Question 1

If $\frac{a}{b} = \frac{4}{5}$ and $\frac{b}{c} = \frac{15}{16}$, then $\frac{c^2 - a^2}{c^2 + a^2}$ would be

Accepted Answer

Concept & Ratio Combination: To evaluate an expression involving multiple variables linked by chained ratios, first combine the ratios into a single relationship between the required variables. For a, b, and c, finding the direct ratio of a : c allows for immediate substitution since the given algebraic expression is homogeneous. Step-by-Step Solution: - Given: (a/b) = (4/5) and (b/c) = (15/16). - Find direct ratio a/c: Multiply the two given ratios to eliminate b. (a/c) = (a/b) × (b/c) = (4/5) × (15/16) - Simplify: (a/c) = (4 × 15/5 × 16) = (1 × 3/1 × 4) = (3/4). - Substitute: Since the expression (c^2 - a^2/c^2 + a^2) is homogeneous (all terms are degree 2), let a = 3 and c = 4. - Calculate: - Numerator: c^2 - a^2 = 4^2 - 3^2 = 16 - 9 = 7 - Denominator: c^2 + a^2 = 4^2 + 3^2 = 16 + 9 = 25 - Result: The ratio is (7/25). Exam Strategy & Shortcut: When you see chained fractions like a/b and b/c, and you need a relation between a and c, immediately multiply them. Recognize the Pythagorean triplet numbers (3, 4, 5) which makes squaring (16 - 9 = 7 and 16 + 9 = 25) instant mental math. Common Pitfall: A common mistake is attempting to make b the same in both ratios (e.g., changing 4/5 to 12/15) and then using a=12, b=15, c=16. While mathematically correct, squaring 12 and 16 takes much longer than using the reduced ratio a=3, c=4. Final Answer: Therefore, the correct answer is (7/25).

Question 2

If $x : y = 3 : 4$ and $a : b = 1 : 2$, then the value of $\frac{2xa + yb}{3yb - 4xa}$ is

Accepted Answer

Concept & Independent Ratio Substitution: When evaluating an expression that consists of products of variables from two independent, unrelated ratios, you can directly substitute the relative values of each variable as long as the expression is homogeneous with respect to both sets of variables. Step-by-Step Solution: - Given: x : y = 3 : 4 and a : b = 1 : 2. - Substitute: Let x = 3, y = 4 and a = 1, b = 2. - Target Expression: (2xa + yb/3yb - 4xa) - Calculate individual terms: - xa = 3 × 1 = 3 \implies 2xa = 2 × 3 = 6 - yb = 4 × 2 = 8 \implies yb = 8 - 3yb = 3 × 8 = 24 - 4xa = 4 × 3 = 12 - Substitute into fraction: - Numerator: 6 + 8 = 14 - Denominator: 24 - 12 = 12 - Simplify: (14/12) = (7/6). Exam Strategy & Shortcut: Since every term in the numerator and denominator is a product of one variable from the first set (x, y) and one from the second set (a, b), the proportionality constants (k and m) will appear as km in every term and factor out completely. Plug in the numbers immediately: (6+8)/(24-12) = 14/12 = 7/6. Common Pitfall: Mixing up the variables during substitution (e.g., multiplying x with b instead of a) is the most frequent error here due to reading too quickly. Final Answer: Therefore, the correct answer is (7/6).

Question 3

If $a : b : c = 2 : 3 : 4$, then $\frac{1}{a} : \frac{1}{b} : \frac{1}{c}$ is equal to

Accepted Answer

Concept & Reciprocal Ratios: To find the ratio of the reciprocals of given numbers, write the reciprocals as fractions and then multiply each fraction by the Least Common Multiple (LCM) of their denominators to clear the fractions and obtain an integer ratio. Step-by-Step Solution: - Given: a : b : c = 2 : 3 : 4. - Form reciprocal ratio: (1/a) : (1/b) : (1/c) = (1/2) : (1/3) : (1/4). - Find LCM: Find the Least Common Multiple of the denominators 2, 3, and 4. The LCM is 12. - Clear fractions: Multiply each term of the ratio by 12. - First term: (1/2) × 12 = 6 - Second term: (1/3) × 12 = 4 - Third term: (1/4) × 12 = 3 - Result: The integer ratio is 6 : 4 : 3. Exam Strategy & Shortcut: A fast shortcut to find the reciprocal ratio of A : B : C is to hide one number and multiply the other two. - For a: hide 2, multiply 3 × 4 = 12. - For b: hide 3, multiply 2 × 4 = 8. - For c: hide 4, multiply 2 × 3 = 6. - Ratio is 12 : 8 : 6, which simplifies to 6 : 4 : 3. Common Pitfall: A classic trap is to assume that the reciprocal ratio simply reverses the order of the original numbers (i.e., choosing 4 : 3 : 2). This is false for three or more numbers. Final Answer: Therefore, the correct answer is 6 : 4 : 3.

Question 4

If $a : (b + c) = 1 : 3$ and $c : (a + b) = 5 : 7$, then $b : (a + c)$ is equal to (S.S.C., 2006)

Accepted Answer

Concept & Equating Total Parts: When given multiple ratios involving the same set of variables partitioned differently (like a vs b+c and c vs a+b), the sum of all variables (a+b+c) must be constant. By making the total number of parts in both ratios equal, you can directly compare and find the individual values of the variables. Step-by-Step Solution: - Given Ratios: 1. a : (b + c) = 1 : 3 2. c : (a + b) = 5 : 7 - Calculate Total Parts: - For ratio 1, total parts = 1 + 3 = 4. - For ratio 2, total parts = 5 + 7 = 12. - Equalize Total Parts: - The LCM of 4 and 12 is 12. - Multiply the first ratio by 3 to make its total parts 12: a : (b + c) = (1 × 3) : (3 × 3) = 3 : 9. - Identify Individual Values: - Now, total parts = 12. - From the modified first ratio, a = 3. - From the second ratio, c = 5. - Find the Missing Variable: - We know a + b + c = 12. - 3 + b + 5 = 12 \implies b + 8 = 12 \implies b = 4. - Calculate Target Ratio: - We need b : (a + c). - Substitute the values: 4 : (3 + 5) = 4 : 8 = 1 : 2. Exam Strategy & Shortcut: Whenever a question gives part-to-rest ratios for the same total pool, immediately look at the sums (1+3=4 and 5+7=12). Scale the ratios so the sums match. Once matched, pluck the independent variables directly (a=3, c=5), subtract from the total to get the last one (b=4), and form your final ratio. Common Pitfall: A common mistake is trying to set up complex simultaneous algebraic equations (e.g., 3a = b+c and 7c = 5a+5b), which is highly prone to calculation errors and wastes precious time compared to simply scaling the ratio parts. Final Answer: Therefore, the correct answer is 1 : 2.

Question 5

If $a : b = 3 : 4$, $b : c = 4 : 7$, then $\frac{a + b + c}{c}$ is equal to (Hotel Management, 2010)

Accepted Answer

Concept & Chaining Ratios: When linking two separate ratios that share a common term (in this case, b), check if the value of that common term is identical in both ratios. If it is, you can directly combine them into a single continuous ratio a : b : c. Step-by-Step Solution: - Given Ratios: a : b = 3 : 4 and b : c = 4 : 7. - Combine Ratios: Since b is represented by 4 in both ratios, we can directly link them. a : b : c = 3 : 4 : 7. - Assign Proportional Values: Let a = 3, b = 4, and c = 7. (Because the target expression is a ratio itself, the constant multiplier k will cancel out). - Evaluate Expression: - Target: (a + b + c/c) - Substitute: (3 + 4 + 7/7) - Simplify: (14/7) = 2. Exam Strategy & Shortcut: Notice that b=4 in both given conditions. Instantly write down a=3, b=4, c=7. Add them up in your head to get 14, and divide by c (7). The answer is 2. This should be solvable mentally in under 5 seconds. Common Pitfall: Overcomplicating the problem by introducing variables like a=3k, b=4k is unnecessary when evaluating a homogeneous fractional expression, as it just adds extra writing. Final Answer: Therefore, the correct answer is 2.

Question 6

If $a : b = c : d$, then $\frac{ma + nc}{mb + nd}$ is equal to

Accepted Answer

Concept & The Addendo Property of Proportions: If two ratios are equal ((a/b) = (c/d)), then any linear combination of their numerators divided by the identical linear combination of their denominators will yield the exact same ratio. Step-by-Step Solution: 1. Given: (a/b) = (c/d) = k (where k is a constant multiplier). 2. Express the numerators in terms of denominators and the constant: a = bk c = dk 3. Substitute these into the target expression (ma + nc/mb + nd): (m(bk) + n(dk)/mb + nd) 4. Factor out the common constant k in the numerator: (k(mb + nd)/mb + nd) 5. As long as mb + nd 
eq 0, the binomials cancel out, leaving: k 6. Since we defined k = (a/b) (which also equals (c/d)), the expression simplifies to a : b. Exam Strategy & Shortcut: Recognize standard mathematical theorems. The theorem states that if (a/b) = (c/d), then (pa + qc/pb + qd) = (a/b) for any constants p and q. Thus, the answer is immediately a : b without any calculation required. Common Pitfall: Do not assume that the answer involves m and n. Because the coefficients m and n are applied symmetrically to both the numerator and the denominator, they act as weights that ultimately factor out of the final proportion. Final Answer: Therefore, the correct answer is a : b.

Question 7

Let $\frac{a}{b} : \frac{b}{a} = x : y$. If $(x - y) = \left\{ \frac{a}{b} + \frac{b}{a} \right\}$, then $x$ is equal to

Accepted Answer

Concept & Algebraic Manipulation of Proportions: When dealing with variables represented as a ratio (e.g., x:y), express one variable in terms of the other using the given proportion before substituting it into the subsequent algebraic equation. Step-by-Step Solution: 1. Given Proportion: (x/y) = (\frac{a/b)}{(b/a)} = (a/b) \cdot (a/b) = (a^2/b^2). 2. This implies x = k \cdot a^2 and y = k \cdot b^2 for some proportionality constant k. 3. Substitute x and y into the second given equation: (x - y) = (a/b) + (b/a) k(a^2) - k(b^2) = (a^2 + b^2/ab) k(a^2 - b^2) = (a^2 + b^2/ab) 4. Solve for the constant k: k = (a^2 + b^2/ab(a^2 - b^2)) 5. The question asks for the value of x. Substitute k back into x = k \cdot a^2: x = ( (a^2 + b^2/ab(a^2 - b^2)) ) \cdot a^2 x = (a(a^2 + b^2)/b(a^2 - b^2)) 6. Compare this result with the given options. None of the options (a), (b), or (c) match this expression. Exam Strategy & Shortcut: By inspecting the degree of the variables, x/y yields terms of degree 2 (a^2/b^2), and the substitution involves adding/subtracting terms. The final term x must retain complex cubic or higher algebraic combinations of a and b, meaning simple linear terms like (a+b/a) will definitely be incorrect. Choose "None of these" confidently. Common Pitfall: Assuming x = (a/b) and y = (b/a) directly. While (x/y) equals the ratio of those terms, x and y themselves can be scaled by any constant multiplier k. The constraint equation dictates what that scale must be. Final Answer: Therefore, the correct answer is None of these.

Question 8

If $\frac{a + b}{c} = \frac{b + c}{a} = \frac{c + a}{b} = k$, then $k$ is equal to

Accepted Answer

Concept & Addendo Property: The Addendo property of proportions states that if a series of ratios are equal, such as (x/y) = (p/q) = (m/n) = k, then the sum of all numerators divided by the sum of all denominators is also equal to the same constant k, provided the sum of denominators is not zero. Step-by-Step Solution: 1. Given: (a + b/c) = (b + c/a) = (c + a/b) = k 2. Apply the property of equal ratios (Addendo): k = \frac{Sum of numerators}{Sum of denominators} 3. Add the numerators and denominators together: k = ((a + b) + (b + c) + (c + a)/c + a + b) 4. Combine like terms in the numerator: k = (2a + 2b + 2c/a + b + c) 5. Factor out the 2: k = (2(a + b + c)/a + b + c) 6. Assuming a + b + c 
eq 0, the (a + b + c) terms cancel out: k = 2 Exam Strategy & Shortcut: Whenever a symmetric cyclic equation like this appears in competitive exams, assume simple non-zero equal values for the variables to solve it instantly. Let a = 1, b = 1, and c = 1. Substitute them into any one of the fractions: (1 + 1/1) = (2/1) = 2. Therefore, k = 2. Common Pitfall: A common mistake is expanding this into complex simultaneous equations or quadratic forms which wastes immense time. Recognizing the symmetry and either adding them or plugging in 1 is the optimal path. Final Answer: Therefore, the correct answer is 2.

Question 9

If $a : b = 2 : 3$ and $b : c = 4 : 5$, then $(a + b) : (b + c)$ is equal to

Accepted Answer

Concept & Combining Ratios: To evaluate an expression involving variables from two different ratios, you must first merge them into a single continuous ratio (a : b : c). This is done by equating the common variable (in this case, b) by finding the Least Common Multiple (LCM) of its corresponding values in both ratios. Step-by-Step Solution: 1. Given: a : b = 2 : 3 b : c = 4 : 5 2. The common term is b. In the first ratio, b = 3. In the second, b = 4. 3. Find the LCM of 3 and 4, which is 12. 4. Multiply both ratios to make b = 12: First ratio: Multiply by 4 \implies a : b = (2 \cdot 4) : (3 \cdot 4) = 8 : 12 Second ratio: Multiply by 3 \implies b : c = (4 \cdot 3) : (5 \cdot 3) = 12 : 15 5. Now, combine them into a single ratio: a : b : c = 8 : 12 : 15 6. We can now assign proportional values: a = 8, b = 12, and c = 15. 7. Substitute these into the requested expression (a + b) : (b + c): (8 + 12) : (12 + 15) 20 : 27 Exam Strategy & Shortcut: Use the "Reverse N" or block multiplication method to combine a:b and b:c instantly. Write them stacked: 2 : 3 4 : 5 Multiply down the left column (2 \cdot 4 = 8), diagonally up (4 \cdot 3 = 12), and down the right column (3 \cdot 5 = 15) to get 8 : 12 : 15. Then plug the numbers in directly. Common Pitfall: A frequent error is adding the values from the disjointed ratios directly, such as taking a=2, b=3 for the first part and b=4, c=5 for the second part, resulting in mismatched scaling. Always unify the ratios first. Final Answer: Therefore, the correct answer is 20 : 27.

Question 10

If $a : b = c : d = e : f = 1 : 2$, then $(3a + 5c + 7e) : (3b + 5d + 7f)$ is equal to

Accepted Answer

Concept & Proportional Scaling: When multiple ratios are equal to a constant, any linear combination of the numerator terms divided by the exact same linear combination of the denominator terms maintains the identical ratio. This is a direct consequence of factoring out the constant multiplier. Step-by-Step Solution: 1. Given: (a/b) = (c/d) = (e/f) = (1/2) 2. From this, we can express the numerator variables in terms of their respective denominators: a = (1/2)b c = (1/2)d e = (1/2)f 3. We need to evaluate the expression (3a + 5c + 7e/3b + 5d + 7f). 4. Substitute the values of a, c, and e into the numerator: = (3(\frac{1/2)b) + 5((1/2)d) + 7((1/2)f)}{3b + 5d + 7f} 5. Factor out the common multiplier (1/2) from the numerator terms: = (\frac{1/2)(3b + 5d + 7f)}{3b + 5d + 7f} 6. Cancel out the common binomial factor (3b + 5d + 7f): = (1/2) 7. Expressing this as a ratio gives 1 : 2. Exam Strategy & Shortcut: Recognize the structural symmetry. If (a/b) = (c/d) = (e/f) = k, then (ma + nc + pe/mb + nd + pf) = k for any scalar values m, n, p. Because the coefficients 3, 5, 7 perfectly match top and bottom, the ratio remains exactly the same as the original given ratio: 1 : 2. Common Pitfall: Students often waste time trying to assign arbitrary individual values to a, b, c, d, e, f (like a=1, b=2, c=2, d=4...) and calculating the heavy arithmetic 3(1) + 5(2) + 7(3)... which is prone to addition errors. Rely on the algebraic property instead. Final Answer: Therefore, the correct answer is 1 : 2.

Question 11

Directions: These questions are based on the following information: Given that $a : b = 5 : 3$ and $b : c = 2 : 5$.  Which of the following is true?

Accepted Answer

Concept & Combining Ratios: To compare the relative sizes of variables from different ratios, you must first merge them into a single, continuous ratio (a : b : c). This is achieved by equating the common variable (in this case, b) by finding the Least Common Multiple (LCM) of its corresponding values. Step-by-Step Solution: 1. Given: a : b = 5 : 3 and b : c = 2 : 5. 2. The common variable is b. In the first ratio, b = 3. In the second, b = 2. 3. Find the LCM of 3 and 2, which is 6. 4. Multiply the ratios to normalize b to 6: First ratio: Multiply by 2 \implies a : b = (5 \cdot 2) : (3 \cdot 2) = 10 : 6 Second ratio: Multiply by 3 \implies b : c = (2 \cdot 3) : (5 \cdot 3) = 6 : 15 5. Combine them into a single ratio: a : b : c = 10 : 6 : 15 6. We can now assign proportional values: a = 10k, b = 6k, and c = 15k (where k > 0). 7. Comparing the values: 15k > 10k > 6k, which translates directly to c > a > b. *(Note: While b a > b perfectly mirrors the strictly descending magnitude order standard in these specific question sets.)* Exam Strategy & Shortcut: Use the block multiplication method: Stack the ratios and multiply down-left, cross-up, and down-right. 5 : 3 2 : 5 a = 5 \cdot 2 = 10 b = 3 \cdot 2 = 6 c = 3 \cdot 5 = 15 Ratio is 10 : 6 : 15. Visually, 15 > 10 > 6 \implies c > a > b. Common Pitfall: Attempting to compare a and c directly from their isolated ratios (assuming a=5 and c=5, so they are equal) without properly unifying the linking variable b first. Final Answer: Therefore, the correct answer is c > a > b.

Question 12

Directions: These questions are based on the following information: Given that $a : b = 5 : 3$ and $b : c = 2 : 5$.  If $c = 50$, the value of $a + b + c$ will be

Accepted Answer

Concept & Proportional Distribution: Once a unified ratio is established, finding the absolute value of one component allows you to determine the scale factor (constant k). This scale factor can then be applied to the sum of the ratio parts to find the total sum efficiently. Step-by-Step Solution: 1. Given: From previous calculations, the unified ratio is a : b : c = 10 : 6 : 15. 2. Let the actual values be a = 10k, b = 6k, and c = 15k. 3. We are given that c = 50. Set up the equation: 15k = 50 k = (50/15) = (10/3) 4. The requested expression is the sum a + b + c: a + b + c = 10k + 6k + 15k = 31k 5. Substitute the exact value of k into the sum expression: 31 \cdot ((10/3)) = (310/3) 6. Convert the fraction to a decimal to check the ranges: (310/3) = 103.333... 7. Analyzing the options, 103.33 is strictly greater than 103 and less than 105. Exam Strategy & Shortcut: Do not waste time calculating the individual absolute values of a and b before adding. Sum the ratio proportions (10 + 6 + 15 = 31) and multiply by the multiplier ((50/15)) in one single step: 31 \cdot (50/15) = 103.33. Common Pitfall: Converting the scale factor k = (10/3) into a rounded decimal like 3.33 too early. Doing so yields 31 \cdot 3.33 = 103.23, which could cause hesitation when distinguishing between narrow bounds, especially if rounding errors push the value into an incorrect bucket. Always keep fractions until the final step. Final Answer: Therefore, the correct answer is more than 103 but less than 105.

Question 13

Directions: These questions are based on the following information: Given that $a : b = 5 : 3$ and $b : c = 2 : 5$.  $(c - a)$ will be equal to I. $10(a + c)$ II. $10a + 25b$

Accepted Answer

Concept & Algebraic Evaluation of Ratios: To determine if statements regarding variables in a ratio are true, express all variables in terms of a single proportionality constant (k) and substitute them into the given algebraic expressions to test for equivalence. Step-by-Step Solution: 1. Given: The combined continuous ratio is a : b : c = 10 : 6 : 15. 2. Let the values be represented as a = 10k, b = 6k, and c = 15k. 3. First, evaluate the target expression (c - a): (c - a) = 15k - 10k = 5k 4. Now, evaluate Statement I: 10(a + c) 10(10k + 15k) = 10(25k) = 250k Since 5k 
eq 250k, Statement I is not equal to (c - a). It is false. 5. Evaluate Statement II: 10a + 25b 10(10k) + 25(6k) = 100k + 150k = 250k Since 5k 
eq 250k, Statement II is also not equal to (c - a). It is false. 6. Because neither algebraic statement evaluates to 5k, both are false. Exam Strategy & Shortcut: Test logic using the simplest integers. Let a=10, b=6, c=15. The target (c-a) equals 5. Statement I is 10(10+15) = 250. Statement II is 10(10) + 25(6) = 100 + 150 = 250. Since 5 
eq 250, both are demonstrably false. Common Pitfall: A student might notice that Statement I (250k) and Statement II (250k) are mathematically equal to each other and mistakenly select "Both I and II are true", forgetting that the question specifically asks if they are equal to the initial expression (c - a). Final Answer: Therefore, the correct answer is Both I and II are false.

Question 14

If $a + b : b + c : c + a = 6 : 7 : 8$ and $a + b + c = 14$, then the value of $c$ is

Accepted Answer

Concept & Proportionality: When given a ratio of sums, introduce a common multiplier to represent the actual values. From there, sum the expressions to find the total in terms of the multiplier and equate it to the given total sum. Step-by-Step Solution: 1. Let the expressions be proportional to a constant k. a + b = 6k b + c = 7k c + a = 8k 2. Add all three equations together: (a + b) + (b + c) + (c + a) = 6k + 7k + 8k 2(a + b + c) = 21k 3. We are given that a + b + c = 14. Substitute this into the equation: 2(14) = 21k 28 = 21k k = (28/21) = (4/3) 4. We need to find the value of c. We know that (a + b + c) = 14. 5. From our first equation, we know a + b = 6k. Let's find its value: a + b = 6 × ((4/3)) = 8 6. Substitute (a + b) into the total sum equation: 8 + c = 14 c = 14 - 8 = 6 Exam Strategy & Shortcut: Instead of solving for all variables individually, directly find the value of the missing variable by subtracting the sum of the other two variables from the total sum. Since c is missing from (a+b), finding (a+b) gets you to the answer in one step. Common Pitfall: A common mistake is assuming a+b = 6, b+c = 7, c+a = 8 literally, without using a constant k. This would falsely lead to 2(a+b+c) = 21 \Rightarrow a+b+c = 10.5, which contradicts the given condition a+b+c = 14. Final Answer: Therefore, the correct answer is 6.

Question 15

The third proportional to $38$ and $15$ is

Accepted Answer

Concept & Formula: When finding the third proportional to two numbers a and b, the numbers are considered to be in continued proportion. This means the ratio of the first to the second is equal to the ratio of the second to the third (the unknown x). a : b :: b : x \implies (a/b) = (b/x) Step-by-Step Solution: 1. Identify the given values: a = 38 and b = 15. Let the third proportional be x. 2. Set up the continued proportion equation: (38/15) = (15/x) 3. Cross-multiply to solve for x: 38 × x = 15 × 15 4. Isolate x by dividing by 38: x = (15 × 15/38) 5. Look at the provided options. The options are in uncalculated fraction form, so we do not need to compute 225 / 38. Exam Strategy & Shortcut: The formula for the third proportional of a and b is simply x = (b^2/a). You can jump straight to this formula: square the second number and divide by the first number. Thus, (15^2/38) or (15 × 15/38). Common Pitfall: Confusing the third proportional with the third term in a sequence of three different given numbers. For two numbers, the middle number is always repeated to form the proportion a:b::b:x. Final Answer: Therefore, the correct answer is (15 × 15/38).

Question 16

The mean proportional between $234$ and $104$ is

Accepted Answer

Concept & Formula: The mean proportional x between two numbers a and b is the geometric mean of the numbers, calculated as: x = \sqrt{a × b} Step-by-Step Solution: 1. Let a = 234 and b = 104. The mean proportional is x = \sqrt{234 × 104}. 2. To avoid large multiplications, find the prime factorization of each number. 234 = 2 × 117 = 2 × 9 × 13 = 2 × 3^2 × 13 104 = 8 × 13 = 2^3 × 13 3. Multiply the factorizations inside the square root: x = \sqrt{(2 × 3^2 × 13) × (2^3 × 13)} 4. Combine powers of the same base: x = \sqrt{2^4 × 3^2 × 13^2} 5. Take the square root by halving the exponents: x = 2^2 × 3^1 × 13^1 = 4 × 3 × 13 6. Calculate the final value: x = 12 × 13 = 156 7. Since 156 is not among the specific options 12, 39, or 54, the answer is "None of these". Exam Strategy & Shortcut: Look at the unit digits. 234 × 104 ends in 4 × 4 = 16, so the product ends in 6. The square root of a number ending in 6 must end in 4 or 6. The given numerical options are 12 (ends in 2, square ends in 4), 39 (ends in 9, square ends in 1), and 54 (ends in 4, square ends in 6). We can quickly check 54^2 \approx 2500, but 234 × 104 is roughly 234 × 100 \approx 23400. The options are way too small. Common Pitfall: Multiplying 234 by 104 to get 24336 and then struggling to find the square root of a large 5-digit number without a calculator. Always factorize first. Final Answer: Therefore, the correct answer is None of these.

Question 17

The mean proportional between $0.02$ and $0.32$ is

Accepted Answer

Concept & Formula: The mean proportional x between two quantities a and b is given by the geometric mean formula: x = \sqrt{a × b} When dealing with decimals, convert them to fractions for easier and safer calculation. Step-by-Step Solution: 1. Identify the values: a = 0.02 and b = 0.32. 2. Express the decimals as fractions: a = (2/100) and b = (32/100) 3. Substitute into the mean proportional formula: x = \sqrt{(2/100) × (32/100)} 4. Multiply the numerators and denominators together: x = \sqrt{(64/10000)} 5. Extract the square roots of the numerator and denominator independently: x = \frac{\sqrt{64}}{\sqrt{10000}} = (8/100) 6. Convert back to decimal format: x = 0.08 Exam Strategy & Shortcut: Multiply the significant figures: 2 × 32 = 64. The square root of 64 is 8. Since there are 4 decimal places in total between the two original numbers (2 in 0.02 and 2 in 0.32), their square root will have exactly half that amount, which is 2 decimal places. Thus, 8 becomes 0.08. Common Pitfall: Miscounting the decimal places in the final step. Many students mistakenly answer 0.8 (which would only be correct if the original product had exactly two decimal places, not four). Final Answer: Therefore, the correct answer is 0.08.

Question 18

The mean proportional between $(3 + \sqrt{2})$ and $(12 - \sqrt{32})$ is

Accepted Answer

Concept & Formula: The mean proportional between two terms a and b is \sqrt{ab}. When terms involve surds (radicals), simplify them first by pulling out common factors to reveal potential conjugate pairs. Step-by-Step Solution: 1. Let a = (3 + \sqrt{2}) and b = (12 - \sqrt{32}). 2. Simplify the second term b. Note that \sqrt{32} = \sqrt{16 × 2} = 4\sqrt{2}. So, b = 12 - 4\sqrt{2}. 3. Factor out the common 4 from b: b = 4(3 - \sqrt{2}) 4. Set up the mean proportional equation x = \sqrt{a × b}: x = \sqrt{(3 + \sqrt{2}) × 4(3 - \sqrt{2})} 5. Rearrange to group the conjugate pairs together: x = \sqrt{4 × (3 + \sqrt{2})(3 - \sqrt{2})} 6. Use the difference of squares identity (A+B)(A-B) = A^2 - B^2 on the conjugate pair: (3 + \sqrt{2})(3 - \sqrt{2}) = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7 7. Substitute this back into the square root equation: x = \sqrt{4 × 7} = \sqrt{4} × \sqrt{7} = 2\sqrt{7} Exam Strategy & Shortcut: Recognize immediately that 12 - \sqrt{32} hides a multiple of the conjugate of 3 + \sqrt{2}. Extracting 4 transforms it to 4(3 - \sqrt{2}). The product of conjugates (a+b)(a-b) quickly yields an integer (9-2=7). Thus, \sqrt{4 × 7} = 2\sqrt{7} can be calculated almost entirely mentally. Common Pitfall: Attempting to multiply (3 + \sqrt{2})(12 - \sqrt{32}) directly using FOIL without simplifying \sqrt{32} first. This leads to a messy expression like 36 - 3\sqrt{32} + 12\sqrt{2} - \sqrt{64} which is highly prone to arithmetic errors. Final Answer: Therefore, the correct answer is 2\sqrt{7}.

Question 19

Find the two numbers whose mean proportion is $12$ and the third proportional is $324$. (R.R.B., 2006)

Accepted Answer

Concept & Strategy: Let the two unknown numbers be a and b. We are given two conditions: 1. The mean proportion of a and b is \sqrt{ab} = 12 \implies ab = 144. 2. The third proportional to a and b is (b^2/a) = 324 \implies b^2 = 324a. Instead of solving complex algebraic cubic equations, the fastest and most reliable method is to back-solve by testing the multiple-choice options against these two conditions. Step-by-Step Solution: 1. Let's solve algebraically first to prove the concept. We have two equations: (i) a × b = 144 \implies a = (144/b) (ii) b^2 = 324a 2. Substitute a from equation (i) into equation (ii): b^2 = 324 × ((144/b)) 3. Multiply both sides by b: b^3 = 324 × 144 4. Break into prime factors to find the cube root easily: 324 = 18^2 = (2 × 3^2)^2 = 2^2 × 3^4 144 = 12^2 = (2^2 × 3)^2 = 2^4 × 3^2 b^3 = (2^2 × 3^4) × (2^4 × 3^2) = 2^6 × 3^6 5. Take the cube root by dividing exponents by 3: b = 2^2 × 3^2 = 4 × 9 = 36 6. Find a using equation (i): a = (144/36) = 4 The numbers are exactly 4 and 36. Exam Strategy & Shortcut: Option Elimination (Back-Solving): Condition 1: Mean proportion is 12, so the product of the numbers must be 12^2 = 144. Test options: (a) 6 × 8 = 48 (Incorrect) (b) 4 × 36 = 144 (Possible) (c) 3 × 24 = 72 (Incorrect) Condition 2: Third proportional is 324. Let's test option (b) for this condition: (36^2/4) = (1296/4) = 324. This matches perfectly. Common Pitfall: Trying to solve b^3 = 324 × 144 = 46656 without factorization. Finding the cube root of 46656 manually is extremely time-consuming and prone to computational errors. Back-solving from the options is much safer. Final Answer: Therefore, the correct answer is 4 and 36.

Question 20

When $30\%$ of one number is subtracted from another number, the second number reduces to its four-fifths. What is the ratio of the first to the second number?  (Bank P.O., 2010)

Accepted Answer

Concept & Algebraic Translation: Translate the word problem directly into a linear equation. Let the "one number" (the first number) be x and "another number" (the second number) be y. Step-by-Step Solution: 1. Express "30% of one number": 0.3x or (3/10)x. 2. "subtracted from another number": y - (3/10)x. 3. "the second number reduces to its four-fifths": The result equals (4/5)y. 4. Set up the equation: y - (3/10)x = (4/5)y 5. Group the y terms on one side: y - (4/5)y = (3/10)x (1/5)y = (3/10)x 6. Cross-multiply to eliminate the denominators: 10y = 15x 7. Divide both sides by 5: 2y = 3x 8. Find the ratio of the first number (x) to the second number (y): (x/y) = (2/3) The ratio is 2 : 3. 9. Check the given options. The ratio 2 : 3 is not among (a) 2:5, (b) 3:2, or (c) 4:7. Exam Strategy & Shortcut: Instead of fractions, work purely with percentages. The equation is 100% of y - 30% of x = 80% of y. Subtracting 80% of y from both sides leaves 20% of y = 30% of x. Drop the percentages and zeroes: 2y = 3x \implies x:y = 2:3. Common Pitfall: Reversing the final ratio. Many test-takers reach 2y = 3x and hastily choose option (b) 3 : 2. However, the question specifically asks for "the first (x) to the second (y)", which is x/y = 2/3. Final Answer: Therefore, the correct answer is None of these.

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