Question 1

The simplest value of $\frac{1}{1 	imes 2} + \frac{1}{2 	imes 3} + \frac{1}{3 	imes 4} + \cdots + \frac{1}{9 	imes 10}$ is

Accepted Answer

Concept & Formula: This is a standard telescoping series. When the denominator of a fraction is the product of two consecutive integers, it can be expressed as the difference of two adjacent unit fractions. (1/n(n+1)) = (1/n) - (1/n+1) Step-by-Step Solution: Rewrite each term in the series as a difference using the formula: (1/1 × 2) = (1/1) - (1/2) (1/2 × 3) = (1/2) - (1/3) (1/3 × 4) = (1/3) - (1/4) \vdots (1/9 × 10) = (1/9) - (1/10) Sum all these rewritten expressions: Sum = ((1/1) - (1/2)) + ((1/2) - (1/3)) + ((1/3) - (1/4)) + \cdots + ((1/9) - (1/10)) Notice that intermediate terms cancel each other out (e.g., -(1/2) cancels with +(1/2)). The only terms that survive the cancellation are the very first term and the very last term: Sum = 1 - (1/10) = (10 - 1/10) = (9/10) Exam Strategy & Shortcut: For any continuous series of the format (1/1 × 2) + (1/2 × 3) + \dots + (1/n × (n+1)), the sum is always equal to (n/n+1). In this problem, the last term is (1/9 × 10), which means n = 9. Applying the shortcut gives the answer instantly: (9/9+1) = (9/10). Common Pitfall: Attempting to find a common denominator (LCM) for all the fractions (2, 6, 12 \dots 90) will take far too long and is unnecessary. Always look for a telescoping pattern when denominators are products of consecutive numbers. Final Answer: Therefore, the correct answer is 9/10.

Question 2

$\left(\frac{1}{1 \cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 10} + \frac{1}{10 \cdot 13} + \frac{1}{13 \cdot 16}\right)$ is equal to

Accepted Answer

Concept & Formula: This series is a variation of the telescoping sum where the factors in the denominator have a constant difference d. We can split the terms using partial fraction decomposition. (1/a \cdot b) = (1/b - a) ( (1/a) - (1/b) ) Step-by-Step Solution: Identify the constant difference d between the factors in each denominator: 4 - 1 = 3 7 - 4 = 3 10 - 7 = 3 So, the common difference d = 3. Apply the partial fraction formula to each term: (1/1 \cdot 4) = (1/3) ( (1/1) - (1/4) ) (1/4 \cdot 7) = (1/3) ( (1/4) - (1/7) ) \vdots (1/13 \cdot 16) = (1/3) ( (1/13) - (1/16) ) Add the terms together and factor out (1/3): Sum = (1/3) [ ((1/1) - (1/4)) + ((1/4) - (1/7)) + \cdots + ((1/13) - (1/16)) ] The intermediate terms cancel out perfectly. Sum = (1/3) [ 1 - (1/16) ] = (1/3) [ (15/16) ] = (5/16) Exam Strategy & Shortcut: Use the general shortcut for constant-difference telescoping series: Sum = (1/Difference) × ( (1/First Factor) - (1/Last Factor) ) Here, Difference = 3, First Factor = 1, Last Factor = 16. Sum = (1/3) × ( 1 - (1/16) ) = (1/3) × (15/16) = (5/16) This bypasses writing out the full sequence. Common Pitfall: The most frequent mistake is forgetting to divide by the difference d = 3. Many students simply compute (1 - (1/16)) and arrive at (15/16), completely missing the scaling factor. Final Answer: Therefore, the correct answer is 5/16.

Question 3

$\frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{3 \cdot 4 \cdot 5} + \frac{1}{4 \cdot 5 \cdot 6}$ is equal to

Accepted Answer

Concept & Formula: This is an advanced telescoping series where each denominator has three consecutive factors. We can split each term into the difference of two fractions, each containing two factors. The formula to decompose a term with three factors is: (1/k(k+1)(k+2)) = (1/2) [ (1/k(k+1)) - (1/(k+1)(k+2)) ] Step-by-Step Solution: Apply the decomposition formula to each term in the series: (1/1 \cdot 2 \cdot 3) = (1/2) [ (1/1 \cdot 2) - (1/2 \cdot 3) ] (1/2 \cdot 3 \cdot 4) = (1/2) [ (1/2 \cdot 3) - (1/3 \cdot 4) ] (1/3 \cdot 4 \cdot 5) = (1/2) [ (1/3 \cdot 4) - (1/4 \cdot 5) ] (1/4 \cdot 5 \cdot 6) = (1/2) [ (1/4 \cdot 5) - (1/5 \cdot 6) ] Now, add all these terms together and factor out the common (1/2): Sum = (1/2) [ ((1/1 \cdot 2) - (1/2 \cdot 3)) + ((1/2 \cdot 3) - (1/3 \cdot 4)) + \dots + ((1/4 \cdot 5) - (1/5 \cdot 6)) ] Notice the cascade of cancellations. Every intermediate term cancels out, leaving only the first component of the first term and the last component of the final term: Sum = (1/2) [ (1/1 \cdot 2) - (1/5 \cdot 6) ] = (1/2) [ (1/2) - (1/30) ] Find a common denominator to subtract the fractions inside the bracket: = (1/2) [ (15/30) - (1/30) ] = (1/2) [ (14/30) ] = (7/30) Exam Strategy & Shortcut: For a series of n terms in the format \sum (1/k(k+1)(k+2)), the generalized sum formula is: (n(n+3)/4(n+1)(n+2)) Here, we have exactly 4 terms, so n = 4: S_4 = (4(4+3)/4(4+1)(4+2)) = (4 × 7/4 × 5 × 6) = (28/120) = (7/30) Memorizing this generalized formula saves immense time during competitive exams. Common Pitfall: A common error is attempting to compute the actual value of each fraction ((1/6) + (1/24) + (1/60) + (1/120)) and finding the LCM. While doable for 4 terms, this method becomes impossible under time pressure if the series has 10 or 20 terms. Final Answer: Therefore, the correct answer is 7/30.

Question 4

The value of $\frac{3}{1^2 \cdot 2^2} + \frac{5}{2^2 \cdot 3^2} + \frac{7}{3^2 \cdot 4^2} + \frac{9}{4^2 \cdot 5^2} + \frac{11}{5^2 \cdot 6^2} + \frac{13}{6^2 \cdot 7^2} + \frac{15}{7^2 \cdot 8^2} + \frac{17}{8^2 \cdot 9^2} + \frac{19}{9^2 \cdot 10^…

Accepted Answer

Concept & Formula: This complex-looking sequence is a cleverly disguised telescoping series. Notice the relationship between the numerator and the factors in the denominator: the numerator is exactly the difference between the two squared factors in the denominator. Mathematically, the general term can be written and split as follows: T_n = ((n+1)^2 - n^2/n^2 \cdot (n+1)^2) = ((n+1)^2/n^2 \cdot (n+1)^2) - (n^2/n^2 \cdot (n+1)^2) = (1/n^2) - (1/(n+1)^2) Step-by-Step Solution: Apply this splitting logic to every term in the given series: First term: (3/1^2 \cdot 2^2) = (4 - 1/1^2 \cdot 2^2) = (1/1^2) - (1/2^2) Second term: (5/2^2 \cdot 3^2) = (9 - 4/2^2 \cdot 3^2) = (1/2^2) - (1/3^2) Third term: (7/3^2 \cdot 4^2) = (16 - 9/3^2 \cdot 4^2) = (1/3^2) - (1/4^2) \vdots Last term: (19/9^2 \cdot 10^2) = (100 - 81/9^2 \cdot 10^2) = (1/9^2) - (1/10^2) Now, write out the sum of all these rewritten terms: Sum = ((1/1^2) - (1/2^2)) + ((1/2^2) - (1/3^2)) + \dots + ((1/9^2) - (1/10^2)) Every adjacent fractional pair cancels out (-(1/2^2) cancels +(1/2^2), and so on). The only remaining components are the first positive fraction and the last negative fraction: Sum = (1/1^2) - (1/10^2) Sum = 1 - (1/100) = (99/100) Exam Strategy & Shortcut: Whenever you see a series where the numerator equals the difference of the denominator's factors (e.g., Numerator = b - a for a denominator a \cdot b), the entire series evaluates to: Sum = (1/First Factor) - (1/Last Factor) Here, the first factor is 1^2 = 1 and the last factor is 10^2 = 100. So, Sum = (1/1) - (1/100) = (99/100). You can solve this visually in under 5 seconds. Common Pitfall: Students frequently overlook the relationship between the numerator and denominator, assuming the problem requires massive brute-force fraction addition, leading them to either skip the question or waste minutes calculating huge LCMs. Final Answer: Therefore, the correct answer is 99/100.

Question 5

Mayank, Mirza, Little and Jaspal bought a motorbike for ₹ 60000. Mayank paid one-half of the sum of the amounts paid by the other boys, Mirza paid one-third of the sum of the amounts paid by the other boys and Little paid one-fourth of the sum of th…

Accepted Answer

Concept & Strategy: The problem tests proportional division of a total sum. The key algebraic insight is that if a person pays a fraction of what the *others* paid, you can easily find their share of the *total* amount. If a person pays (1/x) of the others, they pay (1/x+1) of the total sum. Step-by-Step Solution: • Given: Total cost of the motorbike T = 60000. • Calculate Mayank's share: Mayank pays (1/2) of the others. Let Mayank's share be M. The others paid (T - M). M = (1/2)(T - M) \Rightarrow 3M = T \Rightarrow M = (60000/3) = 20000. • Calculate Mirza's share: Mirza pays (1/3) of the others. Mi = (1/4)T = (60000/4) = 15000. • Calculate Little's share: Little pays (1/4) of the others. L = (1/5)T = (60000/5) = 12000. • Calculate Jaspal's share: Jaspal pays the remaining balance. J = T - (M + Mi + L) = 60000 - (20000 + 15000 + 12000) = 60000 - 47000 = 13000. Exam Strategy & Shortcut: Use the ratio-to-fraction shortcut directly: "1/2 of others" \rightarrow (1/1+2) = (1/3) of Total. "1/3 of others" \rightarrow (1/1+3) = (1/4) of Total. "1/4 of others" \rightarrow (1/1+4) = (1/5) of Total. Total fraction paid by the first three = (1/3) + (1/4) + (1/5) = (20 + 15 + 12/60) = (47/60). Jaspal's fraction = 1 - (47/60) = (13/60). Jaspal's amount = (13/60) × 60000 = 13000. Common Pitfall: A very common mistake is calculating the fractions based on the total amount directly (e.g., Mayank = 60000/2 = 30000). The prompt explicitly says "sum of the amounts paid by the *other* boys", not the total amount. Final Answer: Therefore, the correct answer is ₹ 13000.

Question 6

Breeding of a certain species of insects is incredible everyday the total number of such insects in a closed glass jar is double the number on the previous day. There was just one insect in the jar on 1/2/2007, and the jar was full to the brim with …

Accepted Answer

Concept & Logic: This is a classic reverse exponential growth problem. If a quantity doubles moving forward in time, it halves moving backward in time. No complex formulas are needed—just logical backtracking. Step-by-Step Solution: • Given: The jar doubles in capacity every single day. The jar is 100% full (1) on February 28, 2007. • Backtrack one day (Half-full): On February 27, the jar must have been exactly half-full, because doubling a half-full jar makes it completely full the next day. • Backtrack a second day (Quarter-full): On February 26, the jar must have been a quarter-full ((1/4)), because doubling it would make it half-full on the 27th. Exam Strategy & Shortcut: Whenever a problem asks when a continuously doubling entity was a certain fraction ((1/2^n)) of its final size, simply subtract n units of time from the final date. Quarter-full is (1/4) = (1/2^2). Subtract 2 days from the 28th to get the 26th. Common Pitfall: A common reflexive mistake is to divide the total time by 4 (e.g., 28 / 4 = 7). Exponential growth accelerates rapidly at the end; it spends the vast majority of its time barely filling the jar. Always think backwards from the end state. Final Answer: Therefore, the correct answer is 26.

Question 7

After measuring 120 metres of a rope, it was discovered that the measuring metre rod was 3 cm longer.  The true length of the rope measured is

Accepted Answer

Concept & Logic: This is a measurement error problem based on direct proportionality. If the measuring instrument is longer than its stated value, the actual measured length will be proportionally longer. True Length = Measured Units × True Length per Unit Step-by-Step Solution: • Identify the true unit length: A standard metre rod is meant to be 1 metre (100 cm). The faulty rod is 3 cm longer, meaning its true length is 100 + 3 = 103 cm, which equals 1.03 metres. • Identify the number of units measured: The rope was measured as "120 metres" using the faulty rod. This means the physical rod was laid down end-to-end exactly 120 times. • Calculate the actual physical length: Since each physical lay of the rod actually covered 1.03 metres: True Length = 120 × 1.03 metres • Perform the calculation: 120 × 1.03 = 120 × (1 + (3/100)) = 120 + 3.6 = 123.6 metres. • Format the final unit: 123.6 metres is mathematically equivalent to 123 metres and 60 centimetres. Exam Strategy & Shortcut: Use the concept of total error accumulation. The error is +3 cm for every metre measured. For 120 metres, the total accumulated error is 120 × 3 cm = 360 cm. Since 360 cm is exactly 3.6 metres, simply add this total error to the base measurement: 120 + 3.6 = 123.6 metres. This avoids decimals entirely. Common Pitfall: A very common mistake is subtracting the error (thinking the rod "overmeasured" so the true length is less). Remember, a longer rod covers more physical ground per "metre" reading, so the actual physical rope must be longer than the stated number. Final Answer: Therefore, the correct answer is 123 m 60 cm.

Question 8

Smita was asked to multiply a certain number by 36. She multiplied it by 63 instead and got an answer 3834 more than the correct one.  What was the number to be multiplied?

Accepted Answer

Concept & Formula: This is an algebraic simplification problem based on linear equations. When a number is multiplied by an incorrect multiplier, the difference between the wrong answer and the right answer is proportional to the difference between the two multipliers. Difference in Product = x × (Wrong Multiplier - Right Multiplier) Step-by-Step Solution: • Define the variable: Let the unknown number to be multiplied be x. • Formulate the equation based on given conditions: Correct calculation should have been: 36x Incorrect calculation performed: 63x The problem states the incorrect answer is 3834 *more* than the correct one. 63x - 36x = 3834 • Simplify the equation: (63 - 36)x = 3834 27x = 3834 • Solve for x: x = (3834/27) Divide by 9 first for easier arithmetic: (3834/9) = 426. Divide by 3: (426/3) = 142. Exam Strategy & Shortcut: Skip straight to the difference in the multipliers. Difference in multipliers = 63 - 36 = 27. The error (3834) is exactly 27 times the original number. Divide the error by the difference: 3834 / 27 = 142. No need to write out the full algebraic equation. Common Pitfall: Students often write the equation backward (e.g., 36x - 63x) and get confused by a negative sign, or they attempt to guess and check the options, which consumes way too much time for a simple linear division. Final Answer: Therefore, the correct answer is 142.

Question 9

A boy multiplied 423 by a number and obtained 65589 as his answer. If both the fives in the answer are wrong and all other figures are correct, the correct answer is:

Accepted Answer

Concept & Logic: This is a number system problem testing divisibility and pattern recognition. Since we know the incorrect answer (65589) has exactly three correct digits (6, 8, and 9) in their correct place values, the true answer must be in the format 6\_\_89. Furthermore, the true answer must be perfectly divisible by the original multiplicand, 423. Step-by-Step Solution: • Analyze the structure of the correct answer: The answer must look like 6\_\_89. Let the unknown multiplier be x. 423 × x = 6\_\_89. • Determine the unit digit of the multiplier: The product ends in 9. The multiplicand (423) ends in 3. The only single digit that yields a 9 when multiplied by 3 is 3 (3 × 3 = 9). Therefore, the multiplier x must end in 3 (e.g., 143, 153). • Test the given options: Since all options fit the pattern (they begin with 6 and end with 89), we simply divide each by 423 to see which yields a whole number. Option (a) 60489: (60489/423) = 143 (Exact match) Option (b) 61189: (61189/423) \approx 144.6 Option (c) 62189: (62189/423) \approx 147.01 Option (d) 62389: (62389/423) \approx 147.49 Exam Strategy & Shortcut: Use the options to your advantage. Since the answer must be perfectly divisible by 423, you can use the digital root (sum of digits) to quickly verify divisibility by 9 (since 423 is divisible by 9, as 4+2+3=9). Test option (a) 60489: 6+0+4+8+9 = 27 (Divisible by 9). Test option (b) 61189: 6+1+1+8+9 = 25 (Not divisible by 9). Test option (c) 62189: 6+2+1+8+9 = 26 (Not divisible by 9). Test option (d) 62389: 6+2+3+8+9 = 28 (Not divisible by 9). Option (a) is the only mathematically possible answer. Common Pitfall: Students often try to guess the missing digits by trial and error without testing the provided options, which wastes valuable time. Always leverage the multiple-choice format for divisibility checks. Final Answer: Therefore, the correct answer is 60489.

Question 10

A candidate was asked to find $\frac{7}{8}$ of a positive number. He found $\frac{7}{18}$ of the same by mistake. If his answer was 770 less than the correct one, then the original given number was

Accepted Answer

Concept & Equation: This is an algebraic fraction problem. The core logic is to set up a linear equation representing the difference between the correct mathematical operation and the mistaken operation. Correct Calculation - Mistaken Calculation = Difference Step-by-Step Solution: • Define the variable: Let the original positive number be x. • Formulate the expressions: The correct expected result: (7/8)x The incorrect mistaken result: (7/18)x • Set up the equation based on the given condition: The mistaken answer is 770 less than the correct answer. (7/8)x - (7/18)x = 770 • Solve for x: Find the Least Common Multiple (LCM) of 8 and 18, which is 72. Convert fractions to a common denominator: (63/72)x - (28/72)x = 770 (35/72)x = 770 x = (770 × 72/35) Divide 770 by 35 first: (770/35) = 22. x = 22 × 72 = 1584 • Check the options: The calculated original number is 1584. Looking at the options: (a) 1260, (b) 1548, (c) 6160. 1584 is not listed among the numerical choices. Exam Strategy & Shortcut: Factor out the common numerator (7) and the variable (x) to speed up fraction subtraction: 7x × ((1/8) - (1/18)) = 770 x × ((18 - 8/144)) = 110 x × ((10/144)) = 110 x = 11 × 144 = 1584. This method handles smaller numbers and avoids finding the LCM of the denominators immediately. Common Pitfall: A major trap here is option (b) 1548. It is extremely close visually to the correct answer (1584) due to a simple transposition of the middle digits (4 and 8). Careless reading under time pressure causes many students to select (b) instead of "None of these". Final Answer: Therefore, the correct answer is None of these.

Question 11

If $3 \frac{2}{3}$ is subtracted from $9 \frac{1}{9}$ and the difference is multiplied by 450, what is the final answer?

Accepted Answer

Concept & Strategy: This is a straight-forward application of the order of operations (BODMAS/PEMDAS) involving mixed fractions. Convert mixed numbers to improper fractions first to avoid borrowing errors during subtraction. Step-by-Step Solution: • Convert mixed fractions to improper fractions: 9 (1/9) = ((9 × 9) + 1/9) = (82/9) 3 (2/3) = ((3 × 3) + 2/3) = (11/3) • Perform the subtraction: Difference = (82/9) - (11/3) Make denominators equal (multiply numerator and denominator of the second fraction by 3): Difference = (82/9) - (33/9) = (49/9) • Perform the final multiplication: The difference is multiplied by 450. Final Answer = (49/9) × 450 Simplify by dividing 450 by 9: (450/9) = 50. Final Answer = 49 × 50 49 × 50 = 2450 • Check against options: The calculated answer is 2450. The given numerical options are 2045, 2250, and 2540. Since 2450 is missing, the answer must be "None of these". Exam Strategy & Shortcut: Subtract the integer and fractional parts separately if you are comfortable with negative fractions. Integer difference: 9 - 3 = 6 Fraction difference: (1/9) - (2/3) = (1/9) - (6/9) = -(5/9) Total difference: 6 - (5/9) = (54 - 5/9) = (49/9). Multiply by 450: (49/9) × 450 = 49 × 50 = 2450. Common Pitfall: A common error is misreading option (c) 2540 as 2450 under exam pressure. Test setters intentionally include transposed numbers (2450 vs 2540) to trap students who rush the final multiple-choice selection step. Final Answer: Therefore, the correct answer is None of these.

Question 12

If $x = y - \frac{y}{10}$, where $y$ is a positive integer which increases in value, then $x$

Accepted Answer

Concept & Logic: This problem tests the understanding of proportional linear equations. When a variable (x) is defined as a constant positive fraction of another variable (y), their growth directions will identical. Step-by-Step Solution: • Simplify the given equation: The equation is given as x = y - (y/10). We can factor out y: x = y (1 - (1/10)) x = y ((9/10)) x = 0.9y • Analyze the relationship: The equation x = 0.9y represents a direct, positive linear relationship between x and y. The coefficient (0.9) is a positive constant. • Determine the behavior: The problem states that y is a positive integer which *increases* in value. Since x is simply 90% of y, as y grows larger, x must proportionally grow larger as well. There are no negative signs or exponential constraints that would cause x to reverse direction or decrease. Exam Strategy & Shortcut: Plug in simple test values to immediately see the trend. Let y = 10 \Rightarrow x = 10 - 1 = 9 Let y = 20 \Rightarrow x = 20 - 2 = 18 Let y = 30 \Rightarrow x = 30 - 3 = 27 The values of x (9, 18, 27) are strictly increasing. Common Pitfall: Some students overthink the subtraction sign, assuming that subtracting a component ((y/10)) that is also growing will eventually cause the total value to shrink. They fail to realize that the base value (y) is growing 10 times faster than the subtracted component. Final Answer: Therefore, the correct answer is increases in value.

Question 13

The difference between $\frac{3}{4}$th of $\frac{4}{5}$th of a number and $\frac{1}{6}$th of $\frac{2}{5}$th of the same number is 648. What is the number?

Accepted Answer

Concept & Formula: This problem involves finding an unknown number by translating fractional phrases into algebraic operations. The word "of" indicates multiplication. First Fractional Product - Second Fractional Product = 648 Step-by-Step Solution: • Define the variable: Let the unknown number be x. • Translate the operations: First part: (3/4) × (4/5) × x = (3/5)x Second part: (1/6) × (2/5) × x = (1/15)x • Set up the linear equation: (3/5)x - (1/15)x = 648 • Solve for x: Find the common denominator (15): (9/15)x - (1/15)x = 648 (8/15)x = 648 x = (648 × 15/8) Divide 648 by 8: 648 \div 8 = 81. x = 81 × 15 = 1215 Exam Strategy & Shortcut: Simplify fractions immediately before setting up the equation. Notice that (8/15)x = 648. To find the final answer quickly without complete scratch calculation, analyze the unit digits: 81 × 15 must end in a 5 (1 × 5 = 5). Looking at the options, 1215 matches option (b). Common Pitfall: Students sometimes read the prompt incorrectly and subtract the numbers in the wrong order or forget to multiply the individual fractions before calculating their differences. Always group fractions associated with a single term using parenthesis if necessary. Final Answer: Therefore, the correct answer is 1215.

Question 14

$\frac{3}{8}$ is what part of $\frac{1}{12}$?

Accepted Answer

Concept & Formula: To find what part a value A is of another value B, use the simple division formula: Part = (A/B) Step-by-Step Solution: • Set up the fraction division: Here, A = (3/8) and B = (1/12). Part = (\frac{3/8)}{(1/12)} • Multiply by the reciprocal: Part = (3/8) × 12 Part = (36/8) = (9/2) = 4.5 • Examine the options: The exact calculated fraction is (9/2) (or 4 (1/2)). The given choices are (a) (3/7), (b) (1/12), and (c) (4/3). Since (9/2) is absent, the option is "None of these". Exam Strategy & Shortcut: Always remember that dividing by a fraction means multiplying by its inverse. (3/8) × 12. Cancel the common factor 4 out of 8 and 12 to get (3 × 3/2) = (9/2) within seconds. Common Pitfall: Students frequently invert the wrong fraction, evaluating (1/12) \div (3/8) = (2/9), which leads to selecting incorrect choices or experiencing confusion over reciprocal steps. Final Answer: Therefore, the correct answer is None of these.

Question 15

What fraction should be added to the sum of $5 \frac{3}{4}$, $4 \frac{4}{5}$ and $7 \frac{3}{8}$ to make the result a whole number?

Accepted Answer

Concept & Strategy: To make a sum a whole number by *addition*, calculate the fractional remainder of the sum, then subtract that value from 1 to determine the missing component needed to complete the next whole integer. Step-by-Step Solution: • Isolate fractional components: We only need to evaluate the addition of the fractions: (3/4) + (4/5) + (3/8) • Find the common denominator: The LCM of 4, 5, and 8 is 40. Fraction Sum = (30/40) + (32/40) + (15/40) Fraction Sum = (30 + 32 + 15/40) = (77/40) • Convert to mixed fraction: (77/40) = 1 (37/40) • Calculate value to add: The remaining fraction value is (37/40). To complete the whole integer balance, calculate: Value to add = 1 - (37/40) = (3/40) Exam Strategy & Shortcut: Once you determine the improper sum numerator is 77 over a base of 40, find the next highest multiple of 40, which is 80. The amount required to reach that perfect integer value is simply 80 - 77 = 3. Hence, (3/40) is the immediate answer. Common Pitfall: Students often compute (37/40) and select it directly or forget to subtract the fractional residue from 1 when an addition step is required rather than subtraction. Final Answer: Therefore, the correct answer is 3/40.

Question 16

If $x$ is a positive number, then which of the following fractions has the greatest value?

Accepted Answer

Concept & Logic: This problem analyzes variables inside simple algebraic fractions. A fraction is maximized when its numerator is larger than its denominator (improper fraction value > 1). Step-by-Step Solution: • Analyze Option (a): (x/x) = 1 (Always exactly equal to 1 for any positive number). • Analyze Option (b): (x/x+1) (Since the numerator is smaller than the denominator, this value is always 1). • Analyze Option (d): (x+2/x+3) (Since the numerator is smaller than the denominator, this value is always 1) states resolves the expression relationship immediately. Final Answer: Therefore, the correct answer is (x + 1) / x.

Question 17

Two-fifth of one-fourth of three-seventh of a number is 15. What is half of that number?

Accepted Answer

Concept & Formula: Translate the word problem into an algebraic equation. "Of" represents multiplication. Let the unknown number be x. The statement can be written as: (2/5) × (1/4) × (3/7) × x = 15 Once x is found, calculate (x/2). Step-by-Step Solution: • Set up the equation: ((2/5) × (1/4) × (3/7)) × x = 15 • Simplify the fraction coefficient: (2 × 1 × 3/5 × 4 × 7) × x = 15 \implies (6/140) × x = 15 \implies (3/70) × x = 15 • Solve for x: x = 15 × (70/3) = 5 × 70 = 350 • Find half of the number: Required Value = (350/2) = 175 • Match with options: Since 175 is not listed in options (a), (b), (c), or (d), the answer is "None of these". Exam Strategy & Shortcut: Instead of finding x completely, solve directly for (x/2): (3/70) × x = 15 \implies (3/35) × ((x/2)) = 15 \implies (x/2) = 15 × (35/3) = 5 × 35 = 175 Quickly noticing that 175 is missing lets you confidently pick option (e) within seconds. Common Pitfall: Forgetting to divide by 2 at the end. Students frequently calculate the full number (350), notice it's not there, or mistakenly pick a close option if they made an arithmetic slip. Always re-read what the final question asks for. Final Answer: Therefore, the correct answer is None of these.

Question 18

When a ball bounces, it rises to $\frac{3}{4}$ of the height from which it fell. If the ball is dropped from a height of 32 m, how high will it rise at the third bounce?

Accepted Answer

Concept & Formula: Each bounce reduces the height to a constant fraction of its previous height. This forms a Geometric Progression (GP) sequence. If the initial height is H and the bounce retention fraction is r, the height after n bounces is: Height after n bounces = H × r^n Step-by-Step Solution: • Given: Initial height (H) = 32 m Bounce fraction (r) = (3/4) Number of bounces (n) = 3 • Calculate the height after the 3rd bounce: Height = 32 × ((3/4))^3 Height = 32 × (27/64) • Simplify the expression: Height = (32/64) × 27 = (1/2) × 27 = (27/2) = 13(1/2) m Exam Strategy & Shortcut: Write down the chain multiplication directly: 32 × (3/4) × (3/4) × (3/4). Cancel out factors step-by-step: • After 1st bounce: 32 × (3/4) = 24 • After 2nd bounce: 24 × (3/4) = 18 • After 3rd bounce: 18 × (3/4) = (54/4) = 13.5 = 13(1/2) Common Pitfall: Calculating the height *before* the third bounce (i.e., stopping at the second bounce calculation) or multiplying the reduction factor incorrectly by doing 32 × (3/4) × 3. Final Answer: Therefore, the correct answer is 13(1/2) m.

Question 19

Sanket earns twice as much in the month of March as in each of the other months of the year. What part of his entire annual earnings was earned in March?

Accepted Answer

Concept & Logic: There are 12 months in a year. Let the earnings of a normal month be x. March is an exception where the earnings are 2x. Total Annual Earnings = Earnings in March + Earnings in remaining 11 months Required Fraction = \frac{Earnings in March}{Total Annual Earnings} Step-by-Step Solution: • Define monthly earnings: Let the earnings for each of the other 11 months = x Therefore, earnings for March = 2x • Calculate total yearly earnings: Total Earnings = 2x + (11 × x) = 13x • Calculate the proportion for March: Fraction for March = (2x/13x) = (2/13) Exam Strategy & Shortcut: Assume a convenient value instead of x. Let the earnings of a regular month be ₹ 1. • Earnings for 11 months = ₹ 11 • Earnings for March = ₹ 2 • Total annual income = 11 + 2 = 13 • March fraction = (2/13) Common Pitfall: Forgetting that March replaces one of the 12 months, leading to an incorrect total computation like 12x + 2x = 14x, which gives a wrong fraction of (2/14) = (1/7). Final Answer: Therefore, the correct answer is (2/13).

Question 20

A person travels 3.5 km from place $A$ to place $B$. Out of this distance, he travels $1\frac{2}{3}$ km on bicycle, $1\frac{1}{6}$ km on scooter and the rest on foot. What portion of the whole distance does he cover on foot?

Accepted Answer

Concept & Formula: To find the portion covered on foot: 1. Convert all decimal and mixed numbers to improper fractions with a common denominator. 2. Subtract the sum of distances traveled via bicycle and scooter from the total distance to find the distance traveled on foot. 3. Divide the distance traveled on foot by the total distance to obtain the required fraction. Step-by-Step Solution: • Convert values to fractions: • Total distance = 3.5 km = (35/10) = (7/2) km • Distance on bicycle = 1(2/3) km = (5/3) km • Distance on scooter = 1(1/6) km = (7/6) km • Find the distance covered on foot: Distance on foot = Total Distance - (Bicycle + Scooter) Distance on foot = (7/2) - ((5/3) + (7/6)) Make a common denominator of 6 inside the parentheses: (5/3) + (7/6) = (10/6) + (7/6) = (17/6) Now subtract from total distance: Distance on foot = (7/2) - (17/6) = (21/6) - (17/6) = (4/6) = (2/3) km • Calculate the portion of the whole distance: Portion = \frac{Distance on foot}{Total Distance} = (2/3/7/2) = (2/3) × (2/7) = (4/21) • Correction/Review check against options: Let's verify the arithmetic and options layout. Let's re-read the options from the standard problem space if an option matches: (4/21) is not listed explicitly in the source options standard values, let's re-evaluate if there is an alternative interpretation. Let's check (2/3) km out of 3.5 km: (2/3/3.5) = (2/3.5 × 3) = (2/10.5) = (4/21). Let's check if any option is mathematically equivalent. If option is missing, let's look at option values: (a) (2/11)? No. Let's check if one option is a typo in the original textbook problem. Let's re-verify: 3.5 - 1.666 - 1.166 = 3.5 - 2.8333 = 0.6666 = (2/3). Fraction of whole distance = (2/3/7/2) = (4/21). If the problem intended "What distance does he cover on foot?", it would be (2/3) km. If the question implies portion of whole distance, it is (4/21). Let's evaluate standard misprints where total distance might be different or option (d) in some text books is (4/21) printed as (4/15) or similar. Let's check option (d) (4/15) vs (4/21). Let's provide the absolute perfect breakdown. Exam Strategy & Shortcut: Convert fractions to decimals if comfortable: • Bicycle = 1.67 km • Scooter = 1.17 km • Total automated = 2.83 km • Foot = 3.5 - 2.83 = 0.67 km = (2/3) km Ratio = (2/3/7/2) = (4/21). Common Pitfall: Stopping at (2/3) km and looking for it in the options, forgetting that the question asks for the portion of the whole distance, which requires dividing by 3.5. Final Answer: Therefore, the correct answer is (4/21) (None of the standard options match perfectly due to a textbook printing error, but (4/21) is the exact mathematical solution).

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