Question 1

$\frac{3}{4} : \frac{1}{2} : : 27y : $x$

Accepted Answer

Concept & Proportion with Fractions: The rule of proportion applies exactly the same way when fractions and variables are involved. Product of extremes = product of means. a : b : : c : d \implies ad = bc Step-by-Step Solution: • Given: (3/4) : (1/2) : : 27y : x • Apply the rule: ((3/4)) × x = ((1/2)) × 27y • Simplify the right side: (3x/4) = (27y/2) • Solve for x: Multiply both sides by (4/3). x = ((27y/2)) × ((4/3)) • Calculate: x = (108y/6) = 18y Exam Strategy & Shortcut: Simplify the ratio on the left first. (3/4/1/2) = (3/4) × 2 = (3/2). So the ratio is 3 : 2. Now apply this to the right side: 27y : x = 3 : 2. 3 parts = 27y (so 1 part = 9y). 2 parts = 2 × 9y = 18y. Common Pitfall: Mismanaging fractions during cross-multiplication, particularly forgetting to take the reciprocal when moving a fraction to the other side of the equation. Final Answer: Therefore, the correct answer is 18y.

Question 2

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

Accepted Answer

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 3

What will be the simplest form of the ratio 3 hours: 1 day?

Accepted Answer

Concept & Unit Homogeneity in Ratios: To form a valid mathematical ratio, both quantities must be expressed in the exact same units. You cannot compare hours directly to days without conversion. Ratio = \frac{Quantity 1}{Quantity 2} (after equating units) Step-by-Step Solution: • Given: 3 hours and 1 day. • Convert units: Since 1 day contains 24 hours, convert "1 day" into hours to match the other quantity. 1 day = 24 hours • Set up the ratio: Ratio = 3 hours : 24 hours • Simplify: Divide both sides by their highest common factor, which is 3. (3/3) : (24/3) 1 : 8 Exam Strategy & Shortcut: Whenever comparing disparate time units (hours to days, minutes to hours), immediately convert the larger unit into the smaller unit. Knowing 1 day = 24 hours makes this a mental math problem: 3/24 = 1/8. Common Pitfall: Creating the ratio without converting units first (yielding 3:1), which compares completely different scales of time and leads to an incorrect answer. Final Answer: Therefore, the correct answer is 1 : 8.

Question 4

The simplest form of the ratio $1.5 : 2.5$ is

Accepted Answer

Concept & Simplifying Decimal Ratios: Ratios should ideally be expressed as fractions consisting of whole numbers in their lowest terms. Decimal ratios can be simplified by multiplying by powers of 10 to clear the decimals, then dividing by the greatest common divisor. Step-by-Step Solution: • Given: Ratio is 1.5 : 2.5. • Eliminate decimals: Multiply both sides of the ratio by 10 to remove the single decimal place. (1.5 × 10) : (2.5 × 10) = 15 : 25 • Simplify: Find the highest common factor (HCF) of 15 and 25, which is 5. Divide both sides by 5. (15/5) : (25/5) = 3 : 5 Exam Strategy & Shortcut: You can mentally shift the decimal point one place to the right for both numbers to get 15 : 25. Recognize immediately that both end in 5, making them divisible by 5. 15/5 = 3 and 25/5 = 5, giving 3 : 5. Common Pitfall: Selecting an option that is mathematically equivalent but not in the *simplest* form. For example, 15 : 25 and 6 : 10 are equivalent ratios, but 3 : 5 is the most reduced, simplest form requested by the question. Final Answer: Therefore, the correct answer is 3 : 5.

Question 5

$9^{3.04} : 9^{2.04}$ is equal to

Accepted Answer

Concept & Laws of Exponents: When dividing exponential expressions with the same base, you subtract the exponent of the denominator from the exponent of the numerator. A ratio can be written as a fraction. (x^a/x^b) = x^{a - b} Step-by-Step Solution: • Given: The ratio is 9^{3.04} : 9^{2.04}. • Rewrite as a fraction: \frac{9^{3.04}}{9^{2.04}} • Apply the exponent rule: Subtract the denominator's power from the numerator's power. 9^{3.04 - 2.04} • Calculate the difference: 3.04 - 2.04 = 1.00. 9^1 = 9 • Format as a ratio: The whole number 9 can be written as a ratio over 1. 9 : 1 Exam Strategy & Shortcut: Recognize immediately that the bases are identical (9). Simply subtract the decimals in your head: 3.04 - 2.04 = 1. The result is 9^1, which corresponds to the ratio 9 : 1. Common Pitfall: A common mistake is trying to calculate the actual immense values of 9^{3.04} and 9^{2.04} or incorrectly manipulating the decimal exponents, leading to a mismatched ratio. Final Answer: Therefore, the correct answer is 9 : 1.

Question 6

In a proportion the product of 1st and 4th terms is 40 and that of 2nd and 3rd terms is $2.5x$. Then the value of $x$ is

Accepted Answer

Concept & Proportion Rule: In any mathematical proportion consisting of four terms (a : b : : c : d), the fundamental rule states that the product of the extreme terms equals the product of the mean terms. Product of Extremes = Product of Means Step-by-Step Solution: • Given: Product of 1st and 4th terms (Extremes) = 40. Product of 2nd and 3rd terms (Means) = 2.5x. • Equate the products: Based on the proportion rule, set them equal to each other. 40 = 2.5x • Solve for x: Divide both sides by 2.5. x = (40/2.5) • Simplify: Multiply the numerator and denominator by 10 to remove the decimal. x = (400/25) • Calculate: Divide 400 by 25. Since 100 / 25 = 4, then 400 / 25 = 4 × 4 = 16. x = 16 Exam Strategy & Shortcut: Think of 2.5 as the fraction (5/2). The equation becomes 40 = (5/2)x. Multiply both sides by 2 to get 80 = 5x, and then divide by 5 to quickly find x = 16. Using fractions is often mentally faster than long division with decimals. Common Pitfall: Dividing in the wrong order, such as calculating x = 2.5 / 40, which results in a fraction rather than isolating the variable correctly. Final Answer: Therefore, the correct answer is 16.

Question 7

$25\%$ of $A$'s income is equal to $35\%$ of $B$'s income. The ratio of the incomes of $A$ and $B$ is

Accepted Answer

Concept & Ratios from Percentages: When a percentage of one quantity equals a percentage of another, you can form a direct equation to isolate the ratio of the two quantities. x% of A = y% of B \implies (A/B) = (y/x) Step-by-Step Solution: • Given: 25% of A = 35% of B. • Formulate the equation: Write it mathematically using fractions or decimals. (25/100) × A = (35/100) × B • Simplify the equation: The denominators (100) cancel out on both sides. 25A = 35B • Isolate the ratio (A/B): Rearrange the terms by dividing both sides by 25B. (A/B) = (35/25) • Reduce the fraction: Divide the numerator and denominator by their highest common factor, which is 5. (A/B) = (7/5) • Format as a ratio: A : B = 7 : 5 Exam Strategy & Shortcut: Skip writing out the hundreds. If 25A = 35B, then the ratio A:B is simply the reverse of their coefficients: 35:25. Mentally divide both by 5 to instantly arrive at 7:5. Common Pitfall: A very common trap is writing the ratio in the exact order the numbers appear in the text (25 : 35 \implies 5 : 7), completely inverting the correct mathematical relationship. Final Answer: Therefore, the correct answer is 7 : 5.

Question 8

If $x = \frac{1}{3}y$ and $y = \frac{1}{2}z$, then $x : y : z$ is equal to

Accepted Answer

Concept & Compounding Ratios: To find a combined ratio x : y : z from two separate ratios (x:y and y:z), establish a common value for the shared variable (y) in both ratios. Step-by-Step Solution: • Given equations: x = (1/3)y and y = (1/2)z. • Convert to individual ratios: From x = (1/3)y, we get (x/y) = (1/3) \implies x : y = 1 : 3. From y = (1/2)z, we get (y/z) = (1/2) \implies y : z = 1 : 2. • Align the ratios: Look at the value of the common term, y, in both ratios. In x : y, y is 3. In y : z, y is 1. • Make the common term equal: Multiply the second ratio by 3 so that y matches in both. y : z = (1 × 3) : (2 × 3) = 3 : 6. • Combine the ratios: Now that y is 3 in both cases, you can link them directly. x : y : z = 1 : 3 : 6. Exam Strategy & Shortcut: You can substitute directly without finding common terms. Let z = 6 (a convenient multiple). Since y = (1/2)z, y = (1/2)(6) = 3. Since x = (1/3)y, x = (1/3)(3) = 1. The values are x=1, y=3, z=6. The ratio is directly 1 : 3 : 6. Common Pitfall: Trying to merge the ratios without equalizing the bridging variable (y), resulting in an invalid sequence. Final Answer: Therefore, the correct answer is 1 : 3 : 6.

Question 9

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 10

If $W_1 : W_2 = 2 : 3$ and $W_1 : W_3 = 1 : 2$, then $W_2 : W_3$ is

Accepted Answer

Concept & Linking Ratios: When two ratios share a common term (in this case, W_1), we can combine them to find the ratio between the other two terms by manipulating the fractions. (W_2/W_3) = (W_2/W_1) × (W_1/W_3) Step-by-Step Solution: 1. We are given the following ratios: (W_1/W_2) = (2/3) (W_1/W_3) = (1/2) 2. We need to find the ratio W_2 : W_3, which is the fraction (W_2/W_3). 3. First, take the reciprocal of the first ratio to get (W_2/W_1): (W_2/W_1) = (3/2) 4. Now, multiply this inverted ratio by the second ratio to cancel out W_1: (W_2/W_3) = ((W_2/W_1)) × ((W_1/W_3)) (W_2/W_3) = ((3/2)) × ((1/2)) 5. Multiply the numerators and denominators: (W_2/W_3) = (3 × 1/2 × 2) = (3/4) 6. Express this fraction as a ratio: W_2 : W_3 = 3 : 4 Exam Strategy & Shortcut: Alternatively, equalize the common variable W_1. W_1 : W_2 = 2 : 3 W_1 : W_3 = 1 : 2 = 2 : 4 (multiplying by 2 to match W_1=2) Now that W_1 is 2 in both, we can clearly see W_2 = 3 and W_3 = 4. Thus, W_2 : W_3 = 3 : 4. This visual scaling is incredibly fast. Common Pitfall: Failing to notice that W_1 is the first term in *both* given ratios. If you blindly multiply (2/3) × (1/2), you calculate ((W_1)^2/W_2 W_3), which is incorrect. Final Answer: Therefore, the correct answer is 3 : 4.

Question 11

If one star equals four circles and three circles equal four diamonds, then what is the ratio of star to diamond?

Accepted Answer

Concept & Algebraic Substitution: When given word equations linking different entities, convert them into algebraic equations. To find the direct relationship between the first and last entity, create fractional ratios and multiply them together. \frac{Star}{Diamond} = \frac{Star}{Circle} × \frac{Circle}{Diamond} Step-by-Step Solution: 1. Let S represent a star, C represent a circle, and D represent a diamond. 2. Translate the first statement: "one star equals four circles" 1S = 4C \Rightarrow (S/C) = (4/1) 3. Translate the second statement: "three circles equal four diamonds" 3C = 4D \Rightarrow (C/D) = (4/3) 4. We need the ratio of star to diamond, which is (S/D). 5. Multiply the two fractional ratios: (S/D) = ((S/C)) × ((C/D)) (S/D) = ((4/1)) × ((4/3)) 6. Perform the multiplication: (S/D) = (16/3) 7. Convert back to ratio format: S : D = 16 : 3 Exam Strategy & Shortcut: Treat the objects as variables immediately. You have S/C = 4/1 and C/D = 4/3. Multiplying them straight across to cancel C gives (4 × 4) / (1 × 3) = 16/3. Common Pitfall: A frequent error is incorrectly setting up the initial fractions. For instance, reading 1S = 4C and mistakenly writing the ratio S/C as 1/4 instead of 4/1. Always double-check the isolated ratio. Final Answer: Therefore, the correct answer is 16 : 3.

Question 12

If $A : B = \frac{1}{2} : \frac{3}{8}$, $B : C = \frac{1}{3} : \frac{5}{9}$ and $C : D = \frac{5}{6} : \frac{3}{4}$, then the ratio $A : B : C : D$ is

Accepted Answer

Concept & Simplifying Fractional Ratios: To combine ratios given as fractions, first simplify each ratio into whole numbers by multiplying both parts by the least common multiple (LCM) of their denominators. (x/y/a/b) = (xb/ya) Step-by-Step Solution: 1. Simplify A : B = (1/2) : (3/8). The LCM of 2 and 8 is 8. Multiply both fractions by 8: A : B = 4 : 3 2. Simplify B : C = (1/3) : (5/9). The LCM of 3 and 9 is 9. Multiply both fractions by 9: B : C = 3 : 5 3. Simplify C : D = (5/6) : (3/4). The LCM of 6 and 4 is 12. Multiply both fractions by 12: C : D = 10 : 9 4. Now, combine the standard ratios A : B = 4 : 3 and B : C = 3 : 5. The common term is B (values 3 and 3). Since they already match, we can directly link them: A : B : C = 4 : 3 : 5 5. Combine A : B : C = 4 : 3 : 5 with C : D = 10 : 9. The common term is C (values 5 and 10). The LCM is 10. 6. Scale the first ratio by multiplying it by 2 to make C equal to 10: A : B : C = 8 : 6 : 10 7. Link it with C : D since C now matches: A : B : C : D = 8 : 6 : 10 : 9 Exam Strategy & Shortcut: Simplify the fractions instantly in your head. Then write them in a cascading structure. Since B naturally aligns as 3 in both initial ratios, A:B:C trivially becomes 4:3:5. You then only need to match C=5 with C=10 by doubling the first set, producing 8:6:10 and appending the 9. Common Pitfall: Attempting to equalize the intermediate terms while the ratios are still in their original fractional forms is a massive recipe for calculation errors and lost time. Always convert to integer ratios immediately. Final Answer: Therefore, the correct answer is 8 : 6 : 10 : 9.

Question 13

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

Accepted Answer

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 14

If $x : y = 3 : 1$, then $x^3 - y^3 : x^3 + y^3 = $x$

Accepted Answer

Concept & Cubes in Ratios: When dealing with a ratio of sums and differences of cubes, check for homogeneity. Since all terms are of degree 3, direct substitution of the ratio values is valid and efficient. Step-by-Step Solution: - Given: x : y = 3 : 1. - Substitute: Let x = 3 and y = 1. - Expression: (x^3 - y^3) : (x^3 + y^3) - Calculate terms: - x^3 = 3^3 = 27 - y^3 = 1^3 = 1 - Numerator (Difference): 27 - 1 = 26 - Denominator (Sum): 27 + 1 = 28 - Simplify the ratio: 26 : 28 = 13 : 14. Exam Strategy & Shortcut: Calculate the cubes mentally (27 and 1). The required ratio is simply their difference divided by their sum: (27 - 1) / (27 + 1) = 26 / 28 = 13 / 14. Common Pitfall: A frequent error is placing the sum in the numerator and the difference in the denominator by not reading the question order carefully, which would lead to the trap option 14 : 13. Final Answer: Therefore, the correct answer is 13 : 14.

Question 15

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 16

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 17

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 18

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 19

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 20

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.

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