is calculating 75% of 744 rather than correctly setting 744 as the 25% remainder of the unknown total. Final Answer: Therefore, the correct answer is 3600.

Q: The total number of students in a school is 2140. If the number of girls in the school is 1200, then what is the ratio of the total number of boys to the total number of girls in the school? (Bank Recruitment, 2009)

Concept & Ratio: To find the ratio of two quantities, we first need their exact values and then divide them, simplifying to the lowest terms. Ratio = \frac{Value 1}{Value 2} Step-by-Step Solution: • Given total number of students = 2140. • Given number of girls = 1200. • Calculate the number of boys by subtracting girls from the total: 2140 - 1200 = 940. • Set up the ratio of boys to girls: 940 : 1200. • Simplify the ratio by cancelling the common zero (dividing by 10): 94 : 120. • Further simplify by dividing both numbers by their greatest common divisor, which is 2: (94/2) : (120/2) = 47 : 60 Exam Strategy & Shortcut: Calculate the number of boys (940). Notice that 940 ends in a 0, and removing it gives 94. Half of 94 is 47, which is a prime number. Look directly at the given options: only option (c) has 47 as the first term. You can instantly select it without calculating the denominator's simplified form to save time. Common Pitfall: Finding the ratio of girls to boys (60 : 47) or girls to total students instead of the specifically requested boys to girls. Always double-check the order required by the question! Final Answer: Therefore, the correct answer is 47 : 60.

Q: The value of $x$ where $x : 2\frac{1}{3} : : 21 : 50$ is

Concept & Proportion: In a proportion where four terms are related as a : b : : c : d, the product of the extreme terms (outer terms) is equal to the product of the mean terms (inner terms). a × d = b × c Step-by-Step Solution: • Given: The proportion is x : 2(1/3) : : 21 : 50. • Convert mixed fraction: Convert 2(1/3) to an improper fraction: ((2 × 3) + 1/3) = (7/3). • Set up the equation: Use the product of extremes equals product of means rule. x × 50 = (7/3) × 21 • Simplify: 50x = 7 × 7 50x = 49 • Solve for x: x = (49/50) Exam Strategy & Shortcut: Treat the proportion as equivalent fractions: (x/7/3) = (21/50). Simply cross-multiply the 7/3 to the right side: x = (21 × (7/3)/50). The 3 cancels into the 21 giving 7. So, x = (7 × 7/50) = (49/50). Common Pitfall: A frequent mistake is improperly converting the mixed number 2(1/3), or incorrectly pairing the extremes and means, leading to a flipped fraction. Final Answer: Therefore, the correct answer is (49/50).

Q: If $a : b = 7 : 9$ and $b : c = 15 : 7$, then what is $a : c$?

Concept & Compounding Ratios: To find the direct ratio between the first and last variables in a chain of ratios, you can multiply the corresponding fractions. (a/c) = (a/b) × (b/c) Step-by-Step Solution: 1. Convert the given ratios into fractions: (a/b) = (7/9) (b/c) = (15/7) 2. We want to find a : c, which is equivalent to (a/c). 3. Multiply the two fractions together. The b terms will cancel out: (a/c) = ((7/9)) × ((15/7)) 4. Simplify the expression by cancelling the common factor of 7 in the numerator and denominator: (a/c) = (15/9) 5. Reduce the resulting fraction by dividing the numerator and denominator by their greatest common divisor, which is 3: (a/c) = (5/3) 6. Convert back to ratio format: a : c = 5 : 3 Exam Strategy & Shortcut: Write the numbers down and immediately cancel out common values diagonally. In (7/9) × (15/7), the 7s visually cancel out immediately leaving (15/9), which simplifies to 5 : 3 in seconds. Common Pitfall: Attempting to equalize the middle term b in both ratios (finding a common multiple for 9 and 15) is a valid method but takes significantly more time than simple multiplication. Final Answer: Therefore, the correct answer is 5 : 3.

Q: If $a : b = 2 : 5$, then the value of $(2a + 3b) : (7a + 5b)$ is

Concept & Ratio Substitution: When given a ratio of variables like a : b = m : n and asked to find the value of a homogeneous expression (where all terms have the same total degree), we can directly substitute the proportional values a = m and b = n. Step-by-Step Solution: - Given: a : b = 2 : 5. - Substitute: Let a = 2 and b = 5. - Expression: (2a + 3b) : (7a + 5b) - Calculate Numerator: 2(2) + 3(5) = 4 + 15 = 19 - Calculate Denominator: 7(2) + 5(5) = 14 + 25 = 39 - Result: The ratio is 19 : 39. Exam Strategy & Shortcut: Since the degree of every term in the numerator and denominator is 1, we can bypass using a constant k (e.g., a = 2k, b = 5k) because k will completely factor out and cancel. Direct substitution is the fastest path. Common Pitfall: A common mistake is trying to solve for a or b individually, forgetting that we only need their relative sizes. Also, simple arithmetic errors during multiplication or addition can lead to choosing a distractor option. Final Answer: Therefore, the correct answer is 19 : 39.

Question 1

What is the third proportional to the pair of numbers 9 and 15?

Accepted Answer

Introduction / Context: This question tests the concept of third proportional between two numbers, which appears frequently in ratio and proportion chapters. Understanding proportional relationships is important in many aptitude topics, including similarity in geometry, financial comparisons, and mixture problems. The problem itself is short, but it checks whether the student remembers the specific definition of the third proportional and can apply the formula correctly. Given Data / Assumptions: First number a = 9. Second number b = 15. We are asked to find the third proportional c such that a : b = b : c. All numbers are positive real numbers and normal arithmetic rules apply. Concept / Approach: By definition, if a, b, c are in continued proportion, then a : b = b : c. This gives a proportion equation a / b = b / c. Cross multiplication yields a * c = b * b, so c = b^2 / a. Thus to find the third proportional, we simply square the second number and divide by the first number. This is a direct and very fast method that avoids unnecessary steps. Step-by-Step Solution: Given a = 9 and b = 15. For third proportional c, we require 9 : 15 = 15 : c. Write this as 9 / 15 = 15 / c. Cross multiply to get 9 * c = 15 * 15. So 9 * c = 225. Therefore c = 225 / 9 = 25. Verification / Alternative check: Check the ratio condition directly. With c = 25, first ratio 9:15 simplifies to 3:5 when both terms are divided by 3. Second ratio 15:25 simplifies to 3:5 when both terms are divided by 5. Since both ratios reduce to 3:5, the condition 9:15 = 15:25 is satisfied, confirming that 25 is indeed the third proportional. Why Other Options Are Wrong: Option 30 would give a second ratio 15:30, which simplifies to 1:2, not equal to 9:15 which simplifies to 3:5. Option 27 would yield a ratio 15:27 equal to 5:9, again not equal to 3:5, so it does not satisfy the proportionality condition. Option 36 leads to 15:36 which simplifies to 5:12, again different from the required ratio 3:5, so this choice is incorrect. Common Pitfalls: Some learners mistakenly compute c as a * b instead of b^2 / a, confusing the definition of continued proportion. Others mix up the idea of mean proportional (which involves square roots) with third proportional, which uses the square of the middle term. Another frequent error is to perform addition or subtraction on the numbers instead of maintaining the ratio equality through cross multiplication. Final Answer: The third proportional to 9 and 15 is 25.

Question 2

A total amount of Rs. 7,800 is to be divided between A, B and C in the ratio 1/2 : 1/3 : 1/4. What is the share of B (in Rs.)?

Accepted Answer

Introduction / Context: This question involves dividing a given sum of money in a ratio where the ratio terms themselves are fractions. Such questions are slightly more advanced than simple whole number ratios, but they can be solved comfortably by converting the fractional ratio into an equivalent whole number ratio. This technique appears often in aptitude tests involving distribution of money or resources among different people. Given Data / Assumptions: Total amount to be divided = Rs. 7,800. Ratio of shares of A, B and C = 1/2 : 1/3 : 1/4. All three people together receive the full Rs. 7,800, with no money left over. We are asked to find the share of B only. Concept / Approach: When ratio terms are fractions, a common and efficient method is to clear denominators by multiplying every term by the least common multiple of the denominators. This converts the fractional ratio into an equivalent integer ratio. Then we treat those integers as parts of the total. Each person receives (individual parts / sum of all parts) times the total amount. This keeps the relative proportions exactly the same while making calculations easier. Step-by-Step Solution: Original ratio = 1/2 : 1/3 : 1/4. The denominators are 2, 3 and 4. Their least common multiple is 12. Multiply each term in the ratio by 12. A's part becomes (1/2) * 12 = 6, B's part becomes (1/3) * 12 = 4, C's part becomes (1/4) * 12 = 3. So the equivalent integer ratio is 6 : 4 : 3. Total number of parts = 6 + 4 + 3 = 13 parts. Value of one part = 7,800 / 13 = 600. Share of B = 4 parts = 4 * 600 = 2,400 rupees. Verification / Alternative check: We can also compute the shares of A and C for a quick consistency check. A gets 6 * 600 = 3,600 and C gets 3 * 600 = 1,800. Then total amount distributed is 3,600 + 2,400 + 1,800 = 7,800, which matches the given total. This confirms that the integer ratio and part calculation are correct. Why Other Options Are Wrong: Option 3,600 represents the share of A, who has 6 parts, not B who has 4 parts. Option 1,800 corresponds to the share of C with 3 parts, so it does not answer the question about B. Option 1,200 would be equal to 2 parts of 600 each, which does not match the 4 parts required for B according to the derived ratio. Common Pitfalls: Some learners forget to convert the fractional ratio into whole numbers and try to work directly with fractions, which can lead to arithmetic mistakes. Others add the numerators and denominators incorrectly when combining fractions, which distorts the intended ratio. A common conceptual error is to treat 1/2 : 1/3 : 1/4 as 1:2:3 or some other simplified form, which changes the proportional relationships completely. Final Answer: The share of B from the total amount is Rs. 2,400.

Question 3

Bunty has candies and chewing gums in his sweet box in the ratio 7:13. After he eats 8 candies and 11 chewing gums, the ratio becomes 1:2. How many candies does he have now?

Accepted Answer

Introduction / Context: This problem is a good example of how ratios can change when certain items are added or removed, and how to convert those conditions into equations. It involves two categories, candies and chewing gums, and describes their numbers before and after Bunty eats some of them. Such questions test the learner's ability to represent ratios algebraically and solve a simple linear equation. Given Data / Assumptions: Initial ratio of candies to chewing gums = 7 : 13. Bunty eats 8 candies. Bunty eats 11 chewing gums. The new ratio of candies to chewing gums becomes 1 : 2. We need the number of candies remaining after Bunty has eaten some. Concept / Approach: Let the initial numbers of candies and gums be 7x and 13x respectively. After Bunty eats some items, the new counts are adjusted accordingly. The new ratio is given as 1:2, so we equate the algebraic ratio of the new counts to 1:2. This produces a linear equation in x that we can solve. Once x is known, we find the actual counts and in particular the current number of candies. Step-by-Step Solution: Let initial candies = 7x and initial chewing gums = 13x. After eating, candies left = 7x - 8. After eating, chewing gums left = 13x - 11. Given that the new ratio of candies to chewing gums is 1 : 2. So (7x - 8) / (13x - 11) = 1 / 2. Cross multiply: 2 * (7x - 8) = 1 * (13x - 11). Simplify: 14x - 16 = 13x - 11. Therefore x = 5. Candies now = 7x - 8 = 7 * 5 - 8 = 35 - 8 = 27. Verification / Alternative check: With x = 5, initial numbers are 35 candies and 65 chewing gums. After Bunty eats, candies become 27 and chewing gums become 54. The new ratio 27:54 simplifies by dividing both terms by 27, which gives 1:2. This matches the ratio stated in the problem, confirming that our algebraic setup and solution are correct. Why Other Options Are Wrong: Option 65 represents the initial number of chewing gums, not the candies remaining after Bunty eats some of them. Option 35 corresponds to the initial number of candies before any are eaten, not the final number requested. Option 54 is the final number of chewing gums, not the number of candies, so it does not answer the question asked. Common Pitfalls: Sometimes students confuse initial and final quantities and directly set 7:13 equal to 1:2 without accounting for items eaten. Another typical mistake is to subtract the ratio numbers 7 and 13 by 8 and 11 instead of reducing the actual counts 7x and 13x by those numbers. Some learners forget to cross multiply correctly and make algebraic mistakes when solving the resulting equation. Final Answer: After eating some sweets, Bunty now has 27 candies remaining in his box.

Question 4

The sum of three numbers is 162. The ratio of the first number to the second number is 5:7 and the ratio of the second number to the third number is 5:3. What is the value of the second number?

Accepted Answer

Introduction / Context: This question combines two ratio relationships and a given total sum to find individual numbers. It is a common pattern in aptitude tests where you must merge separate ratios into a single combined ratio for all the quantities involved. Once that combined ratio is obtained, the actual values follow by simple proportional reasoning using the total sum. Given Data / Assumptions: There are three numbers whose sum is 162. First : Second = 5 : 7. Second : Third = 5 : 3. All numbers are positive real numbers, typically treated as integers in such questions. We need to find the second number specifically. Concept / Approach: To combine two ratios that share a common term, we express them with a common value for the shared term. Here the second number appears in both ratios, so we scale each ratio so that the second number has the same multiple in both. That gives us a unified ratio for first, second and third together. Then we use the total sum and the total number of parts in the ratio to find the value of one part and hence each number. Step-by-Step Solution: Given first : second = 5 : 7. Given second : third = 5 : 3. Let first = 5a and second = 7a from the first ratio. From the second ratio, let second = 5b and third = 3b. We need the expressions for the second number to match, so set 7a = 5b. Choose 7a = 5b = 35k, so a = 5k and b = 7k. Then first = 5a = 5 * 5k = 25k, second = 7a = 35k, third = 3b = 3 * 7k = 21k. Total sum = 25k + 35k + 21k = 81k = 162. Therefore k = 162 / 81 = 2. Second number = 35k = 35 * 2 = 70. Verification / Alternative check: With k = 2, the three numbers are 50, 70 and 42. Check the ratios: 50:70 simplifies to 5:7 and 70:42 simplifies to 5:3 when divided by 14. Also, the sum 50 + 70 + 42 = 162 matches the given total, confirming that the second number 70 is correct. Why Other Options Are Wrong: Option 35 is only half of the correct second number and would produce a sum much smaller than 162 if used in the ratio pattern derived above. Option 80 does not fit the derived combined ratio, and plugging it in breaks the given relationships between first, second and third numbers. Option 40 is also inconsistent with the required ratios and would not satisfy both ratio conditions simultaneously. Common Pitfalls: Students often add the ratios directly without first matching the common term, which leads to incorrect combined ratios. Another mistake is to set 5:7 and 5:3 directly as 5:7:3 instead of correctly scaling them to align the shared term. Some learners also forget to use the total sum of 162 to scale the combined ratio to actual numbers, leaving the answer only in parts, not in real values. Final Answer: The value of the second number is 70.

Question 5

In an army selection process, the ratio of selected to unselected candidates was 3:1. If 60 fewer candidates had applied and 30 fewer had been selected, the ratio of selected to unselected would have become 5:1. How many candidates originally applie…

Accepted Answer

Introduction / Context: This problem is a classic example of ratio and algebra in selection processes. It deals with the relationship between selected and unselected candidates in an exam or recruitment and describes how that relationship would change if the number of applicants and selected candidates changed. Such questions test the ability to translate ratio statements into equations and handle changes carefully. Given Data / Assumptions: Initial ratio of selected to unselected candidates = 3 : 1. If 60 fewer candidates had applied, and 30 fewer candidates had been selected, the new ratio would have become 5 : 1. All candidates are either selected or unselected, with no other category. We are asked to find the original total number of candidates who applied. Concept / Approach: Let the numbers of selected and unselected candidates be multiples of the given ratio. Use variables to represent these numbers and then apply the changes described in the question. The new ratio after the hypothetical changes provides a second equation. Solving these equations together yields the original numbers. This is a standard algebraic approach for problems involving changing ratios. Step-by-Step Solution: Let initially selected candidates = 3x and unselected candidates = x, based on the ratio 3:1. So total candidates who applied initially = 3x + x = 4x. Under the hypothetical situation, 60 fewer candidates applied, so new total = 4x - 60. Also, 30 fewer candidates are selected, so selected becomes 3x - 30. Then unselected under the new condition = (4x - 60) - (3x - 30) = x - 30. Given new ratio of selected to unselected is 5 : 1, so (3x - 30) / (x - 30) = 5 / 1. Cross multiply: 3x - 30 = 5(x - 30). So 3x - 30 = 5x - 150. Rearrange: 120 = 2x, hence x = 60. Original total candidates = 4x = 4 * 60 = 240. Verification / Alternative check: With x = 60, initially selected = 180 and unselected = 60, giving ratio 3:1. Under the hypothetical change, total becomes 240 - 60 = 180, selected becomes 180 - 30 = 150, unselected becomes 180 - 150 = 30. The new ratio 150:30 simplifies to 5:1, which matches the problem statement, confirming our answer. Why Other Options Are Wrong: Option 480 would make x = 120, which breaks the second ratio condition when the hypothetical changes are applied. Option 120 would give x = 30, but then unselected candidates after change become zero, which is inconsistent with a 5:1 ratio. Option 720 is three times the correct total and would not satisfy both ratio conditions simultaneously when checked. Common Pitfalls: A common mistake is to subtract 60 and 30 directly from the ratio numbers, instead of from the actual totals and selected counts. Students sometimes forget that unselected after the change must be recomputed from the new total minus the new selected, not just from a scaled ratio. Another pitfall is cross multiplying incorrectly or simplifying the equation in a way that leads to negative or non integral values of x. Final Answer: The original number of candidates who applied for the process was 240.

Question 6

Two whole numbers are such that the ratio of the first number to the second number is 4:3 and the square of the first number exceeds the square of the second number by 112. What is the value of the smaller number?

Accepted Answer

Introduction / Context: This question combines the concept of ratios with a condition involving squares of numbers. Problems involving squares often appear in quantitative aptitude because they require both algebraic understanding and careful arithmetic. Here, the ratio of two numbers is given, and a relationship between their squares allows us to find specific values for the numbers. Given Data / Assumptions: Two whole numbers are in the ratio 4 : 3 (first : second). The square of the first number exceeds the square of the second number by 112. Both numbers are whole numbers (non negative integers). We are asked to find the smaller of the two numbers. Concept / Approach: If two numbers are in the ratio 4:3, we can represent them as 4k and 3k for some positive integer k. The condition about squares then becomes an equation involving k. Specifically, the difference of their squares is known. We can use the algebraic identity a^2 - b^2 = (a - b) * (a + b) to simplify calculations and solve for k. Once k is found, we can determine both numbers and identify the smaller one. Step-by-Step Solution: Let the two numbers be 4k and 3k, with 4k being the larger number. Then the square of the first number is (4k)^2 = 16k^2. The square of the second number is (3k)^2 = 9k^2. Given that the square of the first exceeds the square of the second by 112, so 16k^2 - 9k^2 = 112. Simplify the left side: 7k^2 = 112. Therefore k^2 = 112 / 7 = 16. So k = 4 (taking positive root because numbers are whole and positive). The two numbers are then 4k = 4 * 4 = 16 and 3k = 3 * 4 = 12. The smaller number is 12. Verification / Alternative check: Check the condition directly using the found numbers. The ratio 16:12 simplifies by dividing both numbers by 4 to 4:3, matching the given ratio. The squares are 16^2 = 256 and 12^2 = 144. Their difference is 256 - 144 = 112, which matches the problem statement exactly. This confirms that the calculations and algebraic setup are consistent. Why Other Options Are Wrong: Option 3 would produce numbers 4 * 3 = 12 and 3 * 3 = 9, whose squares differ by 144 - 81 = 63, not 112. Option 4 would yield numbers 16 and 12 if taken as k, which confuses the role of k and the actual numbers and does not satisfy the square difference with the correct ratio simultaneously. Option 36 is much larger and does not correspond to 3k when the other number is 4k under the given square difference condition. Common Pitfalls: Some learners set up the equation using the difference of the numbers instead of the difference of their squares, which leads to a different and incorrect equation. Others forget to use the ratio and try to guess numbers directly, resulting in a time consuming and error prone approach. Another common mistake is to treat 16k^2 - 9k^2 as (16 - 9)k instead of (16 - 9)k^2, which changes the algebra completely. Final Answer: The value of the smaller number is 12.

Question 7

Two tangents are drawn to a circle from an external point. Consider the three statements: (1) The tangents subtend equal angles at the centre of the circle. (2) The tangents drawn from the external point to the circle are equal in length. (3) The ta…

Accepted Answer

Introduction / Context: This is a conceptual geometry question about tangents drawn to a circle from an external point. Properties of tangents are very important in school geometry and in aptitude exams because they lead to many angle and length relationships used in further problem solving. Here we are given three separate statements and asked which of them are true for tangents drawn from the same external point to a circle. Given Data / Assumptions: Two tangents are drawn to a circle from one external point P. Statement (1) claims that these tangents subtend equal angles at the centre of the circle. Statement (2) claims that the tangents drawn from the external point to the circle are equal in length. Statement (3) claims that these tangents are equally inclined to the line segment joining the centre to the external point P. We assume the usual Euclidean geometry rules for circles and tangents. Concept / Approach: A key property of tangents from an external point is that the two tangents to a circle are equal in length. In addition, the radii drawn to the points of tangency are perpendicular to the tangents. By using symmetry and triangle congruence, we can show that the angles made at the centre by the radii to the tangency points are equal, and the angles between each tangent and the line joining the external point to the centre are also equal. So we examine each statement using these standard results. Step-by-Step Solution: Let the circle have centre O and the external point be P. Tangents touch the circle at points A and B, so PA and PB are tangents. By a basic theorem, tangents drawn from an external point to a circle are equal, so PA = PB, which supports statement (2). Draw radii OA and OB; these are perpendicular to the tangents at A and B respectively, so OA is perpendicular to PA and OB is perpendicular to PB. Consider triangles OAP and OBP. OP is common, OA = OB (both radii), and PA = PB (tangents from P). By side side side congruence, triangle OAP is congruent to triangle OBP. Therefore corresponding angles at the centre, angle AOB and angle BOA, are equal, confirming statement (1). Also, the angles between OP and each tangent, angle AOP and angle BOP, are equal, showing that the tangents are equally inclined to OP, which validates statement (3). Verification / Alternative check: The configuration is symmetric about the line OP. If we conceptually reflect triangle OAP across OP we get triangle OBP, meaning every geometric feature on one side has an identical counterpart on the other side. This symmetry guarantees equal angles at the centre and equal inclination of tangents to OP, as well as equal tangent lengths. Hence all three statements are consistent with standard circle theorems. Why Other Options Are Wrong: Option 1 and 2 only omits statement (3), but we have seen that the congruent triangles imply equal inclination of each tangent to OP, so statement (3) is also true. Option 2 and 3 only omits the equal angle at the centre, yet congruence of triangles clearly forces the central angles formed by OA and OB to be equal. Option 1 and 3 only omits the equal tangent lengths, which is one of the most fundamental theorems regarding tangents from an external point, so that omission makes the option incomplete. Common Pitfalls: Some learners incorrectly think that only the tangent lengths are equal and do not realise the stronger symmetry that also forces equal central angles. Others confuse the phrase equally inclined with equal length and treat statements (2) and (3) as duplicates, which they are not. Another error is to misinterpret the position of the external point and not draw the radii to the points of tangency, which makes it harder to visualise the congruent triangles and leads to wrong conclusions. Final Answer: All three statements are true; the correct choice is 1, 2 and 3.

Question 8

Of three positive numbers, the ratio of the first to the second is 5:2 and the ratio of the second to the third is 5:4. If the product of the first and the third numbers is 1,800, what is the sum of the three numbers?

Accepted Answer

Introduction / Context: This question links two ratio relationships and a given product of two numbers to find the sum of all three numbers. It is a nice example of how proportional reasoning can be combined with an additional numerical condition to determine exact values rather than just relative sizes. Such problems are common in competitive exams to test understanding of ratios and algebra. Given Data / Assumptions: Three positive numbers are involved. First : Second = 5 : 2. Second : Third = 5 : 4. The product of the first and the third numbers is 1,800. We are asked for the sum of all three numbers. Concept / Approach: We first express the three numbers in terms of a common parameter by combining the two given ratios. Once we obtain a consistent combined ratio for first, second and third, we use the product of the first and third numbers to determine the value of the scaling parameter. After that, we compute each number and finally sum them. This approach avoids guesswork and uses systematic algebra. Step-by-Step Solution: Let the three numbers be A, B and C. From the first ratio, A : B = 5 : 2, so let A = 5x and B = 2x. From the second ratio, B : C = 5 : 4, so let B = 5y and C = 4y. We need the two expressions for B to be equal, so 2x = 5y. Set 2x = 5y = some common value, for convenience take 2x = 5y = 10k, giving x = 5k and y = 2k. Then A = 5x = 5 * 5k = 25k, B = 2x = 10k, and C = 4y = 4 * 2k = 8k. Given that A * C = 1,800, so 25k * 8k = 1,800. Thus 200k^2 = 1,800, which implies k^2 = 1,800 / 200 = 9. Therefore k = 3 (positive because numbers are positive). Now A = 25k = 75, B = 10k = 30, C = 8k = 24. Sum of the three numbers = 75 + 30 + 24 = 129. Verification / Alternative check: Check both ratios and the product. A:B = 75:30 simplifies to 5:2, and B:C = 30:24 simplifies to 5:4, satisfying the ratio conditions. The product A * C = 75 * 24 = 1,800, matching the given data. Therefore the sum 129 is consistent with all parts of the question. Why Other Options Are Wrong: Option 43 is far too small and would correspond to much smaller numbers that cannot have a product of 1,800 when multiplying the first and third. Option 133 would require slightly different scaling values that would break either the ratios or the product condition. Option 119 also does not arise from any integer scaling factor k that simultaneously satisfies both ratios and the given product of 1,800. Common Pitfalls: Some students try to add the ratios directly without first aligning the common term B, which leads to inconsistent numbers. Another error is to forget that the product condition A * C = 1,800 must hold exactly and treat it as approximate, leading to rounding mistakes. Learners also sometimes take k^2 = 9 but then accidentally use k = 9 instead of k = 3, which makes the numbers much larger and inconsistent with the product condition. Final Answer: The sum of the three numbers is 129.

Question 9

In an army selection process, the ratio of selected to unselected candidates was 4:1. If 90 fewer candidates had applied and 20 fewer had been selected, the ratio of selected to unselected would have become 5:1. How many candidates originally applie…

Accepted Answer

Introduction / Context: This is another ratio based selection question, similar in style to other army or recruitment process problems. It describes initial numbers of selected and unselected candidates and then presents a hypothetical change in which some candidates are removed from the pool and fewer candidates are selected. The question tests algebraic modelling of these changes and correct use of ratios. Given Data / Assumptions: Initially, the ratio of selected to unselected candidates is 4 : 1. If 90 fewer candidates had applied, and 20 fewer candidates had been selected, the new ratio would have been 5 : 1. There are only two categories of candidates: selected and unselected. We need to find the original number of candidates who applied. Concept / Approach: We represent the initial numbers of selected and unselected candidates using a variable multiple of the ratio 4:1. Then we adjust those numbers according to the hypothetical changes described, and express the resulting new numbers as another ratio equation. Solving this equation will give the multiple and therefore the original total number of candidates. This method is systematic and avoids guesswork. Step-by-Step Solution: Let initially selected candidates = 4x and unselected candidates = x. Total candidates who originally applied = 4x + x = 5x. If 90 fewer candidates had applied, new total would have been 5x - 90. If 20 fewer had been selected, new selected would have been 4x - 20. Therefore new unselected = (5x - 90) - (4x - 20) = x - 70. Given new ratio selected : unselected = 5 : 1, so (4x - 20) / (x - 70) = 5 / 1. Cross multiply: 4x - 20 = 5(x - 70). So 4x - 20 = 5x - 350. Rearrange: 330 = x, since 5x - 4x = x and -20 + 350 = 330. Original total candidates = 5x = 5 * 330 = 1,650. Verification / Alternative check: With x = 330, initial selected = 1,320 and unselected = 330, giving ratio 4:1. Under the hypothetical change, total = 1,650 - 90 = 1,560, selected = 1,320 - 20 = 1,300, and unselected = 1,560 - 1,300 = 260. The new ratio 1,300:260 simplifies by dividing by 260 to 5:1. This perfectly matches the problem statement, confirming that 1,650 is correct. Why Other Options Are Wrong: Option 3,300 corresponds to taking x = 660, which does not satisfy the second ratio when the changes are applied. Option 825 would give x = 165, leading to non integer or inconsistent unselected counts when the hypothetical changes are considered. Option 4,950 is three times 1,650 and again fails to satisfy the new ratio 5:1 after reductions are applied. Common Pitfalls: Some students subtract 90 and 20 from the ratio terms instead of the actual counts, confusing ratios with real numbers. Others forget to recompute unselected candidates after the changes and instead attempt to scale the ratio directly, which is incorrect. A frequent algebraic error is mishandling signs when bringing terms to one side of the equation, which leads to a wrong value for x. Final Answer: The original number of candidates who applied for the selection process was 1,650.

Question 10

The salary of Gadadhar is 3/2 times the salary of Haamid, and the salary of Sarvesh is 4/3 times the salary of Haamid. What is the ratio of Gadadhar's salary to Sarvesh's salary?

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Introduction / Context: This is a direct ratio comparison problem involving salaries of three people. We are told how the salaries of Gadadhar and Sarvesh relate to the salary of Haamid. The goal is to express the ratio of Gadadhar's salary to Sarvesh's salary. Such questions test the ability to handle fractional multipliers and convert them into a simple ratio between two quantities. Given Data / Assumptions: Let the salary of Haamid be some base amount H. Salary of Gadadhar = 3/2 times H. Salary of Sarvesh = 4/3 times H. We need the ratio of Gadadhar's salary to Sarvesh's salary. All salaries are positive real numbers. Concept / Approach: When two quantities are given as fractional multiples of a third quantity, we can write them in terms of that base variable and then form their ratio directly. Here we write Gadadhar's salary and Sarvesh's salary both in terms of H and simplify the resulting fraction. The base salary H cancels out, leaving a ratio involving only the given fractions. This method is simple and avoids any need to assume actual rupee values. Step-by-Step Solution: Let Haamid's salary be H. Then Gadadhar's salary = (3/2) * H. Sarvesh's salary = (4/3) * H. Required ratio = Gadadhar's salary : Sarvesh's salary. So ratio = (3/2) * H : (4/3) * H. Cancel H from both terms, giving (3/2) : (4/3). Write the ratio as a single fraction: [(3/2) / (4/3)] = (3/2) * (3/4) = 9/8. Therefore the ratio of Gadadhar's salary to Sarvesh's salary is 9 : 8. Verification / Alternative check: We can pick a convenient value for H to check. For example, let H = 6. Then Gadadhar's salary = (3/2) * 6 = 9 and Sarvesh's salary = (4/3) * 6 = 8. Their ratio is 9:8, which matches the derived ratio. This confirms that our algebraic simplification is correct and independent of the particular value chosen for H. Why Other Options Are Wrong: Option 1:2 would imply that Gadadhar earns half as much as Sarvesh, which contradicts the given multipliers 3/2 and 4/3. Option 2:1 would mean Gadadhar earns twice as much as Sarvesh, again inconsistent with the fractional relationship provided. Option 8:9 reverses the correct ratio and would correspond to Sarvesh earning more than Gadadhar, which does not follow from the given data. Common Pitfalls: Some students add the fractions 3/2 and 4/3 instead of forming the required ratio between them. Others forget to cancel the common factor H and mistakenly think the actual value of H matters when it cancels out completely. A few learners invert the ratio and compute (4/3) : (3/2) instead of the required (3/2) : (4/3), thus getting 8:9 instead of 9:8. Final Answer: The ratio of Gadadhar's salary to Sarvesh's salary is 9 : 8.

Question 11

Read the following passage and answer the given questions.  Study the following given pie - charts carefully and answer the questions given below. The first pie chart represents the breakup of the number of employees working in different departments…

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Concept & Logic: To find the required ratio, we need to calculate two values: the total number of male employees in the HR department, and the number of male employees promoted in the HR department. We use the given total values and percentages to find these subsets. Step-by-Step Solution: 1. Find total male employees in HR: Total employees = 6000 Total HR employees = 25% of 6000 = 1500 Ratio of Male:Female in HR = 7:5 Total Male in HR = (7/7+5) × 1500 = (7/12) × 1500 = 7 × 125 = 875 2. Find male employees promoted in HR: Total promoted employees = 1200 Total HR promoted = 15% of 1200 = 180 Given female promoted in HR = 120 Male promoted in HR = Total HR promoted - Female promoted in HR = 180 - 120 = 60 3. Calculate the final ratio: Ratio = (Total male in HR) : (Male promoted in HR) Ratio = 875 : 60 Dividing both sides by 5: Ratio = 175 : 12 Exam Strategy & Shortcut: Calculate only what is asked. You do not need to find the number of female employees in HR since it is not required for the final ratio. Break down percentages efficiently: 25% is 1/4th, so 6000 / 4 = 1500. 15% of 1200 is 1200 × 0.15 = 12 × 15 = 180. Common Pitfall: A common mistake is applying the 7:5 ratio to the promoted employees instead of the total employees, or confusing the total numbers (6000 vs 1200) when taking percentages. Final Answer: Therefore, the correct answer is 175 : 12.

Question 12

Read the following information carefully and answer the given questions.  Some recruitment is done by three companies, i.e., P, Q and R. The number of vacancies in company P, Q, and R are 6, 12 and 2 respectively. The number of candidates, who got i…

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Concept & Logic: The problem provides absolute numerical values for interview calls but only relative ratios and percentages for the number of candidates who originally applied. Step-by-Step Solution: 1. Let the total candidates applied in companies P, Q, and R be A_P, A_Q, and A_R respectively. 2. From the given directions: A_R = 25% of A_Q, which means A_R = 0.25 A_Q. 3. It is also given that A_R = (3/5) of A_P. 4. These equations only allow us to establish a ratio between the applied candidates: A_P : A_Q : A_R = 5 : 12 : 3. 5. Because there is no absolute numerical data linking the number of applied candidates to the number of interview calls in the general directions, the exact value of A_Q cannot be found. Exam Strategy & Shortcut: Always verify if a question asks for an absolute value when only ratios or percentages are provided. If no absolute anchor is given for that specific group, the answer will invariably be "Cannot be determined". Common Pitfall: Wasting time trying to solve for variables that lack sufficient constraints, or erroneously assuming the number of applied candidates equals the number of interview calls. Final Answer: Therefore, the correct answer is Can't be determined.

Question 13

Read the following information carefully and answer the given questions.  Some recruitment is done by three companies, i.e., P, Q and R. The number of vacancies in company P, Q, and R are 6, 12 and 2 respectively. The number of candidates, who got i…

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Concept & Strategy: First, determine the total number of interview calls for Company P by multiplying its vacancies by the calls per vacancy. Then, apply the given percentages and ratios to separate the total pool into interviewed females and hired females. Step-by-Step Solution: 1. Find the total number of interview calls for company P: 6 vacancies × 60 calls per vacancy = 360 total calls. 2. Calculate the number of female candidates who got an interview call: 40% of 360 = 0.4 × 360 = 144 females. 3. Determine the total available posts in company P, which is 6. 4. The ratio of posts allotted to males and females is 2:1. The number of female posts (jobs) is: (1/2+1) × 6 = (1/3) × 6 = 2 posts. 5. Calculate the females who did not get the job by subtracting hired females from interviewed females: 144 - 2 = 142. Exam Strategy & Shortcut: You do not need to solve for x (from the Q and R data) to answer this specific question. Company P's data for interview calls and vacancies is completely independent and given in absolute numbers. Stick strictly to Company P's metrics. Common Pitfall: Wasting time calculating x from the 320 difference equation when it has no bearing on Company P's calculations. Final Answer: Therefore, the correct answer is 142.

Question 14

Read the following information carefully and answer the given questions.  Some recruitment is done by three companies, i.e., P, Q and R. The number of vacancies in company P, Q, and R are 6, 12 and 2 respectively. The number of candidates, who got i…

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Concept & Logic: Leverage the newly provided absolute difference to find the exact number of applied candidates using their predefined ratios. Then, use the interview calls equation to find x and subtract total calls from total applicants. Step-by-Step Solution: 1. Establish the ratio of applicants: A_R = (3/5) A_P \implies A_P = (5/3) A_R. 2. Use the given difference to find A_R: A_P - A_R = 160. Substituting A_P, we get (5/3) A_R - A_R = 160 \implies (2/3) A_R = 160 \implies A_R = 240. 3. Find applied candidates for Q (A_Q): Since A_R = 25% of A_Q, we know A_Q = 4 × A_R = 4 × 240 = 960. 4. Calculate x from the interview calls difference: Interview calls for Q = 12 × 6x = 72x. Interview calls for R = 2 × 4x = 8x. The difference is 320. So, 72x - 8x = 320 \implies 64x = 320 \implies x = 5. 5. Find actual interview calls for Q: 72 × 5 = 360. 6. Calculate candidates who did not get a call from Q: Total Applied - Interviewed = 960 - 360 = 600. Exam Strategy & Shortcut: Instead of fractions, work directly with unit ratios. The applicant ratio is A_P : A_Q : A_R = 5 : 12 : 3. The difference between P and R is 5 - 3 = 2 units. We are told 2 units = 160, so 1 unit = 80. Total applied in Q is 12 units = 12 × 80 = 960. This simplifies calculation heavily. Common Pitfall: Forgetting to calculate x based on the general directions and instead using a wrong assumption for the number of interview calls. Final Answer: Therefore, the correct answer is 600.

Question 15

Direction: Answer the questions based on the information given below. The given line graph shows the revenue generated from debit card transactions and cash transactions for five different stores, A, B, C, D and E.  <img    src="/api/images/revenue-…

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Concept & Logic: The problem requires finding the unknown variables p and q using the given conditions, and then calculating specific revenue values to find the required ratio. Step-by-Step Solution: • Find p: According to Note (d), the sum of revenue from debit card transactions for stores 'A' and 'E' is 420 million. • From the graph data, revenue from debit cards for store A is 6p and for store E is 8p. • Therefore, 6p + 8p = 420 • 14p = 420 \Rightarrow p = 30 • Find q: According to Note (d), the difference between the revenue from cash transactions for stores 'C' and 'D' is 48 million. • From the graph data, revenue from cash transactions for store C is 10q and for store D is 8q. • Therefore, 10q - 8q = 48 • 2q = 48 \Rightarrow q = 24 • Calculate Required Values: • Debit card revenue for store A = 6p = 6 × 30 = 180 • Debit card revenue for store B = 9p = 9 × 30 = 270 • Total debit card revenue for stores A and B together = 180 + 270 = 450 • Total revenue for store B worldwide (debit + cash): • Debit card revenue for B = 270 • Cash transaction revenue for B = 7.5q = 7.5 × 24 = 180 • Total revenue for B = 270 + 180 = 450 • Find the Ratio: • Ratio = (Total debit revenue of A and B) : (Total revenue of B) • Ratio = 450 : 450 = 1 : 1 Exam Strategy & Shortcut: Focus calculations only on the variables and specific data points requested by the question. Avoid calculating the revenue for all stores or determining their ratings, as it wastes valuable time. Common Pitfall: A common mistake is misreading the graph values or mixing up the debit card equation with the cash transaction variables. Always double-check the legend mapping. Final Answer: Therefore, the correct answer is 1 : 1.

Question 16

Direction: Study the following information carefully and answer the questions.  The following pie chart shows the distribution of the number of laptops sold by five stores.  <img    src="/api/images/laptop-sales-distribution-pie-chart.png"    alt="P…

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Concept & Ratio Applications: By finding the exact number of laptops sold by Store D and E from the pie chart variables, we can use the given percentages and ratios to find unknown values for an external Store F. Step-by-Step Solution: • Given: z = 14 and y = 56 (solved from base conditions). Store D = (2z - 5)% = 23%. Store E = (8.5z + 1)^\circ = 120^\circ. • Calculation: Total laptops = 7200. • Store D laptops = 0.23 × 7200 = 1656. • Store E laptops = (120/360) × 7200 = 2400. • Deduction: Difference between D and E = 2400 - 1656 = 744. • This difference is 75% less than laptops sold by store F (S_F). Thus, it represents 25% (or (1/4)) of S_F. • 744 = 0.25 × S_F \implies S_F = 744 × 4 = 2976. • The ratio of sold to unsold (U_F) by store F is 62:75. • (2976/U_F) = (62/75) \implies U_F = (2976 × 75/62). • U_F = 48 × 75 = 3600. Exam Strategy & Shortcut: Recognize that "75% less than" means exactly 25% or (1/4) of the value. Multiplying the difference 744 by 4 quickly gives the sold units. For the ratio, directly scaling up 62 to 2976 (a multiplier of 48) allows you to just multiply 75 by 48 (which is 75 × 50 - 75 × 2 = 3750 - 150 = 3600) to find the unsold units mentally. Common Pitfall: A common pitfall is calculating 75% of 744 rather than correctly setting 744 as the 25% remainder of the unknown total. Final Answer: Therefore, the correct answer is 3600.

Question 17

The total number of students in a school is 2140. If the number of girls in the school is 1200, then what is the ratio of the total number of boys to the total number of girls in the school? (Bank Recruitment, 2009)

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Concept & Ratio: To find the ratio of two quantities, we first need their exact values and then divide them, simplifying to the lowest terms. Ratio = \frac{Value 1}{Value 2} Step-by-Step Solution: • Given total number of students = 2140. • Given number of girls = 1200. • Calculate the number of boys by subtracting girls from the total: 2140 - 1200 = 940. • Set up the ratio of boys to girls: 940 : 1200. • Simplify the ratio by cancelling the common zero (dividing by 10): 94 : 120. • Further simplify by dividing both numbers by their greatest common divisor, which is 2: (94/2) : (120/2) = 47 : 60 Exam Strategy & Shortcut: Calculate the number of boys (940). Notice that 940 ends in a 0, and removing it gives 94. Half of 94 is 47, which is a prime number. Look directly at the given options: only option (c) has 47 as the first term. You can instantly select it without calculating the denominator's simplified form to save time. Common Pitfall: Finding the ratio of girls to boys (60 : 47) or girls to total students instead of the specifically requested boys to girls. Always double-check the order required by the question! Final Answer: Therefore, the correct answer is 47 : 60.

Question 18

The value of $x$ where $x : 2\frac{1}{3} : : 21 : 50$ is

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Concept & Proportion: In a proportion where four terms are related as a : b : : c : d, the product of the extreme terms (outer terms) is equal to the product of the mean terms (inner terms). a × d = b × c Step-by-Step Solution: • Given: The proportion is x : 2(1/3) : : 21 : 50. • Convert mixed fraction: Convert 2(1/3) to an improper fraction: ((2 × 3) + 1/3) = (7/3). • Set up the equation: Use the product of extremes equals product of means rule. x × 50 = (7/3) × 21 • Simplify: 50x = 7 × 7 50x = 49 • Solve for x: x = (49/50) Exam Strategy & Shortcut: Treat the proportion as equivalent fractions: (x/7/3) = (21/50). Simply cross-multiply the 7/3 to the right side: x = (21 × (7/3)/50). The 3 cancels into the 21 giving 7. So, x = (7 × 7/50) = (49/50). Common Pitfall: A frequent mistake is improperly converting the mixed number 2(1/3), or incorrectly pairing the extremes and means, leading to a flipped fraction. Final Answer: Therefore, the correct answer is (49/50).

Question 19

If $a : b = 7 : 9$ and $b : c = 15 : 7$, then what is $a : c$?

Accepted Answer

Concept & Compounding Ratios: To find the direct ratio between the first and last variables in a chain of ratios, you can multiply the corresponding fractions. (a/c) = (a/b) × (b/c) Step-by-Step Solution: 1. Convert the given ratios into fractions: (a/b) = (7/9) (b/c) = (15/7) 2. We want to find a : c, which is equivalent to (a/c). 3. Multiply the two fractions together. The b terms will cancel out: (a/c) = ((7/9)) × ((15/7)) 4. Simplify the expression by cancelling the common factor of 7 in the numerator and denominator: (a/c) = (15/9) 5. Reduce the resulting fraction by dividing the numerator and denominator by their greatest common divisor, which is 3: (a/c) = (5/3) 6. Convert back to ratio format: a : c = 5 : 3 Exam Strategy & Shortcut: Write the numbers down and immediately cancel out common values diagonally. In (7/9) × (15/7), the 7s visually cancel out immediately leaving (15/9), which simplifies to 5 : 3 in seconds. Common Pitfall: Attempting to equalize the middle term b in both ratios (finding a common multiple for 9 and 15) is a valid method but takes significantly more time than simple multiplication. Final Answer: Therefore, the correct answer is 5 : 3.

Question 20

If $a : b = 2 : 5$, then the value of $(2a + 3b) : (7a + 5b)$ is

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Concept & Ratio Substitution: When given a ratio of variables like a : b = m : n and asked to find the value of a homogeneous expression (where all terms have the same total degree), we can directly substitute the proportional values a = m and b = n. Step-by-Step Solution: - Given: a : b = 2 : 5. - Substitute: Let a = 2 and b = 5. - Expression: (2a + 3b) : (7a + 5b) - Calculate Numerator: 2(2) + 3(5) = 4 + 15 = 19 - Calculate Denominator: 7(2) + 5(5) = 14 + 25 = 39 - Result: The ratio is 19 : 39. Exam Strategy & Shortcut: Since the degree of every term in the numerator and denominator is 1, we can bypass using a constant k (e.g., a = 2k, b = 5k) because k will completely factor out and cancel. Direct substitution is the fastest path. Common Pitfall: A common mistake is trying to solve for a or b individually, forgetting that we only need their relative sizes. Also, simple arithmetic errors during multiplication or addition can lead to choosing a distractor option. Final Answer: Therefore, the correct answer is 19 : 39.

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