Question 1

If $5a + 3b : 2a - 3b = 23 : 5$, then the value of $a : b$ is

Accepted Answer

Concept & Cross-Multiplication: To find the base ratio of two variables from a fractional equation, express the ratio as a fraction and use cross-multiplication to isolate the variables on opposite sides of the equation. Step-by-Step Solution: - Given: (5a + 3b/2a - 3b) = (23/5) - Cross-multiply: 5(5a + 3b) = 23(2a - 3b) - Expand both sides: 25a + 15b = 46a - 69b - Group like terms: - Move terms with b to the left: 15b + 69b = 84b - Move terms with a to the right: 46a - 25a = 21a - Equation: 84b = 21a - Solve for a/b: (a/b) = (84/21) - Simplify: (a/b) = (4/1) = 4 : 1. Exam Strategy & Shortcut: You can quickly group terms mentally: (23 \cdot 2 - 5 \cdot 5)a = (23 \cdot 3 + 5 \cdot 3)b \implies 21a = 84b \implies a : b = 84 : 21 = 4 : 1. This saves writing down the intermediate expansion steps. Common Pitfall: A common pitfall is incorrectly managing the negative signs during expansion, such as writing -69b as +69b, which drastically changes the resulting ratio. Final Answer: Therefore, the correct answer is 4 : 1.

Question 2

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 3

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 4

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 5

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 6

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 7

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 8

The ratio of urea and potash in a mixed fertilizer is 7 : 3. Express the quantity of urea present as percentage of the total amount of fertilizer.

Accepted Answer

Concept & Ratio to Percentage: To convert a part of a ratio into a percentage of the whole, you must first find the total number of parts in the ratio. Then, divide the specific part by the total parts and multiply by 100. Percentage = ( \frac{Part}{Total Parts} ) × 100 Step-by-Step Solution: 1. Identify the ratio of urea to potash: 7 : 3. 2. Calculate the total number of parts in the mixture: Total Parts = 7 (urea) + 3 (potash) = 10 parts 3. Identify the number of parts that represent urea, which is 7. 4. Calculate the percentage of urea in the total mixture: Percentage of Urea = ((7/10)) × 100 Percentage of Urea = 0.7 × 100 = 70% Exam Strategy & Shortcut: Since the total parts sum up to exactly 10 (7 + 3), you can instantly scale this to a percentage (out of 100) by multiplying both parts by 10. The ratio 7 : 3 becomes 70 : 30 out of 100. Thus, urea is clearly 70%. Common Pitfall: Students sometimes calculate the percentage based on the other component, e.g., (7/3), instead of the total mixture. Always divide by the sum of the ratio parts when asked for a percentage of the "total amount". Final Answer: Therefore, the correct answer is 70%.

Question 9

In a class the number of girls is 20% more than that of the boys. The strength of the class is 66. If 4 more girls are admitted to the class, the ratio of the number of boys to that of the girls is

Accepted Answer

Concept & Percentage Proportions: To solve problems relating totals and relative percentages, express both quantities in terms of a single variable. The total sum of these expressions equals the overall strength. Total = Boys + Girls Step-by-Step Solution: 1. Let the number of boys in the class be x. 2. The number of girls is 20% more than the boys, which can be written as x + 0.20x = 1.2x. 3. The total strength of the class is 66. Set up the equation: x + 1.2x = 66 4. Solve for x: 2.2x = 66 x = (66/2.2) = 30 5. So, there are 30 boys. The number of girls is 1.2 × 30 = 36. 6. If 4 more girls are admitted, the new number of girls becomes 36 + 4 = 40. 7. The new ratio of boys to girls is: \frac{Boys}{Girls} = (30/40) = (3/4) 8. This corresponds to the ratio 3 : 4. Exam Strategy & Shortcut: Instead of decimals, use proportional parts. Let boys be 100 units. Girls are 120 units. The ratio of Boys:Girls is 100:120 = 5:6. Total parts = 5 + 6 = 11 parts. Since 11 parts = 66 students, 1 part = 6 students. Boys = 5 × 6 = 30. Girls = 6 × 6 = 36. New girls = 36 + 4 = 40. New ratio = 30 : 40 = 3 : 4. Common Pitfall: A common error is adding the 4 new students to the boys' count instead of the girls, or calculating the final ratio backwards as girls to boys (4 : 3). Always pay close attention to the final order requested in the prompt. Final Answer: Therefore, the correct answer is 3 : 4.

Question 10

The sides of a triangle are in the ratio 3 : 4 : 5. The measure of the largest angle of the triangle is

Accepted Answer

Concept & Angle Sum Property: *Note: The original question text contains a typographical error, stating "sides" instead of "angles". A triangle with sides in the ratio 3:4:5 is a right-angled triangle with the largest angle being 90°, which is not among the given options. To match the provided options, we must solve under the assumption that the "angles" are in the ratio 3:4:5.* The sum of all interior angles of a triangle is always 180°. Step-by-Step Solution: • Assume the angles of the triangle are 3x, 4x, and 5x. • By the angle sum property of a triangle: 3x + 4x + 5x = 180^\circ. • 12x = 180^\circ • x = (180^\circ/12) = 15^\circ • The largest angle corresponds to the largest ratio part, which is 5x. • Largest angle = 5 × 15^\circ = 75^\circ. Exam Strategy & Shortcut: Recognize that the total ratio parts are 3 + 4 + 5 = 12. Divide the total sum of angles, 180°, by 12 to find the value of one ratio unit (15^\circ). Instantly multiply by the largest ratio unit (5) for the largest angle: 15 × 5 = 75^\circ. Common Pitfall: Misinterpreting the ratio as sides literally and concluding the triangle is a right triangle. This would lead to an answer of 90°, confusing the student when it does not appear in the multiple-choice options due to the book's misprint. Final Answer: Therefore, the correct answer is 75°.

Question 11

What number has to be added to the terms of 3 : 5 to make the ratio 5 : 6? (R.R.B., 2006)

Accepted Answer

Concept & Ratio Modification: When the same number is added to both terms of a ratio, express the initial terms using a common variable (or just the raw numbers if we are looking for absolute addition to the simplest form) and equate to the new fraction. Step-by-Step Solution: • Let the number to be added to both terms be x. • The original terms are 3 and 5. • After adding x to both, the new terms become (3 + x) and (5 + x). • The new ratio is 5 : 6. Set up the equation: (3 + x/5 + x) = (5/6) • Cross-multiply to solve for x: 6(3 + x) = 5(5 + x) 18 + 6x = 25 + 5x • Rearrange the equation to isolate x: 6x - 5x = 25 - 18 x = 7 • The number that must be added is 7. Exam Strategy & Shortcut: Use Option Elimination. Just add the options to 3 and 5 and see which one yields the fraction (5/6). Test (a) 6: (3+6/5+6) = (9/11). Incorrect. Test (b) 7: (3+7/5+7) = (10/12) = (5/6). Correct! This avoids algebra entirely. Common Pitfall: A common mistake is adding the variable to the numerator and denominator separately without treating them as a cohesive binomial during cross-multiplication. Final Answer: Therefore, the correct answer is 7.

Question 12

The least whole number which when subtracted from both the numerator and the denominator of the fractional number 6 : 7, gives a ratio less than 16 : 21, is

Accepted Answer

Concept & Fractional Inequalities: When a constant is subtracted from both the numerator and denominator of a proper fraction, the value of the new fraction decreases. We can set up an inequality to find the specific range for this unknown constant. Step-by-Step Solution: 1. Let the least whole number to be subtracted be x. 2. The initial fraction is (6/7). Subtracting x from both gives the expression (6 - x/7 - x). 3. According to the problem, this new ratio must be strictly less than (16/21): (6 - x/7 - x) 0 (since x is a small whole number, testing will verify this), we cross-multiply: 21(6 - x) (14/5) x > 2.8 6. The question asks for the *least whole number*. The smallest integer strictly greater than 2.8 is 3. Exam Strategy & Shortcut: Option Elimination: Plug the given options directly into the expression (6-x/7-x) and compare it to (16/21) (which is approximately 0.761). - Test (a) 2: (6-2/7-2) = (4/5) = 0.8 (Not less than 0.761) - Test (b) 3: (6-3/7-3) = (3/4) = 0.75 (Less than 0.761). Valid! Since we need the least whole number and the options are ascending, we stop at 3. Common Pitfall: Treating the inequality as a strict equation (=) and getting 2.8, then incorrectly rounding down to 2 instead of selecting the next whole number that satisfies the *less than* condition. Final Answer: Therefore, the correct answer is 3.

Question 13

The ratio of the students in schools $A, B$ and $C$ is 5 : 4 : 7. If the number of students in the schools are increased by 20%, 25% and 20% respectively, what would be the new ratio of the students in schools $A, B$ and $C$?  (Bank P.O., 2010)

Accepted Answer

Concept & Percentage Increase in Ratios: To find the new ratio after each part of a ratio undergoes a percentage increase, multiply the original ratio parts by their respective growth factors (1 + (r/100)). Step-by-Step Solution: • Assume Initial Values: Let the number of students in schools A, B, and C be 5x, 4x, and 7x. To make percentage calculations easy, assume x = 10. So, the initial students are 50, 40, and 70. • Calculate Increases: • School A new students = 50 × (1 + (20/100)) = 50 × 1.20 = 60. • School B new students = 40 × (1 + (25/100)) = 40 × 1.25 = 50. • School C new students = 70 × (1 + (20/100)) = 70 × 1.20 = 84. • Find the New Ratio: New Ratio = 60 : 50 : 84. • Simplify: Divide all terms by 2. New Ratio = 30 : 25 : 42. Exam Strategy & Shortcut: Skip the variables and multiply the ratio numbers directly by their percentage multipliers: A = 5 × 1.2 = 6 B = 4 × 1.25 = 5 C = 7 × 1.2 = 8.4 Ratio = 6 : 5 : 8.4. Multiply by 10 to remove the decimal \implies 60 : 50 : 84. Simplify by dividing by 2 \implies 30 : 25 : 42. Common Pitfall: Trying to add the percentages directly to the ratio values (e.g., 5 + 0.20 = 5.20) instead of applying the percentage as a multiplier of the base ratio unit. Final Answer: Therefore, the correct answer is 30 : 25 : 42.

Ratio and Proportion Questions

Questions

Practice smarter